Question 1

Which of the following is NOT a required condition for making inference about a population proportion?

Accepted Answer

The sample size must be large enough so that the sampling distribution is approximately normal.

Question 2

A clinical trial tests a new drug on 120 patients, and 30 show improvement. What is the sample proportion $$\hat{p} of $$patients who improved?

Accepted Answer

$$\hat{p} = \frac{30}{120} = 0.25$$

Question 3

Which formula gives the standard error (SE) of the sample proportion $$\hat{p}$$ when $$p is $$unknown?

Accepted Answer

$$SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Question 4

In a significance test for a population proportion, which of the following is the correct formula for the test statistic $$z$$ under the null hypothesis $$H_0$$: p = $$p_0$$?

Accepted Answer

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

Question 5

A study finds that 23 out of 440 arthritis patients experienced adverse symptoms from a pain reliever. What is the 90% large-sample confidence interval for the population proportion $$p$$? (Use $$z^* = 1.645.)$$

Accepted Answer

$$\hat{p} \pm 1.645 \sqrt{\frac{\hat{p}(1-\hat{p})}{440}}$$

Question 6

What is the main advantage of the 'plus four' confidence interval for a population proportion compared to the large-sample method?

Accepted Answer

It is more accurate for small sample sizes.

Question 7

In Mendel's pea experiment, 5474 out of 7324 peas were smooth. What is the sample proportion $$\hat{p} of $$smooth peas?

Accepted Answer

$$\hat{p} = \frac{5474}{7324} \approx 0.747$$

Question 8

Which of the following best describes the p-value in a significance test for a population proportion?

Accepted Answer

The probability, assuming $$H_0 is $$true, of obtaining a test statistic as extreme or more extreme than the observed value.

Question 9

A researcher wants to estimate the proportion of patients with a rare disease with a margin of error no more than 0.01 at 90% confidence. Assuming $$p^* = 0.1$$ and $$z^* = 1.645$$, what sample size $$n is $$required?

Accepted Answer

$$n = \left( \frac{1.645}{0.01} \$$right)^2$$ (0.1)(0.9) \approx 2435$$

Question 10

Which of the following scenarios would violate the conditions for inference about a population proportion?

Accepted Answer

The sample is not randomly selected from the population.

Question 11

For a two-sided alternative hypothesis $$H_1$$: p 
eq $$p_0$$, how is the p-value calculated from the test statistic $$z$$?

Accepted Answer

p-value = $$2P(Z \geq |z|)$$

Question 12

Which of the following is the correct formula for the margin of error $$m in a $$large-sample confidence interval for a population proportion?

Accepted Answer

$$m = z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Question 13

A sample of 1000 people is surveyed, and 400 say they prefer brand A. Using the 'plus four' method, what is the adjusted sample proportion $$	ilde{p}$$?

Accepted Answer

$$	ilde{p} = \frac{400+2}{1000+4} = \frac{402}{1004}$$

Question 14

Why is it often recommended to use $$p^* = 0.5$$ when calculating the required sample size for a desired margin of error?

Accepted Answer

It gives the largest possible sample size, ensuring the margin of error will not be exceeded.

Question 15

A pharmaceutical company wants to estimate the proportion of patients who experience side effects from a new drug. Which of the following best describes the parameter $$p$$?

Accepted Answer

The proportion of all patients in the population who would experience side effects if given the drug.

Question 16

If a sample proportion $$\hat{p} is $$very close to 0 or 1, which of the following is a likely consequence?

Accepted Answer

The normal approximation to the sampling distribution may not be accurate.

Question 17

A researcher wants a 95% confidence interval for a population proportion. Which $$z^*$$ value should be used?

Accepted Answer

$$z^* = 1.96$$

Question 18

Which of the following best explains why the 'plus four' method adds two successes and two failures to the data?

Accepted Answer

To improve the accuracy of the confidence interval, especially for small samples.

Question 19

In the context of genetic inheritance, Mendel's law predicts a population proportion of 0.75 for a dominant trait. If a sample yields $$\hat{p} = 0.747$$ with $$n = 7324$$, what is the null hypothesis for a significance test?

Accepted Answer

$$H_0$$: p = 0.75$$

Inference About a Population Proportion: Hypothesis Testing, Confidence Intervals, and Sample Size

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