Q: A researcher samples 100 individuals from a population where $$p = 0.08. $$What is the standard deviation of the sampling distribution of the sample proportion $$\hat{p}$$?

$$\$$\sqrt{\frac{0.08 \times 0.92}{100}} = 0.0271\$

Q: Which of the following conditions must be met for the sampling distribution of $$\hat{p} to be $$approximately normal?

$$\n p$$ \geq 15$$ and n(1-p$$) \geq 15\$

Q: A bank finds that 8% of its loans are not paid on time. In a random sample of 200 loans, what is the mean and standard deviation of the sampling distribution of $$\hat{p}$$?

Mean: $$0.08$$, Standard deviation: $$\$$\sqrt{\frac{0.08 \times 0.92}{200}} = 0.0192\$

Question 1

Which of the following best describes the difference between a parameter and a statistic?

Accepted Answer

A parameter is a value calculated from the population, while a statistic is a value calculated from a sample.

Question 2

Suppose the population standard deviation is $$\sigma = 12$$ and the sample size is $$n = 36. $$What is the standard error of the sample mean $$\bar{x}$$?

Accepted Answer

$$\$$\sigma_{\bar{x}} = \frac{12}{\sqrt{36}} = 2\$

Question 3

According to the Central Limit Theorem, which of the following statements is TRUE?

Accepted Answer

The sampling distribution of the sample mean $$\bar{x}$$ will be approximately normal if the sample size is large enough, regardless of the population distribution.

Question 4

A researcher samples 100 individuals from a population where $$p = 0.08. $$What is the standard deviation of the sampling distribution of the sample proportion $$\hat{p}$$?

Accepted Answer

$$\$$\sqrt{\frac{0.08 	imes 0.92}{100}} = 0.0271\$

Question 5

Which of the following is NOT an assumption required for the Central Limit Theorem to apply to sample means?

Accepted Answer

The population must be normally distributed.

Question 6

A batch of a chemical product has impurity amounts with $$\mu = 4$$ grams and $$\sigma = 1.5$$ grams. For $$n = 50$$ batches, what is the probability that the sample mean impurity exceeds 4.2 grams? (Assume normality.)

Accepted Answer

First, calculate the standard error: $$\$$\sigma_{\bar{x}} = \frac{1.5}{\sqrt{50}} = 0.212\$$. Then, z$$ = \frac{4.2 - 4}{0.212} = 0.943$$. The probability is 1 - 0.827 = 0.173$.

Question 7

If the sample size increases while the population standard deviation remains constant, what happens to the standard error of the sample mean?

Accepted Answer

It decreases.

Question 8

A population is right-skewed with mean $$\mu = 5.2$$ and $$\sigma = 2.0. $$For a random sample of $$n = 36$$, what is the shape of the sampling distribution of $$\bar{x}$$?

Accepted Answer

It is approximately normal.

Question 9

Which of the following conditions must be met for the sampling distribution of $$\hat{p} to be $$approximately normal?

Accepted Answer

$$
 p$$ \geq 15$$ and n(1-p$$) \geq 15\$

Question 10

A bank finds that 8% of its loans are not paid on time. In a random sample of 200 loans, what is the mean and standard deviation of the sampling distribution of $$\hat{p}$$?

Accepted Answer

Mean: $$0.08$$, Standard deviation: $$\$$\sqrt{\frac{0.08 	imes 0.92}{200}} = 0.0192\$

Question 11

In the context of election polling, why do the sample proportion and population proportion distributions often look similar?

Accepted Answer

Because the sample is random and large enough, the sample proportion is an unbiased estimator of the population proportion.

Question 12

A GMAT test has $$\mu = 500$$ and $$\sigma = 110. $$What is the standard error of the mean for a sample of 100 students?

Accepted Answer

$$\$$\frac{110}{\sqrt{100}} = 11\$

Question 13

Which of the following would have the most uncertainty regarding its results?

Accepted Answer

An individual tester's score.

Question 14

If a sample of 100 people is taken from a population where 6% are at least 75 years old, is the sampling distribution of $$\hat{p}$$ approximately normal?

Accepted Answer

No, because $$n p = 6$$ and $$n(1-p) = 94$$, and $$n p \u003c 15.$$

Question 15

A sample of 300 respondents is taken from a population where $$p = 0.075. $$What is the probability that fewer than 5% are at least 75 years old?

Accepted Answer

First, $$\mu_{\hat{p}} = 0.075$$, $$\sigma_{\hat{p}} = \sqrt{\frac{0.075 	imes 0.925}{300}} = 0.0152$$. $$z = \frac{0.05 - 0.075}{0.0152} = -1.64. $$Probability is $$0.0505.$$

Sampling Distributions for Proportions and Means: Key Concepts and Applications

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