Mean And Variance Of Binomial Distributions Quiz

1) A binomial distribution has n = 50 trials and probability of success p = 0.4. What is the mean (expected value) of this distribution?

10

15

20

25

Mean = np = 50 × 0.4 = 20.

Explanation

Mean = np = 50 × 0.4 = 20.

2) In a binomial distribution, if the mean is 24 and the probability of success is 0.6, how many trials were conducted?

30

40

50

60

Mean = np ⇒ 24 = n × 0.6 ⇒ n = 24/0.6 = 40.

Explanation

Mean = np ⇒ 24 = n × 0.6 ⇒ n = 24/0.6 = 40.

3) A fair coin is flipped 100 times. What is the expected number of heads?

25

40

50

75

Mean = np = 100 × 0.5 = 50.

Explanation

Mean = np = 100 × 0.5 = 50.

4) The variance of a binomial distribution is calculated using the formula σ² = np(1 − p). If n = 80 and p = 0.25, what is the variance?

10

15

20

25

Var = 80 × 0.25 × 0.75 = 15.

Explanation

Var = 80 × 0.25 × 0.75 = 15.

5) A basketball player has a 70% free throw success rate. If she attempts 30 free throws, what is the expected number of successful shots?

15

18

21

24

Mean = np = 30 × 0.7 = 21.

Explanation

Mean = np = 30 × 0.7 = 21.

6) Which of the following statements best describes what the variance tells us about a binomial distribution?

The most likely outcome

The total number of trials

How spread out the distribution is around the mean

The probability of success

Variance measures spread around the mean.

Explanation

Variance measures spread around the mean.

7) A binomial distribution has n = 60 and p = 0.5. What is the standard deviation (round to the nearest tenth)?

3.5

3.9

4.2

4.5

SD = √(np(1−p)) = √(60×0.5×0.5) = √15 ≈ 3.9.

Explanation

SD = √(np(1−p)) = √(60×0.5×0.5) = √15 ≈ 3.9.

8) If a binomial distribution has mean 18 and variance 12.6, what is the probability of success p?

0.2

0.3

0.4

0.5

Var = mean×(1−p) ⇒ 12.6 = 18(1−p) ⇒ 1−p = 0.7 ⇒ p = 0.3.

Explanation

Var = mean×(1−p) ⇒ 12.6 = 18(1−p) ⇒ 1−p = 0.7 ⇒ p = 0.3.

9) A quality control inspector knows that 5% of items produced are defective. If she inspects 200 items, what is the expected number of defective items?

5

10

15

20

Mean = np = 200 × 0.05 = 10.

Explanation

Mean = np = 200 × 0.05 = 10.

10) Two binomial distributions both have n = 100. Distribution A has p = 0.5 and Distribution B has p = 0.9. Which distribution has the larger variance?

Distribution A

Distribution B

They have equal variances

Cannot be determined

Var = np(1−p). A: 25; B: 9 → A is larger.

Explanation

Var = np(1−p). A: 25; B: 9 → A is larger.

11) What is the probability of success (getting a question correct) for each trial?

0.10

0.20

0.25

0.50

One correct out of 4 choices → 1/4 = 0.25.

Explanation

One correct out of 4 choices → 1/4 = 0.25.

12) What is the expected number of questions the student will answer correctly?

5.00

6.25

7.50

10.00

Mean = np = 25 × 0.25 = 6.25.

Explanation

Mean = np = 25 × 0.25 = 6.25.

13) What is the variance of the number of correct answers?

3.75

4.69

5.25

6.25

Var = np(1−p) = 25 × 0.25 × 0.75 = 4.6875 ≈ 4.69.

Explanation

Var = np(1−p) = 25 × 0.25 × 0.75 = 4.6875 ≈ 4.69.

14) What does the variance calculated in question 13 tell us about the student's performance?

The student will definitely get between 4 and 8 questions correct

There is considerable variability in how many questions the student might get correct

The student's score will always be close to the mean

The variance indicates the student will fail

Variance describes variability around the mean, not a guaranteed range.

Explanation

Variance describes variability around the mean, not a guaranteed range.

15) A binomial distribution has variance 16 and p = 0.2. What is the mean of this distribution?

10

15

20

25

Var = np(1−p) = n×0.2×0.8 = 0.16n = 16 ⇒ n = 100 ⇒ mean = np = 100×0.2 = 20.

Explanation

Var = np(1−p) = n×0.2×0.8 = 0.16n = 16 ⇒ n = 100 ⇒ mean = np = 100×0.2 = 20.

16) As the probability of success p approaches 0 or 1, what happens to the variance of a binomial distribution (assuming n stays constant)?

The variance increases

The variance decreases

The variance stays the same

The variance becomes negative

Var = np(1−p) → goes to 0 as p→0 or p→1.

Explanation

Var = np(1−p) → goes to 0 as p→0 or p→1.

17) A factory produces light bulbs, and 8% are defective. If a sample of 150 bulbs is tested, what is the variance in the number of defective bulbs?

9.6

11.04

12.0

13.2

Var = np(1−p) = 150×0.08×0.92 = 11.04.

Explanation

Var = np(1−p) = 150×0.08×0.92 = 11.04.

18) Which binomial distribution would have the maximum variance for a fixed number of trials n?

P = 0.1

P = 0.3

P = 0.5

P = 0.9

Var maximal at p = 0.5 since p(1−p) is maximized at 0.25.

Explanation

Var maximal at p = 0.5 since p(1−p) is maximized at 0.25.

19) What is the expected number of patients who will respond positively to the medication?

60

72

78

84

Mean = np = 120 × 0.65 = 78.

Explanation

Mean = np = 120 × 0.65 = 78.

20) What is the standard deviation of the number of patients who respond positively (round to the nearest tenth)?

4.7

5.2

6.1

7.3

SD = √(np(1−p)) = √(120×0.65×0.35) = √27.3 ≈ 5.2.

Explanation

SD = √(np(1−p)) = √(120×0.65×0.35) = √27.3 ≈ 5.2.

Mean and Variance of Binomial Distributions Quiz