Case Study / Scenario-Based MCQs for Sub-Topics of Topic 16: Statistics & Probability

Question 1. A researcher wants to study the literacy rate among women in a specific district of Uttar Pradesh. They decide to conduct a survey in 10 randomly selected villages within that district and collect information from all women aged 18 and above in those villages. The collected information includes whether they can read and write, their age, and their occupation.

Based on this scenario, which of the following is an example of the 'Raw Data' collected by the researcher?

(A) The calculated overall literacy rate for the 10 villages.

(B) The individual responses from each woman about her literacy status (e.g., 'Yes' or 'No').

(C) A frequency table showing the count of literate and illiterate women in each village.

(D) A graph representing the literacy rate across the different age groups surveyed.

Question 2. In the scenario above, the researcher collects information on the 'occupation' of the women. Which type of variable is 'occupation'?

(A) Quantitative and discrete

(B) Quantitative and continuous

(C) Qualitative

(D) Primary data (but not a variable type)

Question 3. A student is collecting data on the number of cars owned by families in their neighbourhood in Delhi. They visit 30 houses and record the number of cars for each family. For one specific house, they record '2' cars.

In statistical terms, what does the value '2' represent in this context?

(A) The variable being studied.

(B) The data set.

(C) An observation.

(D) The frequency.

Question 4. A newspaper company uses data from the Directorate of Economics & Statistics to report on agricultural production in various states of India over the past decade. The data includes total tons of wheat produced per state per year.

In this scenario, what type of data is the newspaper company using?

(A) Primary data

(B) Raw data

(C) Secondary data

(D) Qualitative data

Question 5. After collecting the raw data on student marks from a test, a teacher arranges the marks in ascending order and then creates a table showing how many students scored between 0-10, 10-20, etc., marks. Finally, she draws a histogram to show the distribution of marks.

Which stages of data handling are explicitly mentioned in this process?

(A) Data Collection, Organization, and Presentation.

(B) Data Collection, Analysis, and Interpretation.

(C) Data Collection and Interpretation only.

(D) Data Analysis and Presentation only.

Question 1. The marks obtained by 20 students in a Statistics test (out of 10) are as follows: 7, 5, 8, 6, 7, 5, 8, 8, 7, 6, 9, 5, 6, 8, 7, 7, 6, 8, 5, 9.

A teacher wants to create an ungrouped frequency distribution table for these marks. What would be the frequency and tally mark for the mark '7'?

(A) Frequency: 4, Tally mark: $||||$

(B) Frequency: 5, Tally mark: $\bcancel{||||}$

(C) Frequency: 6, Tally mark: $\bcancel{||||} |$

(D) Frequency: 5, Tally mark: $|||||$

Question 2. A factory records the daily temperature (in degrees Celsius) at noon for 30 days in May. The data ranges from $30^\circ C$ to $40^\circ C$. The factory decides to create a grouped frequency distribution with class intervals 30-32, 32-34, 34-36, 36-38, 38-40 (exclusive).

Which of the following values represents a lower class limit in this distribution?

(A) 32

(B) 33

(C) 40

(D) 31

Question 3. In the scenario from Question 2, what is the class size (width) of the class intervals used?

(A) $1^\circ C$

(B) $2^\circ C$

(C) $3^\circ C$

(D) $10^\circ C$

Question 4. Consider a frequency distribution table with classes 0-10, 10-20, 20-30, 30-40 and frequencies 5, 8, 12, 7 respectively. The 'less than' cumulative frequency for the class 20-30 would represent the number of observations less than 30.

What is this cumulative frequency?

(A) 5

(B) $5 + 8 = 13$

(C) $5 + 8 + 12 = 25$

(D) $5 + 8 + 12 + 7 = 32$

Question 5. Using the frequency distribution table from Question 4, what is the 'more than' cumulative frequency for the class 10-20? This represents the number of observations greater than or equal to 10.

(A) 8

(B) $8 + 12 + 7 = 27$

(C) $12 + 7 = 19$

(D) $5 + 8 + 12 + 7 = 32$

Question 1. A survey was conducted to find the favourite fruit of 50 students in a school. 15 students liked Mango, 10 liked Banana, 12 liked Apple, and 13 liked Orange. The results are to be represented by a pie chart.

What is the angle of the sector representing 'Mango'?

(A) $(15/50) \times 100 = 30^\circ$

(B) $(15/50) \times 360^\circ = 108^\circ$

(C) $15^\circ$

(D) $(15/50) \times 50 = 15^\circ$

Question 2. A company's annual report shows the production of cars over four quarters. Quarter 1: 500 cars, Quarter 2: 750 cars, Quarter 3: 600 cars, Quarter 4: 900 cars. This data is represented using a single bar graph.

If the bar for Quarter 1 has a height of 2 cm, what would be the height of the bar for Quarter 4?

(A) $(900/500) \times 2 \text{ cm} = 1.8 \times 2 \text{ cm} = 3.6 \text{ cm}$

(B) $(500/900) \times 2 \text{ cm} \approx 1.11 \text{ cm}$

(C) $900 \text{ cm}$

(D) $3.6 \text{ cm}$

Question 3. A school wants to compare the enrollment of boys and girls in Classes 6 to 10 for the current academic year. Which basic chart would be most suitable for this comparison?

(A) Single Bar Graph (showing total enrollment per class)

(B) Pie Chart (showing the boy-girl split for the whole school)

(C) Double Bar Graph (comparing boys' enrollment vs girls' enrollment for each class)

(D) Pictograph (using symbols of boys and girls)

Question 4. A pictograph is used to show the number of bicycles sold by a shop in a week. One symbol of a bicycle represents 25 bicycles. If the shop sold 175 bicycles, how many symbols should be used?

(A) $175 / 25 = 7$ full symbols

(B) $175 \times 25 = 4375$ symbols

(C) 7 symbols

(D) Need to use fractions of symbols

Question 5. A pie chart shows the preferred mode of payment among customers: Cash (40%), Card (35%), UPI (25%). What is the angle of the sector for 'Card' payments?

(A) $35^\circ$

(B) $(35/100) \times 360^\circ = 126^\circ$

(C) $126^\circ$

(D) $360^\circ - 126^\circ - (40/100)\times360^\circ - (25/100)\times360^\circ$

Question 1. The weights (in kg) of 50 students are grouped into the following classes: 40-50, 50-60, 60-70, 70-80, 80-90 (exclusive). The frequencies are 10, 15, 12, 8, 5 respectively. A histogram is drawn for this data.

What does the bar for the class 50-60 represent in this histogram?

(A) The number of students weighing exactly 55 kg.

(B) The number of students weighing between 50 kg (inclusive) and 60 kg (exclusive), which is 15 students.

(C) The average weight of students in that range.

(D) The cumulative number of students weighing less than 60 kg.

Question 2. Using the data from Question 1, a frequency polygon is drawn by joining the midpoints of the tops of the histogram bars. What are the class marks for these intervals?

(A) 40, 50, 60, 70, 80

(B) 50, 60, 70, 80, 90

(C) 45, 55, 65, 75, 85

(D) 10, 15, 12, 8, 5

Question 3. Consider two sections of Class 10 with marks distributions shown by two overlaid frequency polygons on the same graph. Section A's polygon is flatter and more spread out, while Section B's polygon is taller and narrower.

Which conclusion can be drawn about the marks distributions?

(A) Section A has a higher average mark than Section B.

(B) Section B has less variability in marks than Section A.

(C) Section A has less variability in marks than Section B.

(D) Both sections have the same average mark.

Question 4. A company records the time taken (in minutes) by employees to complete a task, grouped into intervals like 0-5, 5-10, 10-15, 15-20. A histogram is created. If the frequency for the 5-10 minute interval is 25, and the class width is 5, what is the height of the bar if the class width is scaled to 1 unit for calculation (i.e., using frequency density)?

(A) 25

(B) 5

(C) $25/5 = 5$ (Frequency Density)

(D) $25 \times 5 = 125$

Question 5. A frequency polygon is drawn for a dataset. The point (65, 12) is plotted on the graph. If the x-axis represents class marks and the y-axis represents frequency, what does this point signify?

(A) 65 observations have a frequency of 12.

(B) The class with class mark 65 has a frequency of 12.

(C) 12 observations have a value less than 65.

(D) The cumulative frequency at 65 is 12.

Question 1. The following table shows the marks obtained by 100 students in an exam:

Marks (Less Than)	20	40	60	80	100
No. of Students (CF)	12	35	65	87	100

A 'less than' ogive is plotted using this data.

What is the total number of students (N) from this ogive?

(A) 20

(B) 12

(C) 100 (The cumulative frequency of the last interval)

(D) 87

Question 2. Using the data from Question 1, where N=100, the Median is estimated from the 'less than' ogive. To estimate the Median, you locate $N/2 = 50$ on the y-axis and find the corresponding value on the x-axis. Looking at the table, a cumulative frequency of 35 is at 40 marks, and 65 is at 60 marks.

The Median mark is expected to be between:

(A) 20 and 40

(B) 40 and 60

(C) 60 and 80

(D) 80 and 100

Question 3. A survey records the daily earnings of workers (in $\textsf{₹}$) grouped into classes. A 'more than' ogive is plotted. The point (200, 80) is on the ogive, and the total number of workers surveyed is 100. What does the point (200, 80) indicate?

(A) 80 workers earn exactly $\textsf{₹}200$ per day.

(B) 80 workers earn less than $\textsf{₹}200$ per day.

(C) 80 workers earn $\textsf{₹}200$ or more per day.

(D) 20 workers earn less than $\textsf{₹}200$ per day.

Question 4. Using the data from Question 3 (Total workers = 100, (200, 80) on 'more than' ogive), how many workers earn less than $\textsf{₹}200$ per day?

(A) 80

(B) $100 - 80 = 20$

(C) 200

(D) Cannot be determined from a 'more than' ogive alone.

Question 5. Consider the intersection point of a 'less than' and 'more than' ogive for a distribution. The coordinates of the intersection point are (45, 75).

What does this point tell us about the distribution?

(A) The Median is 75.

(B) The total frequency is 75.

(C) The Median is 45, and the total frequency is $75 \times 2 = 150$.

(D) The Median is 45, and $N/2$ is 75, so total frequency is 150.

Question 1. Five friends pool their money to buy a gift. The amounts contributed are $\textsf{₹}150, \textsf{₹}200, \textsf{₹}100, \textsf{₹}250, \textsf{₹}300$. They want to know the average contribution per person.

What is the arithmetic mean contribution?

(A) $\textsf{₹}1000 / 5 = \textsf{₹}200$

(B) $\textsf{₹}200$

(C) $\textsf{₹}250$

(D) $\textsf{₹}150$

Question 2. The average marks of 10 students in a test were 75. Later, it was discovered that the marks of one student were wrongly entered as 60 instead of 80.

What is the correct average mark?

(A) Original total marks = $10 \times 75 = 750$. Corrected total marks = $750 - 60 + 80 = 770$. Corrected mean = $770 / 10 = 77$.

(B) 75

(C) $75 + (80-60)/10 = 75 + 20/10 = 77$

(D) 77

Question 3. A company pays its employees hourly wages. The distribution of hours worked by 50 employees in a week is given in a grouped frequency table. The classes are 20-30, 30-40, 40-50, 50-60, and corresponding frequencies are 10, 15, 20, 5. To calculate the average hours worked using the Direct Method, you would first find the class marks.

What is the class mark for the interval 30-40?

(A) 30

(B) 40

(C) $(30+40)/2 = 35$

(D) 35

Question 4. In the scenario from Question 3, what is the sum $\sum f_i x_i$ used in the numerator of the Direct Method formula? (Using class marks: 25, 35, 45, 55)

(A) $10 \times 25 + 15 \times 35 + 20 \times 45 + 5 \times 55 = 250 + 525 + 900 + 275 = 1950$

(B) $25+35+45+55 = 160$

(C) $10+15+20+5 = 50$

(D) 1950

Question 5. Consider two classes, Section A with 40 students having an average mark of 65, and Section B with 60 students having an average mark of 70. What is the average mark for the entire group of 100 students?

(A) $(65+70)/2 = 67.5$

(B) $(40 \times 65 + 60 \times 70) / (40+60) = (2600 + 4200) / 100 = 6800 / 100 = 68$

(C) 68

(D) 70

Question 1. The heights (in cm) of 7 plants are recorded as: 15, 22, 18, 25, 20, 16, 28. To find the Median height, you first arrange the data in order.

What is the ordered dataset?

(A) 15, 16, 18, 20, 22, 25, 28

(B) 28, 25, 22, 20, 18, 16, 15

(C) 15, 18, 20, 22, 25, 28, 16

(D) 20

Question 2. Using the ordered dataset from Question 1 (15, 16, 18, 20, 22, 25, 28), what is the Median height?

(A) 15

(B) 28

(C) 20 (The value at the $(7+1)/2 = 4^{th}$ position)

(D) 20

Question 3. The daily sales (in $\textsf{₹}$) of a shop for 8 days are: 1500, 2000, 1200, 1800, 2500, 1600, 2200, 1900. To find the Median sales, arrange the data in order.

What is the average of the two middle values?

(A) $(1800+1900)/2 = \textsf{₹}1850$ (Ordered: 1200, 1500, 1600, 1800, 1900, 2000, 2200, 2500)

(B) $(1600+1800)/2 = \textsf{₹}1700$

(C) $\textsf{₹}1850$

(D) $\textsf{₹}1900$

Question 4. Consider the grouped frequency table from Question 9: classes 0-10, 10-20, 20-30, 30-40 and frequencies 5, 8, 12, 7. Total frequency $N=32$. To find the Median, locate $N/2 = 16$. The cumulative frequencies are 5, 13, 25, 32.

What is the median class?

(A) 0-10 (CF=5, 16 is not less than 5)

(B) 10-20 (CF=13, 16 is not less than 13)

(C) 20-30 (CF=25, 16 is less than 25, and 16 is greater than CF of previous class, 13)

(D) 30-40

Question 5. Using the median class 20-30 from Question 4 (L=20, N=32, cf of preceding class = 13, frequency of median class = 12, class size h=10), calculate the Median using the formula $M = L + \frac{(N/2 - cf)}{f} \times h$.

(A) $20 + \frac{(16 - 13)}{12} \times 10 = 20 + \frac{3}{12} \times 10 = 20 + \frac{1}{4} \times 10 = 20 + 2.5 = 22.5$

(B) $20 + (16-13)/12 = 20 + 3/12 = 20.25$

(C) 22.5

(D) 25

Question 1. The sizes of shoes sold by a shop on a particular day are recorded: 6, 7, 8, 8, 7, 9, 6, 8, 7, 8, 10, 7, 8, 6, 8.

What is the mode shoe size?

(A) 6 (Frequency 3)

(B) 7 (Frequency 4)

(C) 8 (Frequency 6)

(D) 9 (Frequency 1)

Question 2. Consider the grouped frequency table from Question 9: classes 0-10, 10-20, 20-30, 30-40 and frequencies 5, 8, 12, 7. The modal class is the one with the highest frequency.

What is the modal class?

(A) 0-10

(B) 10-20

(C) 20-30 (Frequency 12 is highest)

(D) 30-40

Question 3. Using the modal class 20-30 from Question 2 (L=20, $f_1$=12, $f_0$=8, $f_2$=7, h=10), calculate the Mode using the formula $Mode = L + \frac{(f_1 - f_0)}{(2f_1 - f_0 - f_2)} \times h$.

(A) $20 + \frac{(12 - 8)}{(2 \times 12 - 8 - 7)} \times 10 = 20 + \frac{4}{(24 - 15)} \times 10 = 20 + \frac{4}{9} \times 10 = 20 + 40/9 \approx 20 + 4.44 = 24.44$

(B) $20 + 4/9 \times 10 = 24.44$

(C) 20

(D) 30

Question 4. For a distribution of monthly incomes in a town, a few individuals have extremely high incomes, causing the distribution to be positively skewed. Based on the typical relationship for positively skewed distributions, which measure of central tendency would be the highest?

(A) Mean

(B) Median

(C) Mode

(D) They would all be equal

Question 5. If the Mean and Median of a moderately skewed distribution are 50 and 48 respectively, what is the estimated Mode using the empirical formula Mode $\approx$ 3 Median - 2 Mean?

(A) $3 \times 48 - 2 \times 50 = 144 - 100 = 44$

(B) 50

(C) 48

(D) 44

Question 1. The daily maximum temperatures (in $^\circ$C) for a week in Jaipur were recorded as: 38, 40, 37, 41, 39, 42, 38. What is the range of the temperatures?

(A) $42 - 37 = 5^\circ C$

(B) $42^\circ C$

(C) $37^\circ C$

(D) $5^\circ C$

Question 2. The scores of 5 students in a test are: 10, 15, 20, 25, 30. The Mean score is $(10+15+20+25+30)/5 = 20$. What is the Mean Deviation from the Mean?

(A) Mean Deviation = $\frac{|10-20| + |15-20| + |20-20| + |25-20| + |30-20|}{5} = \frac{|-10| + |-5| + |0| + |5| + |10|}{5} = \frac{10+5+0+5+10}{5} = \frac{30}{5} = 6$

(B) 5

(C) 6

(D) 20

Question 3. For the data in Question 2 (10, 15, 20, 25, 30), the Median is 20. What is the Mean Deviation from the Median?

(A) Mean Deviation = $\frac{|10-20| + |15-20| + |20-20| + |25-20| + |30-20|}{5} = \frac{10+5+0+5+10}{5} = \frac{30}{5} = 6$

(B) 5

(C) 6

(D) 20

Question 4. Consider two datasets of daily wages in two different small companies: Company A: {300, 310, 320, 330, 340}, Company B: {100, 200, 320, 440, 540}. Both have the same mean ($\textsf{₹}320$). Which company has a larger Range in daily wages?

(A) Company A ($340-300 = \textsf{₹}40$)

(B) Company B ($540-100 = \textsf{₹}440$)

(C) Both have the same Range.

(D) Cannot compare using Range.

Question 5. A dataset of student marks is {50, 60, 70, 80, 90}. If each student is given 5 bonus marks, the new dataset is {55, 65, 75, 85, 95}. How does the Mean Deviation from the Mean change?

(A) It increases by 5.

(B) It decreases by 5.

(C) It remains unchanged. (Original Mean=70, New Mean=75. Deviations |x-70|: 20,10,0,10,20. Mean Dev=12. New Deviations |x-75|: 20,10,0,10,20. Mean Dev=12)

(D) It is multiplied by a factor.

Question 1. The weights (in kg) of 3 students are 40, 50, 60. The mean weight is $(40+50+60)/3 = 50$ kg. What is the sample variance ($s^2$)?

(A) $\frac{(40-50)^2 + (50-50)^2 + (60-50)^2}{3} = \frac{(-10)^2 + 0^2 + 10^2}{3} = \frac{100+0+100}{3} = 200/3 \approx 66.67$

(B) $\frac{(40-50)^2 + (50-50)^2 + (60-50)^2}{3-1} = \frac{100+0+100}{2} = 200/2 = 100$

(C) 50

(D) 100

Question 2. Using the sample variance of 100 from Question 1, what is the sample standard deviation (s)?

(A) $\sqrt{100} = 10$ kg

(B) 10 kg

(C) 100 kg

(D) $\sqrt{200/3}$ kg

Question 3. A dataset of daily profits (in $\textsf{₹}$) for a street vendor is {1000, 1200, 1100, 1300, 1400}. The standard deviation of the profits is calculated. If the vendor doubles their profit each day, the new dataset is {2000, 2400, 2200, 2600, 2800}. How does the standard deviation change?

(A) It doubles. (If original SD is $\sigma$, new SD is $|2|\sigma = 2\sigma$)

(B) It is squared.

(C) It remains unchanged.

(D) It increases by a constant amount.

Question 4. If the Variance of a dataset is 64, what is its Standard Deviation?

(A) 8 (Standard Deviation is the positive square root of Variance)

(B) $\pm 8$

(C) 64

(D) $\sqrt{64} = 8$

Question 5. Two investment options have the following annual returns for the last 5 years: Option A: {5%, 6%, 5%, 6%, 5%}, Option B: {2%, 8%, 5%, 10%, 0%}. Both have the same mean return (5.4%). To understand which option is riskier (more variable), you calculate the variance or standard deviation. Which option is expected to have a higher standard deviation?

(A) Option A (Values are clustered tightly around the mean)

(B) Option B (Values are more spread out)

(C) Both will have the same standard deviation.

(D) Cannot determine without calculating.

Question 1. A small business owner wants to compare the variability in sales between two products. Product X has average daily sales of $\textsf{₹}5000$ with a standard deviation of $\textsf{₹}500$. Product Y has average daily sales of $\textsf{₹}2000$ with a standard deviation of $\textsf{₹}300$. Which product has higher relative variability?

(A) Product X ($CV_X = (500/5000)\times 100 = 10\%$)

(B) Product Y ($CV_Y = (300/2000)\times 100 = 15\%$)

(C) Both have equal relative variability.

(D) Cannot compare due to different average sales.

Question 2. A factory manager compares the consistency of two machines producing screws. Machine A produces screws with an average length of 10 cm and a standard deviation of 0.1 cm. Machine B produces screws with an average length of 5 cm and a standard deviation of 0.08 cm. Which machine is more consistent in terms of screw length?

(A) Machine A ($CV_A = (0.1/10)\times 100 = 1\%$)

(B) Machine B ($CV_B = (0.08/5)\times 100 = 1.6\%$)

(C) Both are equally consistent.

(D) Machine A is more consistent (lower CV).

Question 3. For a distribution, the mean is 150 and the Coefficient of Variation is 10%. What is the Standard Deviation?

(A) $CV = (\sigma / \mu) \times 100$. $10 = (\sigma / 150) \times 100$. $\sigma = (10 \times 150) / 100 = 15$.

(B) 10

(C) 150

(D) 15

Question 4. The weights of packets of rice from two different brands are compared. Brand P has a mean weight of 10 kg and a standard deviation of 0.5 kg. Brand Q has a mean weight of 5 kg and a standard deviation of 0.3 kg. Which brand shows less relative variability in weight?

(A) Brand P ($CV_P = (0.5/10)\times 100 = 5\%$)

(B) Brand Q ($CV_Q = (0.3/5)\times 100 = 6\%$)

(C) Both show equal relative variability.

(D) Brand P (lower CV).

Question 5. Consider a dataset where the data points are {10, -10, 5, -5}. The mean is 0. Can the Coefficient of Variation be meaningfully calculated and interpreted for this dataset?

(A) Yes, calculate SD and divide by the absolute mean.

(B) No, because the mean is zero, and division by zero is undefined.

(C) Yes, CV is always applicable to numerical data.

(D) No, because the data includes negative values.

Question 1. The distribution of incomes in India is generally characterized by a large number of people with low to moderate incomes and a small number of people with very high incomes. What type of skewness would you expect this distribution to have?

(A) Symmetric

(B) Positively skewed (tail is towards higher income values)

(C) Negatively skewed

(D) Bimodal

Question 2. A distribution of exam marks is found to have Mean = 70, Median = 72, and Mode = 75. What is the likely skewness of this distribution?

(A) Positively skewed (Mean < Median < Mode suggests negative skew)

(B) Negatively skewed (Mean < Median < Mode)

(C) Symmetric

(D) Cannot be determined from these values alone.

Question 3. A stock market's daily price changes over a long period are analyzed. The distribution of changes shows a high peak around the mean (zero change) and heavier tails than expected under a normal distribution (meaning more frequent extreme gains or losses). Which type of kurtosis does this distribution exhibit?

(A) Platykurtic

(B) Mesokurtic

(C) Leptokurtic (High peak and fatter tails)

(D) Symmetric

Question 4. For a distribution of the lifespan of a certain electronic component, most components fail relatively early, but a few last for a very long time. This results in a distribution with a peak shifted to the left and a long tail extending to the right. What kind of skewness is this?

(A) Negatively skewed

(B) Symmetric

(C) Positively skewed (Tail on the right)

(D) Bimodal

Question 5. A researcher calculates Bowley's coefficient of skewness for a dataset and gets a value of -0.3. What does this indicate about the distribution?

(A) The distribution is perfectly symmetric.

(B) The distribution is positively skewed.

(C) The distribution is negatively skewed (coefficient is negative).

(D) The distribution is mesokurtic.

Question 1. The scores of 10 students in a test (out of 100), arranged in ascending order, are: 45, 52, 58, 65, 70, 72, 78, 85, 90, 95. What is the first quartile (Q1)?

(A) 52

(B) 58

(C) 56.5

(D) 65

Question 2. Using the scores from Question 1 (45, 52, 58, 65, 70, 72, 78, 85, 90, 95), what is the third quartile (Q3)?

(A) 78

(B) 85

(C) $P_{75}$ Position $\approx (10+1) \times 75/100 = 8.25^{th}$ position. Value = $8^{th} + 0.25 \times (9^{th} - 8^{th}) = 85 + 0.25 \times (90-85) = 85 + 0.25 \times 5 = 85 + 1.25 = 86.25$. Using n/4 method: $3n/4=7.5^{th}$ pos = average of 7th and 8th = (78+85)/2 = 81.5. Let's go with the (n+1)/4 method result as 86.25 or check options for closest.

(D) 86.25

Question 3. Using the quartiles calculated for the data in Question 1 (assuming Q1=56.5, Q3=86.25), what is the Interquartile Range (IQR)?

(A) $86.25 - 56.5 = 29.75$

(B) $56.5$

(C) $86.25$

(D) $29.75$

Question 4. In a list of 20 ordered salaries, what is the position of the $80^{th}$ percentile ($P_{80}$)?

(A) $80^{th}$ position

(B) $0.80 \times 20 = 16^{th}$ position (for certain methods)

(C) $(20+1) \times 80/100 = 21 \times 0.8 = 16.8^{th}$ position (Using (n+1)k/100 method)

(D) $16.8^{th}$ position

Question 5. A student's score in a competitive exam is at the 90th percentile. What does this mean?

(A) The student scored 90 marks.

(B) The student performed better than approximately 90% of the other students who took the exam.

(C) The student answered 90% of the questions correctly.

(D) The student's score is exactly 90% of the maximum possible score.

Question 1. Data on ice cream sales and temperature for a city are collected over several weeks. A scatter diagram is plotted, and the points show a general trend where higher temperatures are associated with higher ice cream sales. What type of correlation is this likely to be?

(A) Zero correlation

(B) Negative correlation

(C) Positive correlation

(D) Perfect negative correlation

Question 2. A researcher studies the relationship between hours of television watched per week and the number of hours spent exercising per week among college students. The scatter diagram shows a pattern where students who watch more TV tend to exercise less. What type of correlation is this likely to be?

(A) Positive correlation

(B) Negative correlation

(C) Zero correlation

(D) Perfect positive correlation

Question 3. A company analyzes the relationship between the amount spent on advertising and the resulting sales revenue. The data is plotted on a scatter diagram. If the points are very close to forming a straight line that goes upwards from left to right, what would you expect the Karl Pearson's correlation coefficient ($r$) to be?

(A) Close to -1

(B) Close to 0

(C) Close to +1 (Strong positive linear relationship)

(D) Exactly -1

Question 4. A wine critic ranks 10 different wines based on taste, and a chemist ranks the same 10 wines based on a chemical composition score. To measure the agreement between the critic's ranking and the chemist's ranking, which correlation coefficient would be most appropriate?

(A) Karl Pearson's coefficient

(B) Spearman's Rank Correlation Coefficient (Used for ranked data)

(C) Coefficient of Variation

(D) Mean Deviation

Question 5. A study finds a strong positive correlation (r = 0.9) between shoe size and reading ability in children aged 5-10. What is the most likely explanation for this correlation?

(A) Larger feet cause better reading ability.

(B) Better reading ability causes larger feet.

(C) Shoe size and reading ability are likely correlated due to a common factor, such as age (older children have larger feet and better reading skills).

(D) There is no real relationship, the correlation is a coincidence.

Question 1. A box contains 5 green pens and 7 blue pens. A pen is drawn randomly from the box.

What is the sample space for this experiment?

(A) {Green, Blue} (Representing the colours)

(B) {5 Green, 7 Blue} (Representing the counts)

(C) {Pen 1, Pen 2, ..., Pen 12} (If pens are distinguishable)

(D) {Green, Blue}

Question 2. In the scenario from Question 1, what is the probability of drawing a green pen using the classical definition?

(A) $5/12$ (Number of green pens / Total pens)

(B) $7/12$

(C) $5/7$

(D) $1/2$

Question 3. A bag contains slips of paper numbered from 1 to 20. A slip is drawn randomly. Let A be the event of drawing an even number, and B be the event of drawing a number greater than 15.

What is the intersection event $A \cap B$?

(A) Numbers that are both even AND greater than 15: {16, 18, 20}

(B) Numbers that are even OR greater than 15

(C) Numbers that are less than or equal to 15

(D) {16, 17, 18, 19, 20}

Question 4. A company manufactures bulbs. In a batch of 1000 bulbs, 20 are found to be defective. If a bulb is chosen randomly from this batch, what is the experimental probability of choosing a defective bulb?

(A) $20 / 1000 = 0.02$

(B) $20 / 980$

(C) 0.02

(D) $1/2$

Question 5. Consider rolling a standard six-sided die. Event A is 'getting a number less than 3' ({1, 2}). Event B is 'getting a number greater than 4' ({5, 6}). Are events A and B mutually exclusive?

(A) Yes, because their intersection is empty ($A \cap B = \emptyset$).

(B) No, because $P(A)+P(B) < 1$.

(C) Yes, because they are simple events.

(D) No, because they are both possible outcomes.

Question 1. In a group of 50 students, 30 like Cricket (C) and 25 like Football (F). 10 students like both. If a student is chosen randomly, what is the probability that they like either Cricket or Football?

(A) $P(C \cup F) = P(C) + P(F) - P(C \cap F)$. $P(C) = 30/50 = 0.6$, $P(F) = 25/50 = 0.5$, $P(C \cap F) = 10/50 = 0.2$. $P(C \cup F) = 0.6 + 0.5 - 0.2 = 0.9$.

(B) $0.6 + 0.5 = 1.1$

(C) $0.6 \times 0.5 = 0.3$

(D) 0.9

Question 2. The probability that a student passes Mathematics is 0.8, and the probability that they pass Physics is 0.7. The probability that they pass both is 0.6. What is the probability that they pass neither subject?

(A) $P(\text{Pass Math } \cup \text{ Pass Phys}) = P(M \cup P) = P(M) + P(P) - P(M \cap P) = 0.8 + 0.7 - 0.6 = 0.9$. $P(\text{Pass neither}) = P(M' \cap P') = P((M \cup P)') = 1 - P(M \cup P) = 1 - 0.9 = 0.1$.

(B) $1 - (0.8+0.7) = -0.5$

(C) 0.6

(D) 0.1

Question 3. If A and B are two events such that $P(A) = 0.3$, $P(B) = 0.4$, and $P(A \cap B) = 0.1$. What is the probability of A or B occurring ($P(A \cup B)$)?

(A) $0.3 + 0.4 = 0.7$

(B) $0.3 + 0.4 - 0.1 = 0.6$

(C) 0.6

(D) 0.1

Question 4. The probability that it will rain tomorrow is 0.3. The probability that it will be cloudy but not rain is 0.2. Are the events "rain tomorrow" and "cloudy but not rain tomorrow" mutually exclusive?

(A) Yes, because it cannot rain and be cloudy but not rain at the same time.

(B) No, because they are both weather events.

(C) Yes, if the sample space is clearly defined.

(D) No, you need more information.

Question 5. A bag contains 10 tickets numbered 1 to 10. A ticket is drawn at random. Event A is drawing an even number, Event B is drawing a number greater than 7. What is the probability of drawing an even number or a number greater than 7?

(A) $P(A) = 5/10 = 0.5$, $P(B) = 3/10 = 0.3$. $A=\{2,4,6,8,10\}$, $B=\{8,10\}$, $A \cap B = \{8,10\}$. $P(A \cap B) = 2/10 = 0.2$. $P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.5 + 0.3 - 0.2 = 0.6$.

(B) $0.5 + 0.3 = 0.8$

(C) 0.6

(D) 0.2

Question 1. A survey found that 40% of people read newspaper A, 50% read newspaper B, and 20% read both. If a person is chosen randomly and it is known that they read newspaper B, what is the probability that they also read newspaper A?

(A) $P(A|B) = P(A \cap B) / P(B) = 0.20 / 0.50 = 0.4$

(B) $P(B|A) = P(A \cap B) / P(A) = 0.20 / 0.40 = 0.5$

(C) 0.4

(D) 0.2

Question 2. A bag contains 3 red and 4 blue marbles. Two marbles are drawn without replacement. What is the probability that the second marble drawn is red, given that the first marble drawn was blue?

(A) $P(\text{2nd Red | 1st Blue})$. After drawing 1 blue, 3 red and 3 blue remain (total 6). $P(\text{2nd Red | 1st Blue}) = 3/6 = 1/2$.

(B) $3/7$

(C) $4/7$

(D) $1/2$

Question 3. In a class, 60% of students are girls and 40% are boys. 30% of girls have long hair, and 50% of boys have long hair. If a student is chosen randomly and has long hair, what is the probability that the student is a girl? (This requires Bayes' Theorem logic, but can be solved with conditional prob if structured appropriately).

Let G = Girl, B = Boy, L = Long Hair. $P(G)=0.6, P(B)=0.4, P(L|G)=0.3, P(L|B)=0.5$. We need $P(G|L)$. Using $P(G|L) = P(G \cap L) / P(L)$. $P(G \cap L) = P(L|G)P(G) = 0.3 \times 0.6 = 0.18$. $P(L) = P(L|G)P(G) + P(L|B)P(B) = 0.18 + 0.5 \times 0.4 = 0.18 + 0.20 = 0.38$. $P(G|L) = 0.18 / 0.38 = 18/38 = 9/19 \approx 0.474$.

(A) $0.3 / 0.6 = 0.5$

(B) $0.6 \times 0.3 = 0.18$

(C) $0.18 / 0.38 \approx 0.474$

(D) 0.3

Question 4. Two cards are drawn simultaneously from a well-shuffled deck of 52 playing cards. What is the probability that both cards are Kings, given that at least one of the cards is a King?

Let A = Both cards are Kings, B = At least one card is a King. We need $P(A|B) = P(A \cap B) / P(B)$. $A \cap B$ is the event that both are Kings (since if both are kings, at least one is a king), so $A \cap B = A$. $P(A)$ = Probability of drawing 2 Kings out of 4 = $\binom{4}{2} / \binom{52}{2} = 6 / (52 \times 51 / 2) = 6 / 1326 = 1/221$. $P(B) = 1 - P(\text{No King})$. $P(\text{No King}) = \binom{48}{2} / \binom{52}{2} = (48 \times 47 / 2) / 1326 = (24 \times 47) / 1326 = 1128 / 1326 = 188/221$. $P(B) = 1 - 188/221 = 33/221$. $P(A|B) = (1/221) / (33/221) = 1/33$.

(A) $1/221$

(B) $1/13$

(C) $1/33$

(D) $2/52$

Question 5. If $P(A) = 0.6$, $P(B) = 0.5$, and $P(A|B) = 0.4$. Find $P(A \cap B)$.

(A) $0.4 \times 0.5 = 0.2$ (Using $P(A \cap B) = P(A|B) \times P(B)$)

(B) $0.6 \times 0.5 = 0.3$

(C) 0.2

(D) $0.6 / 0.4 = 1.5$

Question 1. A bag contains 4 red and 6 blue marbles. A marble is drawn, its colour is noted, and it is *replaced*. Then a second marble is drawn. What is the probability that the first marble is red and the second marble is blue?

(A) $P(\text{1st Red}) = 4/10 = 0.4$. $P(\text{2nd Blue}) = 6/10 = 0.6$ (with replacement, events are independent). $P(\text{1st Red and 2nd Blue}) = P(\text{1st Red}) \times P(\text{2nd Blue}) = 0.4 \times 0.6 = 0.24$.

(B) $0.4 + 0.6 = 1$

(C) $0.4 \times 0.6 = 0.24$

(D) $4/10 \times 5/9$

Question 2. A bag contains 4 red and 6 blue marbles. A marble is drawn, its colour is noted, and it is *not replaced*. Then a second marble is drawn. What is the probability that the first marble is red and the second marble is blue?

(A) $P(\text{1st Red}) = 4/10$. $P(\text{2nd Blue | 1st Red}) = 6/9$ (After drawing 1 red, 3 red and 6 blue remain). $P(\text{1st Red and 2nd Blue}) = P(\text{1st Red}) \times P(\text{2nd Blue | 1st Red}) = (4/10) \times (6/9) = 24/90 = 4/15$.

(B) $4/10 + 6/9$

(C) $4/10 \times 6/10$

(D) $4/15$

Question 3. A box contains 3 coins: one is a fair coin (Heads probability 0.5), one has two heads (Heads probability 1), and one is a biased coin (Heads probability 0.2). A coin is selected randomly and tossed. What is the probability of getting a Head?

Let $C_F$ = Fair Coin, $C_{HH}$ = Two Heads Coin, $C_B$ = Biased Coin. These form a partition. $P(C_F) = P(C_{HH}) = P(C_B) = 1/3$. $P(H|C_F)=0.5$, $P(H|C_{HH})=1$, $P(H|C_B)=0.2$. Using Law of Total Probability: $P(H) = P(H|C_F)P(C_F) + P(H|C_{HH})P(C_{HH}) + P(H|C_B)P(C_B) = (0.5)(1/3) + (1)(1/3) + (0.2)(1/3) = (0.5+1+0.2)/3 = 1.7/3 \approx 0.567$.

(A) $0.5 + 1 + 0.2 = 1.7$

(B) $1.7 / 3 \approx 0.567$

(C) $0.5 \times 1 \times 0.2 = 0.1$

(D) $1/3$

Question 4. If $P(A) = 0.3$ and $P(B) = 0.4$, and $P(A \cap B) = 0.12$. Are events A and B independent?

(A) Yes, because $P(A \cap B) = 0.12$ and $P(A)P(B) = 0.3 \times 0.4 = 0.12$. Since $P(A \cap B) = P(A)P(B)$, they are independent.

(B) No, because $P(A \cap B) \neq 0$.

(C) Yes, because they are not mutually exclusive.

(D) No, because $P(A) + P(B) \neq 1$.

Question 5. Two events A and B are such that $P(A) = 0.6$ and $P(B) = 0.5$. If they are independent, what is $P(A \cup B)$?

(A) $P(A) + P(B) = 0.6 + 0.5 = 1.1$

(B) $P(A) + P(B) - P(A \cap B) = P(A) + P(B) - P(A)P(B) = 0.6 + 0.5 - (0.6)(0.5) = 1.1 - 0.3 = 0.8$.

(C) $0.6 \times 0.5 = 0.3$

(D) 0.8

Question 1. In a factory, machine A produces 60% of the items and machine B produces 40%. Machine A has a 2% defect rate, and machine B has a 1% defect rate. An item is chosen randomly and found to be defective. What is the probability that it was produced by machine A?

Let A = produced by machine A, B = produced by machine B, D = defective. $P(A)=0.6, P(B)=0.4, P(D|A)=0.02, P(D|B)=0.01$. We want $P(A|D)$. Using Bayes' Theorem: $P(A|D) = \frac{P(D|A)P(A)}{P(D)}$. $P(D) = P(D|A)P(A) + P(D|B)P(B) = 0.02 \times 0.6 + 0.01 \times 0.4 = 0.012 + 0.004 = 0.016$. $P(A|D) = \frac{0.012}{0.016} = \frac{12}{16} = \frac{3}{4} = 0.75$.

(A) 0.6

(B) 0.02

(C) 0.75

(D) 0.4

Question 2. A doctor is diagnosing a patient. The patient has a certain symptom (S). The doctor considers two diseases, $D_1$ and $D_2$, which are mutually exclusive and exhaustive possible diagnoses. Based on prior knowledge, $P(D_1)=0.1$ and $P(D_2)=0.9$. The probability of having the symptom given $D_1$ is $P(S|D_1)=0.8$, and given $D_2$ is $P(S|D_2)=0.2$. If the patient has the symptom, what is the probability they have disease $D_1$?

We want $P(D_1|S)$. Using Bayes' Theorem: $P(D_1|S) = \frac{P(S|D_1)P(D_1)}{P(S|D_1)P(D_1) + P(S|D_2)P(D_2)} = \frac{0.8 \times 0.1}{0.8 \times 0.1 + 0.2 \times 0.9} = \frac{0.08}{0.08 + 0.18} = \frac{0.08}{0.26} = \frac{8}{26} = \frac{4}{13} \approx 0.308$.

(A) 0.1

(B) 0.8

(C) 0.308

(D) 0.9

Question 3. In a village, 1% of the population has a rare disease. A test for the disease is 95% accurate (true positive rate) and has a 10% false positive rate. If a person tests positive, what is the probability they actually have the disease?

Let D = Disease, ND = No Disease, TP = Test Positive, TN = Test Negative. $P(D)=0.01$, $P(ND)=0.99$. $P(TP|D)=0.95$. $P(TP|ND)=0.10$. We want $P(D|TP)$. $P(D|TP) = \frac{P(TP|D)P(D)}{P(TP|D)P(D) + P(TP|ND)P(ND)} = \frac{0.95 \times 0.01}{0.95 \times 0.01 + 0.10 \times 0.99} = \frac{0.0095}{0.0095 + 0.099} = \frac{0.0095}{0.1085} \approx 0.0875$.

(A) 0.95

(B) 0.01

(C) $0.0095 / 0.1085 \approx 0.0875$

(D) 0.10

Question 4. A startup company receives funding applications. Historically, 30% of applications come from IT startups ($S_1$), and 70% from non-IT startups ($S_2$). IT startups have a 60% success rate in getting funding (E), while non-IT startups have a 20% success rate. If an application is successful (event E), what is the probability it came from an IT startup ($S_1$)?

$P(S_1)=0.3, P(S_2)=0.7$. $P(E|S_1)=0.6, P(E|S_2)=0.2$. We want $P(S_1|E)$. $P(S_1|E) = \frac{P(E|S_1)P(S_1)}{P(E|S_1)P(S_1) + P(E|S_2)P(S_2)} = \frac{0.6 \times 0.3}{0.6 \times 0.3 + 0.2 \times 0.7} = \frac{0.18}{0.18 + 0.14} = \frac{0.18}{0.32} = \frac{18}{32} = \frac{9}{16} = 0.5625$.

(A) 0.3

(B) 0.6

(C) 0.5625

(D) 0.2

Question 5. A student takes a multiple-choice test with 4 options per question. They know the answer to 60% of the questions. For the remaining 40%, they guess randomly. If the student answers a question correctly, what is the probability that they actually knew the answer?

Let K = Knew answer, G = Guessed. C = Answered Correctly. $P(K)=0.6, P(G)=0.4$. $P(C|K)=1$ (if they knew, they are correct). $P(C|G)=1/4=0.25$ (guessing 1 out of 4 options). We want $P(K|C)$. $P(K|C) = \frac{P(C|K)P(K)}{P(C|K)P(K) + P(C|G)P(G)} = \frac{1 \times 0.6}{1 \times 0.6 + 0.25 \times 0.4} = \frac{0.6}{0.6 + 0.1} = \frac{0.6}{0.7} = 6/7 \approx 0.857$.

(A) 0.6

(B) 1

(C) 0.857

(D) 0.4

Question 1. A factory produces light bulbs. A quality control inspector randomly selects 3 bulbs from a large batch and tests if they are defective. Let X be the number of defective bulbs in the sample of 3.

What type of random variable is X?

(A) Continuous random variable

(B) Qualitative variable

(C) Discrete random variable (Can only be 0, 1, 2, or 3)

(D) Deterministic variable

Question 2. Consider a discrete random variable Y representing the outcome when rolling a single fair six-sided die. The probability distribution is $P(Y=y) = 1/6$ for y = 1, 2, 3, 4, 5, 6.

What is the sum of probabilities for this distribution?

(A) $1/6$

(B) $6 \times (1/6) = 1$ (Sum of probabilities for all possible outcomes must be 1)

(C) 6

(D) $1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 1$

Question 3. A random variable Z represents the time (in minutes) a customer waits in a queue at a bank. This time can be any value within a certain range (e.g., 0 to 10 minutes or more).

What type of random variable is Z?

(A) Discrete random variable

(B) Countable random variable

(C) Continuous random variable (Time is measured on a continuous scale)

(D) Categorical variable

Question 4. The probability distribution of a discrete random variable X is given below:

x	1	2	3
P(X=x)	0.4	0.k	0.3

For this to be a valid probability distribution, the sum of probabilities must be 1.

What is the value of k?

(A) $0.4 + 0.k + 0.3 = 1 \Rightarrow 0.7 + 0.k = 1 \Rightarrow 0.k = 0.3 \Rightarrow k=3$. The probability is $P(X=2)=0.3$.

(B) 4

(C) 3

(D) 0.3

Question 5. Consider a continuous random variable X representing the temperature of a fluid. The Probability Density Function (PDF), $f(x)$, describes the distribution. What is the probability that the temperature is exactly 25 degrees Celsius, i.e., $P(X=25)$?

(A) The value of $f(25)$.

(B) The area under the curve at $x=25$.

(C) 0 (For a continuous variable, the probability of taking any single exact value is 0)

(D) 1

Question 1. A game involves rolling a fair six-sided die. You win $\textsf{₹}10$ if you roll a 6, and lose $\textsf{₹}2$ if you roll any other number. Let X be the amount you win (can be negative).

What is the expected amount you win (Expected Value E(X))?

(A) $P(X=10) = 1/6$, $P(X=-2) = 5/6$. $E(X) = 10 \times (1/6) + (-2) \times (5/6) = 10/6 - 10/6 = 0$.

(B) $\textsf{₹}10 \times 1/6 = \textsf{₹}1.67$

(C) 0

(D) $\textsf{₹}2 \times 5/6 = \textsf{₹}1.67$

Question 2. Consider the random variable X from Question 1 (X can be 10 or -2, with $P(X=10)=1/6, P(X=-2)=5/6$, $E(X)=0$). What is the Variance of X?

First, find $E(X^2) = 10^2 \times (1/6) + (-2)^2 \times (5/6) = 100/6 + 4 \times 5/6 = 100/6 + 20/6 = 120/6 = 20$. $Var(X) = E(X^2) - [E(X)]^2 = 20 - 0^2 = 20$.

(A) $10^2 \times (1/6) + (-2)^2 \times (5/6) = 20$

(B) 0

(C) 20

(D) $\sqrt{20} \approx 4.47$

Question 3. A company's profit (in $\textsf{₹}$ Lakhs) per quarter is a random variable Y with $E(Y) = 5$ and $Var(Y) = 2$. Due to a change in strategy, the profit for next quarter is expected to be $Z = 2Y + 3$. What is the expected profit for next quarter, $E(Z)$?

(A) $E(Z) = E(2Y + 3) = 2 E(Y) + 3 = 2 \times 5 + 3 = 10 + 3 = 13$ Lakhs.

(B) $\textsf{₹}5$ Lakhs

(C) $\textsf{₹}10$ Lakhs

(D) $\textsf{₹}13$ Lakhs

Question 4. Using the scenario from Question 3, what is the variance of the profit for next quarter, $Var(Z)$?

(A) $Var(Z) = Var(2Y + 3) = 2^2 Var(Y) = 4 \times 2 = 8$ (Constant added does not affect variance, multiplication by constant squares the effect).

(B) 2

(C) $2 \times 2 + 3 = 7$

(D) 8

Question 5. The number of accidents per week on a highway section is a random variable X with $E(X) = 1.5$. What is the expected number of accidents in a month (approximately 4 weeks), assuming the same rate and independence between weeks?

Let $X_i$ be the number of accidents in week i. Total accidents in 4 weeks $X_{total} = X_1 + X_2 + X_3 + X_4$. Assuming independence, $E(X_{total}) = E(X_1) + E(X_2) + E(X_3) + E(X_4) = 1.5 + 1.5 + 1.5 + 1.5 = 6$.

(A) 1.5

(B) 6

(C) $1.5 \times 4 = 6$

(D) Cannot be determined without the full distribution.

Question 1. A multiple-choice test has 15 questions, and each question has 4 options, only one of which is correct. If a student guesses randomly on every question, what is the probability of getting a question correct? Let X be the number of questions the student answers correctly.

This scenario fits a Binomial distribution. What are the parameters (n, p) for X?

(A) n=15, p=4

(B) n=15, p=1/4 = 0.25 (Number of trials = 15, Probability of success (guessing correctly) = 1/4)

(C) n=4, p=15

(D) n=15, p=0.25

Question 2. Using the scenario from Question 1 (X ~ B(15, 0.25)), what is the expected number of questions the student will answer correctly by guessing?

(A) Mean = $np = 15 \times 0.25 = 3.75$

(B) 15

(C) 0.25

(D) 3.75

Question 3. A company produces electronic components. The probability that a component is defective is 0.02. A random sample of 100 components is taken. What is the probability that exactly 2 components in the sample are defective?

This is a Binomial distribution with n=100, p=0.02. We want $P(X=2)$. $P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$. $P(X=2) = \binom{100}{2} (0.02)^2 (0.98)^{100-2} = \frac{100 \times 99}{2 \times 1} \times (0.02)^2 \times (0.98)^{98} = 4950 \times 0.0004 \times (0.98)^{98}$. (Calculation is complex without calculator/tables, but the setup matches the formula).

(A) $\binom{100}{2} (0.02)^2 (0.98)^{98}$

(B) $100 \times 0.02 = 2$

(C) $(0.02)^2$

(D) $\binom{100}{2} (0.02)^{98} (0.98)^{2}$

Question 4. A company claims that 80% of its customers are satisfied with its service. In a random sample of 10 customers, what is the probability that exactly 7 are satisfied?

This is B(10, 0.8). We want $P(X=7)$. $P(X=7) = \binom{10}{7} (0.8)^7 (0.2)^{10-7} = \binom{10}{7} (0.8)^7 (0.2)^3$.

(A) $\binom{10}{7} (0.8)^7 (0.2)^3$

(B) $10 \times 0.8 = 8$

(C) $(0.8)^7 (0.2)^3$

(D) $\binom{10}{7} (0.2)^7 (0.8)^3$

Question 5. If a Binomial distribution has n=20 and p=0.1, what is its variance?

(A) Mean = $20 \times 0.1 = 2$. Variance = $np(1-p) = 20 \times 0.1 \times 0.9 = 1.8$.

(B) 2

(C) 1.8

(D) $\sqrt{1.8} \approx 1.34$

Question 1. On average, a call center receives 5 calls per minute. Assuming the calls follow a Poisson process, what is the expected number of calls in a 3-minute interval?

The average rate per minute is $\lambda_{min} = 5$. For a 3-minute interval, the rate is $\lambda_{3min} = 5 \times 3 = 15$. The expected number of calls is the parameter $\lambda$ for that interval.

(A) 5

(B) 15

(C) 3

(D) 5/3

Question 2. The number of defects in a roll of fabric follows a Poisson distribution with an average of 2 defects per roll. What is the probability of finding exactly 3 defects in a randomly selected roll?

Poisson with $\lambda=2$. We want $P(X=3)$. $P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}$. $P(X=3) = \frac{e^{-2} 2^3}{3!} = \frac{e^{-2} \times 8}{6} = \frac{4e^{-2}}{3}$.

(A) $\frac{e^{-2} 2^3}{3!}$

(B) $e^{-2}$

(C) $2 \times 3 = 6$

(D) $\frac{4e^{-2}}{3}$

Question 3. If the average number of typos on a book page is 1.5, and the number of typos follows a Poisson distribution, what is the variance in the number of typos per page?

(A) 1.5 (For a Poisson distribution, Mean = Variance = $\lambda$)

(B) $\sqrt{1.5}$

(C) $1.5^2$

(D) 3

Question 4. A company observes that the number of customer complaints per day follows a Poisson distribution with a mean of 4. What is the probability that the company receives no complaints on a given day?

Poisson with $\lambda=4$. We want $P(X=0)$. $P(X=0) = \frac{e^{-4} 4^0}{0!} = \frac{e^{-4} \times 1}{1} = e^{-4}$.

(A) $\frac{e^{-4} 4^0}{0!}$

(B) $e^{-4}$

(C) 4

(D) 0

Question 5. The number of cars passing a specific point on a highway in an hour can be modeled by a Poisson distribution. If the average number of cars is 60 per hour, what is the average number of cars per minute?

The rate is 60 per hour. To find the rate per minute, divide by 60 minutes/hour. $\lambda_{min} = 60 / 60 = 1$ per minute.

(A) 60

(B) 1

(C) $60/60 = 1$

(D) $60 \times 60 = 3600$

Question 1. The heights of adult males in a city are approximately normally distributed with a mean ($\mu$) of 170 cm and a standard deviation ($\sigma$) of 5 cm. What is the probability that a randomly selected adult male is exactly 170 cm tall?

(A) The highest probability.

(B) The value of the PDF at 170 cm.

(C) 0 (Probability of an exact value for a continuous distribution is 0)

(D) 0.5

Question 2. Using the scenario from Question 1 ($\mu=170$, $\sigma=5$), what is the z-score for an adult male with a height of 175 cm?

(A) $z = (175 - 170) / 5 = 5 / 5 = 1$

(B) -1

(C) 5

(D) 1

Question 3. The scores on a standardized test are normally distributed with a mean of 500 and a standard deviation of 100. What percentage of students scored between 400 and 600?

400 is one SD below the mean (z = (400-500)/100 = -1). 600 is one SD above the mean (z = (600-500)/100 = 1). According to the Empirical Rule, approx 68% falls within $\pm 1$ SD.

(A) Approx 68%

(B) Approx 95%

(C) Approx 99.7%

(D) 50%

Question 4. For the test scores in Question 3 ($\mu=500$, $\sigma=100$), a student scores 700. What is their z-score?

(A) $z = (700 - 500) / 100 = 200 / 100 = 2$

(B) 1

(C) 2

(D) -2

Question 5. A machine fills bags of sugar. The weights of the bags are normally distributed with a mean of $\mu = 1000$ grams and a standard deviation of $\sigma = 5$ grams. What is the probability that a randomly selected bag weighs less than 990 grams?

990 grams is two SD below the mean ($z = (990-1000)/5 = -10/5 = -2$). Using Z-tables for $P(Z \le -2)$ gives a very small probability (approx 0.0228). The question asks for the probability setup.

(A) $P(X < 990)$

(B) $P(Z < (990-1000)/5)$

(C) $P(Z < -2)$

(D) All of the above represent the probability.

Question 1. A market research firm wants to estimate the average monthly expenditure on mobile phones for all young adults (aged 18-30) in Bangalore. They survey 500 young adults from different parts of the city.

In this scenario, what is the population?

(A) The 500 surveyed young adults.

(B) All young adults in Bangalore aged 18-30.

(C) All residents of Bangalore.

(D) The average monthly expenditure.

Question 2. Using the scenario from Question 1, what is the sample?

(A) The 500 surveyed young adults.

(B) All young adults in Bangalore aged 18-30.

(C) The average monthly expenditure calculated from the survey.

(D) The method used to select the 500 young adults.

Question 3. If the market research firm calculates the average monthly expenditure from the 500 surveyed young adults, what is this calculated average called?

(A) A population parameter.

(B) The true average expenditure.

(C) A sample statistic.

(D) The inferential conclusion.

Question 4. The market research firm uses the average expenditure from their sample to estimate the average monthly expenditure for *all* young adults in Bangalore. What branch of statistics does this process belong to?

(A) Descriptive Statistics

(B) Probability Theory

(C) Inferential Statistics

(D) Data Visualization

Question 5. The market research firm decides to use Stratified Sampling. They divide Bangalore's young adults into subgroups based on income levels (Low, Medium, High) and then randomly sample from each subgroup. Why might they choose this method?

(A) To ensure that young adults from each income level are represented in the sample.

(B) To make the sample size smaller.

(C) To avoid random selection entirely.

(D) To study the entire population.

Question 1. A company claims that the average lifespan of their LED bulbs is 10,000 hours. A consumer rights group wants to test this claim. They take a random sample of 50 bulbs and find the average lifespan is 9,800 hours. They want to determine if this sample evidence is strong enough to conclude that the true average lifespan is less than 10,000 hours.

What is the null hypothesis ($H_0$) for this test?

(A) The average lifespan is less than 10,000 hours ($\mu < 10000$).

(B) The average lifespan is equal to 10,000 hours ($\mu = 10000$).

(C) The sample average lifespan is 9,800 hours ($\bar{x} = 9800$).

(D) The average lifespan is greater than 10,000 hours ($\mu > 10000$).

Question 2. Using the scenario from Question 1, what is the alternative hypothesis ($H_1$ or $H_a$) for this test?

(A) The average lifespan is equal to 10,000 hours ($\mu = 10000$).

(B) The sample average lifespan is less than 10,000 hours ($\bar{x} < 10000$).

(C) The average lifespan is less than 10,000 hours ($\mu < 10000$).

(D) The average lifespan is not equal to 10,000 hours ($\mu \neq 10000$).

Question 3. The consumer group sets the level of significance ($\alpha$) for the test at 0.05. After performing the hypothesis test, they obtain a p-value of 0.03.

What decision should they make regarding the null hypothesis?

(A) Fail to reject $H_0$ because the p-value is greater than $\alpha$.

(B) Reject $H_0$ because the p-value is less than or equal to $\alpha$.

(C) Accept $H_0$ because the sample mean is close to 10,000.

(D) The p-value is irrelevant to the decision.

Question 4. In a clinical trial, a new drug is being tested to see if it reduces blood pressure. The null hypothesis is that the drug has no effect (average blood pressure change is zero). The alternative hypothesis is that the drug reduces blood pressure (average blood pressure change is negative). The researchers mistakenly conclude the drug is effective when it actually has no effect.

What type of error have the researchers made?

(A) Type I Error (Rejecting $H_0$ when $H_0$ is true)

(B) Type II Error (Failing to reject $H_0$ when $H_0$ is false)

(C) Correct decision

(D) Statistical power error

Question 5. A company is evaluating a new manufacturing process. They hypothesize that the new process reduces the proportion of defective items compared to the old process (which had a 5% defect rate). They test this hypothesis using a sample from the new process. They set $\alpha = 0.01$. If they fail to reject the null hypothesis (that the defect rate is still 5% or higher), even though the new process actually *does* reduce defects, what type of error have they made?

(A) Type I Error

(B) Type II Error (Failing to reject $H_0$ when $H_0$ is false)

(C) Correct decision

(D) Significance error

Question 1. A university wants to test if the average height of male students is different from the national average for young men, which is reported as 168 cm. They collect a random sample of 25 male students, find the sample mean height is 170 cm, and the sample standard deviation is 5 cm. The population standard deviation is unknown.

Which statistical test is most appropriate to test the hypothesis about the average height?

(A) One-sample Z-test (requires population SD)

(B) Two independent samples t-test

(C) One-sample t-test (comparing sample mean to a known value, population SD unknown)

(D) Paired samples t-test

Question 2. Using the scenario from Question 1, what are the degrees of freedom for the appropriate test?

(A) 25 (Sample size)

(B) $25 - 1 = 24$ (For one-sample t-test, df = n-1)

(C) $25 - 2 = 23$

(D) 168

Question 3. A company introduces a training program to improve employee productivity. They measure the productivity of a random sample of 15 employees *before* and *after* the training program. They want to test if the average productivity has significantly changed.

Which statistical test is most appropriate for this comparison?

(A) One-sample t-test

(B) Two independent samples t-test

(C) Paired samples t-test (Comparing measurements on the same subjects before and after)

(D) Chi-square test

Question 4. A researcher compares the average reaction times of two different groups of participants: Group A ($n_1=12$ participants) and Group B ($n_2=15$ participants), who were given different stimuli. The sample mean reaction time for Group A is 0.5 seconds, and for Group B is 0.6 seconds. The population standard deviations are unknown, but assumed to be roughly equal. They want to test if there is a significant difference in average reaction times between the two groups.

Which statistical test is most appropriate?

(A) One-sample t-test

(B) Paired samples t-test

(C) Two independent samples t-test (Comparing means of two unrelated groups, population SD unknown)

(D) ANOVA

Question 5. Using the scenario from Question 4 (Group A $n_1=12$, Group B $n_2=15$, testing for difference in means), what are the degrees of freedom for the appropriate test (assuming equal variances)?

(A) 12

(B) 15

(C) $12 + 15 - 2 = 25$ (For two independent samples t-test, df = $n_1 + n_2 - 2$)

(D) 27