Q: A medical researcher wants to test if a new vaccine has a different effectiveness rate than the current vaccine, which is 70%. What are the correct null and alternative hypotheses?

$$H_0$$: p = 0.70$$, H_A$$: p \neq 0.70$$

Q: A two-sided test is performed and the z-score is -2.0. What is the correct way to calculate the p-value?

$$p = 2 \times P(Z \u003c -2.0)$$

Question 1

A pharmaceutical company claims that at least 80% of patients recover after using their new drug. A hospital tests this claim by treating 100 patients and finds that 74 recover. What is the correct null hypothesis ($$H_0) $$for a one-proportion z-test?

Accepted Answer

$$H_0$$: p = 0.80$$

Question 2

Which of the following best describes the alternative hypothesis ($$H_A) in a $$one-sided test where a manufacturer wants to show their product's defect rate is less than 5%?

Accepted Answer

$$H_A: p \u003c 0.05$$

Question 3

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 $$p-$$value $$\u003c \alpha$$

Question 4

Which of the following is an assumption required for a one-proportion z-test?

Accepted Answer

The sample size is large enough so that $$np_0$$ \geq 10$$ and n(1$$-$$p_0$$) \geq 10$$

Question 5

A two-sided hypothesis test is being conducted. Which of the following statements is true about the p-value calculation?

Accepted Answer

The p-value is twice the area in one tail beyond the observed test statistic

Question 6

Suppose you are testing $$H_0$$: p = 0.5$$ vs. H_A$$: p \u003e 0.5$$ and you observe a z-score of 2.1. What is the one-sided p-value?

Accepted Answer

$$p = P(Z \u003e 2.1)$$

Question 7

A city claims that 60% of its residents recycle. A survey of 200 residents finds that 110 recycle. What is the value of the test statistic for a one-proportion z-test?

Accepted Answer

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

Question 8

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

Accepted Answer

A scientist wants to know if the proportion of smokers in a city is different from the national average

Question 9

A student performs a one-proportion z-test and gets a p-value of 0.12. If $$\alpha = 0.05$$, what is the correct interpretation?

Accepted Answer

There is not enough evidence to reject the null hypothesis at the 0.05 significance level

Question 10

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 11

A medical researcher wants to test if a new vaccine has a different effectiveness rate than the current vaccine, which is 70%. What are the correct null and alternative hypotheses?

Accepted Answer

$$H_0$$: p = 0.70$$, H_A$$: p 
eq 0.70$$

Question 12

In a one-proportion z-test, which of the following would violate the assumptions of the test?

Accepted Answer

The sample size is so small that $$np_0$$ \u003c 10$$

Question 13

A two-sided test is performed and the z-score is -2.0. What is the correct way to calculate the p-value?

Accepted Answer

$$p = 2 	imes P(Z \u003c -2.0)$$

Question 14

A school claims that 90% of its students graduate on time. A random sample of 50 students finds that 41 graduated on time. What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{41}{50}$$

Question 15

Which of the following best explains why causation cannot be concluded from a significant hypothesis test result in an observational study?

Accepted Answer

Confounding variables may explain the observed association

Question 16

A one-proportion z-test is performed with $$n = 100$$, $$p_0 = 0.4$, and $$\hat{p} = 0.5$$. What is the value of the standard error used in the test statistic?

Accepted Answer

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

Question 17

A company wants to show that their new process reduces the defect rate below 10%. Which of the following is the correct alternative hypothesis?

Accepted Answer

$$H_A: p \u003c 0.10$$

Question 18

If a one-proportion z-test is performed and the p-value is 0.001, which of the following is the most appropriate conclusion at $$\alpha = 0.05$$?

Accepted Answer

There is strong evidence against the null hypothesis

Question 19

A researcher is testing $$H_0$$: p = 0.3$$ vs. H_A$$: p \u003e 0.3$$ and observes a sample proportion $$\hat{p} = 0.38$$ with n = 150$. What is the first step in performing the one-proportion z-test?

Accepted Answer

Check that $$np_0$$ \geq 10$$ and n(1$$-$$p_0$$) \geq 10$$

Question 20

Which of the following is NOT a correct conclusion from a hypothesis test?

Accepted Answer

We accept the null hypothesis

Hypothesis Testing: One-Proportion Z-Test and Statistical Significance

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