Q: A geneticist claims that 25% of children inherit a certain trait from their parents. In a random sample of 40 children, what is the variance of the sample proportion $$\hat{p}$$?

$$\frac{0.25 \times 0.75}{40}$$

Question 1

Which of the following best describes a random sample in the context of statistical inference?

Accepted Answer

A sample where every individual in the population has an equal chance of being selected, and the measurements on individuals are independent random variables.

Question 2

Suppose $$X is a $$Bernoulli random variable with parameter $$p. $$What is the variance of $$X$$?

Accepted Answer

$$p(1-p)$$

Question 3

A researcher is interested in the proportion of defective items produced by a factory. If the probability that an item is defective is $$p = 0.02$$, and a random sample of $$n = 100$$ items is taken, what is the expected value (mean) of the sample proportion $$\hat{p}$$?

Accepted Answer

$$0.02$$

Question 4

Which of the following is NOT an assumption required for the binomial setting?

Accepted Answer

The probability of success changes from trial to trial.

Question 5

Suppose $$Y \sim \mathrm{Bin}(n, p). $$What is the standard deviation of $$Y$$?

Accepted Answer

$$\sqrt{n p (1-p)}$$

Question 6

A geneticist claims that 25% of children inherit a certain trait from their parents. In a random sample of 40 children, what is the variance of the sample proportion $$\hat{p}$$?

Accepted Answer

$$\frac{0.25 	imes 0.75}{40}$$

Question 7

If $$X_1$$, $$X_2$$, \ldots, X_n$$ are independent Bernoulli random variables with parameter p$, what is the distribution of Y$$ = $$X_1$$ + $$X_2$$ + \cdots + X_n$$?

Accepted Answer

Binomial distribution with parameters $$n$$ and $$p$$

Question 8

A medical test for a disease has a 5% chance of a false positive. If 20 healthy individuals are tested, what is the probability that exactly 2 will test positive? (Let $$p = 0.05)$$

Accepted Answer

$$\binom{20}{2} (0.05)^2$ (0.95$$)^{18}$$

Question 9

Which of the following statements about the sample proportion $$\hat{p} is $$TRUE?

Accepted Answer

The variance of $$\hat{p}$$ decreases as the sample size $$n$$ increases.

Question 10

Suppose $$Y \sim \mathrm{Bin}(10, 0.4). $$What is $$P(Y = 3)$$?

Accepted Answer

$$\binom{10}{3} (0.4)^3$ (0.6)^7$

Question 11

A researcher doubles the sample size from $$n = 10 to$$ $$n = 20$$ when estimating a population proportion $$p = 0.25. By $$what factor does the variance of the sample proportion $$\hat{p}$$ change?

Accepted Answer

It is halved.

Question 12

Which of the following is the correct formula for the probability mass function of a Bernoulli random variable $$X$$ with parameter $$p$$?

Accepted Answer

$$P(X = x) = p^x (1-p$$)^{1-x}$$ for x = 0$$, 1$$

Question 13

In a clinical trial, 8 out of 20 patients respond positively to a new drug. What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$0.4$$

Question 14

Which of the following best explains why the sample proportion $$\hat{p} is $$considered a random variable?

Accepted Answer

Because it is calculated from a random sample and will vary from sample to sample.

Question 15

Suppose $$Y \sim \mathrm{Bin}(n, p). $$Which of the following is the mean of the sample proportion $$\hat{p} = \frac{Y}{n}$$?

Accepted Answer

$$p$$

Question 16

A researcher wants to estimate the proportion of voters who support a new policy. Which of the following would result in a more precise (less variable) estimate of the population proportion?

Accepted Answer

Increasing the sample size.

Question 17

If $$Y \sim \mathrm{Bin}(n, p)$$, which of the following expressions gives the probability that $$Y = y$$?

Accepted Answer

$$P(Y = y) = \binom{n}{y} p^y (1-p$$)^{n-y}$$

Question 18

Which of the following real-world scenarios is best modeled by a binomial distribution?

Accepted Answer

Counting the number of patients who recover after receiving a new treatment, where each patient has the same probability of recovery.

Question 19

Suppose a researcher observes $$Y \sim \mathrm{Bin}(n, p)$$ and wants to find the probability that the sample proportion $$\hat{p}$$ equals $$k/n$$ for some integer $$k. $$Which formula should be used?

Accepted Answer

$$P(\hat{p} = \frac{k}{n}) = \binom{n}{k} p^k (1-p$$)^{n-k}$$

Distribution of the Sample Proportion: Binomial and Bernoulli Distributions

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