Question 1

A researcher wants to estimate the proportion of adults in a city who support a new public transportation initiative. Which of the following conditions must be met for the sampling distribution of the sample proportion $$\hat{p} to be $$approximately normal?

Accepted Answer

The sample size $$n$$ must be large enough so that $$np \geq 10$$ and $$n(1-p) \geq 10.$$

Question 2

Suppose you collect a random sample of 100 patients and find that 60% have a certain genetic marker. Construct a 95% confidence interval for the population proportion $$p. $$Which formula should you use?

Accepted Answer

$$\hat{p} \pm z^* \sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$$

Question 3

A pharmaceutical company wants to determine the mean recovery time for a new drug. Which condition is necessary for the sampling distribution of the sample mean $$\bar{x} to be $$approximately normal?

Accepted Answer

The sample size is large ($$n \geq 30) or $$the population is normally distributed.

Question 4

A 95% confidence interval for a population mean is calculated as $$(12.3, 15.7). $$What is the margin of error?

Accepted Answer

$$1.7$$

Question 5

A hospital administrator wants to estimate the proportion of patients who are readmitted within 30 days. She wants the margin of error to be no more than $$0.03 at a 95$$% confidence level. If no preliminary estimate of $$p is $$available, what value should she use for $$p in $$the sample size formula?

Accepted Answer

$$p = 0.5$$

Question 6

Which of the following best describes the null hypothesis $$H_0 in a $$one-sample $$t-$$test for a population mean?

Accepted Answer

$$H_0$$: \mu = \$$mu_0$$

Question 7

A clinical trial tests whether a new medication lowers blood pressure more than the standard treatment. The $$p-$$value for the test is $$0.04$$ and the significance level is $$\alpha = 0.05. $$What is the correct conclusion?

Accepted Answer

Reject the null hypothesis; there is evidence the new medication is more effective.

Question 8

A researcher wants to compare the mean test scores of students from two different schools. Which hypothesis test should be used?

Accepted Answer

Two-sample $$t-$$test for the difference in means

Question 9

Which of the following scenarios represents paired data?

Accepted Answer

Measuring the blood pressure of patients before and after taking a medication

Question 10

A 95% confidence interval for the difference in means between two treatments is $$(-2.1, 0.5). $$What does this interval suggest?

Accepted Answer

There is not enough evidence to conclude a significant difference between the treatments at the 5% level.

Question 11

A paired sample $$t-$$test is appropriate when:

Accepted Answer

Each observation in one sample can be paired with a unique observation in the other sample.

Question 12

Which formula is used to calculate the test statistic for a one-sample $$z-$$test for a population proportion?

Accepted Answer

$$z = \dfrac{\hat{p} - p_0$}{\sqrt{\dfrac{p_0$(1-p_0$)}{n}}}$$

Question 13

A researcher wants to determine the sample size needed to estimate a population mean within a margin of error $$E at a 95$$% confidence level. Which formula should be used if the population standard deviation $$\sigma is $$known?

Accepted Answer

$$n = \left( \dfrac{z^* \sigma}{E} \$$right)^2$$

Question 14

If the $$p-$$value of a hypothesis test is $$0.12$$ and the significance level is $$0.05$$, what is the correct conclusion?

Accepted Answer

Fail to reject the null hypothesis.

Question 15

Which of the following statements about the margin of error in a confidence interval is correct?

Accepted Answer

Increasing the sample size decreases the margin of error.

Question 16

A study compares the mean cholesterol levels before and after a new diet in the same group of patients. Which statistical test is most appropriate?

Accepted Answer

Paired sample $$t-$$test

Question 17

Which of the following is NOT a requirement for constructing a confidence interval for a population mean when the population standard deviation is unknown?

Accepted Answer

The population standard deviation must be known.

Question 18

A 90% confidence interval for a population proportion is $$(0.45, 0.55). $$What does this mean?

Accepted Answer

We are 90% confident that the true population proportion is between 0.45 and 0.55.

Question 19

Which formula is used to construct a confidence interval for the difference in two population means (assuming equal variances and independent samples)?

Accepted Answer

$$(\bar{x}_1 - \bar{x}_2) \pm t^* s_p \sqrt{\dfrac{1}{n_1$} + \dfrac{1}{n_2$}}$$

Study Guide: Sampling Distributions, Confidence Intervals, and Hypothesis Testing

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