Q: A city claims that 60% of its residents recycle. You survey 100 residents and find that 54 recycle. What is the value of the test statistic for a one-proportion z-test?

$$z = \frac{0.54 - 0.6}{\sqrt{\frac{0.6 \times 0.4}{100}}}$$

Question 1

A pharmaceutical company claims that at least 80% of patients recover after using their new drug. Which of the following correctly states the null and alternative hypotheses for testing this claim?

Accepted Answer

$$H_0$$: p \geq 0.8$$; H_A$$: p \u003c 0.8$$

Question 2

A researcher performs a one-proportion z-test and obtains a p-value of 0.03. If the significance level $$\alpha is 0.05$$, what is the correct conclusion?

Accepted Answer

Reject the null hypothesis because $$0.03 \u003c 0.05.$$

Question 3

Which of the following best describes the difference between a one-sided and a two-sided hypothesis test?

Accepted Answer

A one-sided test only considers deviations in one direction from the null hypothesis, while a two-sided test considers deviations in both directions.

Question 4

Suppose you calculate a z-score of 2.1 for a one-proportion z-test. What is the two-sided p-value associated with this z-score?

Accepted Answer

$$2 	imes P(Z \u003e 2.1)$$

Question 5

Which of the following is NOT an assumption for performing a one-proportion z-test?

Accepted Answer

The population standard deviation is known.

Question 6

A city claims that 60% of its residents recycle. You survey 100 residents and find that 54 recycle. What is the value of the test statistic for a one-proportion z-test?

Accepted Answer

$$z = \frac{0.54 - 0.6}{\sqrt{\frac{0.6 	imes 0.4}{100}}}$$

Question 7

A two-sided hypothesis test is performed at $$\alpha = 0.01$$ and the p-value is 0.008. What is the correct interpretation?

Accepted Answer

There is strong evidence against the null hypothesis.

Question 8

Which of the following scenarios requires a two-sided test?

Accepted Answer

Testing whether the proportion of defective products is different from 5%.

Question 9

A student claims that the proportion of left-handed students in their university is less than 10%. You collect a random sample and perform a one-proportion z-test. Which of the following is the correct alternative hypothesis?

Accepted Answer

$$H_A: p \u003c 0.10$$

Question 10

If the p-value is greater than the significance level $$\alpha$$, what should you conclude?

Accepted Answer

Fail to reject the null hypothesis.

Question 11

A medical researcher wants to test if a new vaccine is more effective than the current one, which has a success rate of 70%. Which of the following is the correct set of hypotheses?

Accepted Answer

$$H_0$$: p = 0.70$$; H_A$$: p \u003e 0.70$$

Question 12

Which of the following is a common mistake when interpreting the p-value?

Accepted Answer

Believing the p-value is the probability that the null hypothesis is true.

Question 13

A one-proportion z-test is performed and the test statistic is $$z = -1.5. $$For a one-sided test with $$H_A: p \u003c p_0$, what is the p-value?

Accepted Answer

$$P(Z \u003c -1.5)$$

Question 14

Which of the following best describes the purpose of a significance level $$\alpha in $$hypothesis testing?

Accepted Answer

It is the probability of rejecting the null hypothesis when it is actually true.

Question 15

A company wants to know if the proportion of customers satisfied with their service has changed from last year's 85%. Which test should they use?

Accepted Answer

A two-sided one-proportion z-test.

Question 16

Which of the following is a correct conclusion if the p-value is 0.12 and $$\alpha = 0.05$$?

Accepted Answer

There is not enough evidence to reject the null hypothesis.

Question 17

In a one-proportion z-test, which formula is used to calculate the standard error of the sample proportion under the null hypothesis?

Accepted Answer

$$SE = \sqrt{\frac{p_0$ (1 - p_0$)}{n}}$$

Question 18

A study finds a statistically significant result at $$\alpha = 0.05. $$Which of the following is a correct interpretation?

Accepted Answer

The observed data are unlikely under the null hypothesis, at the 5% significance level.

Question 19

Which of the following is a real-world example where a one-proportion z-test would be appropriate?

Accepted Answer

Testing if the proportion of voters supporting a candidate differs from 50%.

Question 20

A researcher wants to test if the rate of a rare disease in a population is higher than the known rate of 0.1%. Which is the correct alternative hypothesis?

Accepted Answer

$$H_A: p \u003e 0.001$$

Hypothesis Testing and One-Proportion Z-Test: Key Concepts and Applications

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