Q: Given a sample proportion $$\hat{p} = 0.35$$ from a sample of $$n = 100$$, and a population proportion $$p = 0.39$$, what is the formula for the standardized test statistic $$z$$?

$$z = \frac{0.35 - 0.39}{\sqrt{\frac{0.39 \times 0.61}{100}}}$$

Question 1

Which of the following is a necessary condition for using the z-test for a population proportion?

Accepted Answer

Both $$np \geq 5$$ and $$n(1-p) \geq 5$$ must be satisfied.

Question 2

A researcher claims that more than 60% of college students own a laptop. In a random sample of 200 students, 130 report owning a laptop. At the 5% significance level, what is the correct null hypothesis ($$H_0$$)?

Accepted Answer

$$H_0$$: p \leq 0.60$$

Question 3

Given a sample proportion $$\hat{p} = 0.35$$ from a sample of $$n = 100$$, and a population proportion $$p = 0.39$$, what is the formula for the standardized test statistic $$z$$?

Accepted Answer

$$z = \frac{0.35 - 0.39}{\sqrt{\frac{0.39 	imes 0.61}{100}}}$$

Question 4

In hypothesis testing for a population proportion, what does the P-value represent?

Accepted Answer

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

Question 5

A sample of 11,400 adults yields 2,900 successes. What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{2,900}{11,400}$$

Question 6

If the significance level $$\alpha is 0.01 $$and the P-value is 0.03, what is the correct conclusion?

Accepted Answer

Fail to reject the null hypothesis.

Question 7

Which of the following best describes a Type I error in the context of hypothesis testing for proportions?

Accepted Answer

Rejecting a true null hypothesis.

Question 8

A left-tailed z-test for a population proportion is being conducted at the 1% significance level. If the calculated z-value is -1.5 and the critical value is -2.33, what is the correct decision?

Accepted Answer

Fail to reject the null hypothesis.

Question 9

Which formula is used to calculate the standard error of the sample proportion in a z-test for proportions?

Accepted Answer

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

Question 10

A researcher wants to test if the proportion of people who exercise regularly is different from 0.5. Which of the following alternative hypotheses should be used?

Accepted Answer

$$H_1$$: p 
eq 0.5$$

Question 11

In a two-tailed test, if the area in the right tail for the observed z-value is 0.04, what is the P-value?

Accepted Answer

$$0.08$$

Question 12

Which of the following scenarios is most appropriate for using a z-test for a population proportion?

Accepted Answer

Testing whether the proportion of defective products in a factory is less than 5%.

Question 13

A company claims that at least 80% of its products pass quality control. Which of the following is the correct null hypothesis?

Accepted Answer

$$H_0$$: p \geq 0.80$$

Question 14

If a sample of 150 people contains 45 who prefer a new product, what is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{45}{150}$$

Question 15

Which of the following best describes the rejection region in a left-tailed z-test at the 5% significance level?

Accepted Answer

z \u003c -1.645

Hypothesis Testing for Proportions: Study Notes

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