Q: Suppose a sample of 100 candies contains 22 blue candies. The company claims 30% of candies are blue. What is the value of the test statistic $$z$$ for this test?

$$z = \dfrac{0.22 - 0.3}{\sqrt{\dfrac{0.3 \times 0.7}{100}}} = -1.746$$

Question 1

Which of the following is the correct formula for the test statistic $$z$$ when testing a claim about a population proportion $$p$$ using the normal approximation method?

Accepted Answer

$$z = \dfrac{\hat{p} - p}{\sqrt{\dfrac{pq}{n}}}$$

Question 2

A researcher wants to test the claim that less than 30% of adults have sleepwalked. Which of the following correctly states the null and alternative hypotheses?

Accepted Answer

$$H_0$$: p = 0.30;\quad $$H_1$$: p \u003c 0.30$$

Question 3

Which of the following is NOT a requirement for using the normal approximation method to test a claim about a population proportion?

Accepted Answer

The population must be normally distributed.

Question 4

Suppose a sample of 100 candies contains 22 blue candies. The company claims 30% of candies are blue. What is the value of the test statistic $$z$$ for this test?

Accepted Answer

$$z = \dfrac{0.22 - 0.3}{\sqrt{\dfrac{0.3 	imes 0.7}{100}}} = -1.746$$

Question 5

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 6

A right-tailed hypothesis test for a population proportion yields a test statistic $$z = 1.25$$ and a significance level $$\alpha = 0.05. $$What is the correct conclusion?

Accepted Answer

Fail to reject the null hypothesis because $$z = 1.25 is $$less than the critical value $$z_{0.05} = 1.645.$$

Question 7

Which of the following best describes the meaning of 'fail to reject the null hypothesis' in the context of hypothesis testing?

Accepted Answer

There is not sufficient evidence to support the alternative hypothesis.

Question 8

A study finds that 29.2% of 19,136 adults have sleepwalked. Test the claim that fewer than 30% of adults have sleepwalked at $$\alpha = 0.05. $$The calculated $$z is$$ $$-2.41. $$What is the correct interpretation?

Accepted Answer

There is sufficient evidence to support the claim that fewer than 30% of adults have sleepwalked.

Question 9

Which of the following is a correct requirement for using the normal approximation to the binomial for hypothesis testing about a proportion?

Accepted Answer

Both $$np \geq 5$$ and $$nq \geq 5$$ must be satisfied.

Question 10

A company claims that at least 50% of users enable two-factor authentication. In a random sample of 926 users, 482 say they do. What is the alternative hypothesis if you want to test the claim that most users enable two-factor authentication?

Accepted Answer

$$H_1$$: p \u003e 0.5$$

Question 11

If the P-value for a hypothesis test is 0.008 and the significance level is $$\alpha = 0.05$$, what is the correct decision?

Accepted Answer

Reject the null hypothesis.

Question 12

Which of the following scenarios best illustrates a two-tailed test for a population proportion?

Accepted Answer

Testing if the proportion of blue candies is different from 30%.

Question 13

In the context of hypothesis testing, what does the significance level $$\alpha$$ represent?

Accepted Answer

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

Question 14

A sample of 926 users yields $$\hat{p} = 0.52. If $$the null hypothesis is $$p = 0.5$$, what is the value of $$q$$ used in the test statistic formula?

Accepted Answer

$$q = 1 - p = 0.5$$

Question 15

Which of the following statements is TRUE regarding the interpretation of hypothesis test results?

Accepted Answer

Failing to reject the null hypothesis does not prove it is true.

Question 16

A hypothesis test for a population proportion yields a test statistic $$z = -1.746$$ and a two-tailed critical value of $$\pm 1.96. $$What is the correct conclusion?

Accepted Answer

Fail to reject the null hypothesis because $$-1.746 is $$within $$-1.96$$ and $$1.96.$$

Question 17

Which of the following is a correct step in the P-value method for hypothesis testing?

Accepted Answer

Compare the P-value to the significance level $$\alpha to $$decide whether to reject the null hypothesis.

Question 18

A left-tailed test yields a test statistic $$z = -2.41$$ and a P-value of 0.008. If $$\alpha = 0.05$$, what is the correct conclusion?

Accepted Answer

Reject the null hypothesis and support the alternative hypothesis.

Question 19

Which of the following is the correct symbolic form for the alternative hypothesis when testing the claim that 'most' users (more than half) use two-factor authentication?

Accepted Answer

$$H_1$$: p \u003e 0.5$$

8.2

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