Q: An entrepreneur wants to open a restaurant if more than 15% of local residents are interested. In a survey of 500 people, 90 express interest. What is the value of the test statistic $$z$$ for $$H_0$$: p = 0.15$$?

$$z = \frac{0.18 - 0.15}{\sqrt{\frac{0.15 \times 0.85}{500}}}$$

Question 1

A medical clinic found that 30% of its patients had illnesses of a psychosomatic nature. A psychologist wants to test if this percentage has changed. She examines a random sample of 80 patients and finds that 36 have such illnesses. What is the correct null hypothesis for this test?

Accepted Answer

$$H_0$$: p = 0.30$$

Question 2

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

Accepted Answer

Rejecting the null hypothesis when it is true.

Question 3

A frozen orange juice producer claims that 20% of all orange juice drinkers prefer its product. A competitor samples 200 drinkers and finds that 34 prefer the producer's brand. What is the value of the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{34}{200} = 0.17$$

Question 4

Which formula is used to calculate the test statistic $$z$$ for testing a population proportion $$p$$?

Accepted Answer

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

Question 5

A new cancer screening method is tested on 70 patients known to have the disease. It detects cancer in 64 patients. The current method detects cancer in 85% of such patients. What is the alternative hypothesis if researchers hope the new method is better?

Accepted Answer

$$H_a: p \u003e 0.85$$

Question 6

What does the p-value represent in a hypothesis test?

Accepted Answer

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

Question 7

A consumer group suspects that fewer than 60% of graduate students have made a purchase online. They sample 80 students and find 42 have done so. What is the correct alternative hypothesis?

Accepted Answer

$$H_a: p \u003c 0.60$$

Question 8

If the significance level $$\alpha is $$set at 0.05, what is the correct decision rule for a hypothesis test?

Accepted Answer

Reject $$H_0 if $$the p-value is less than 0.05.

Question 9

An entrepreneur wants to open a restaurant if more than 15% of local residents are interested. In a survey of 500 people, 90 express interest. What is the value of the test statistic $$z$$ for $$H_0$$: p = 0.15$$?

Accepted Answer

$$z = \frac{0.18 - 0.15}{\sqrt{\frac{0.15 	imes 0.85}{500}}}$$

Question 10

Which of the following is NOT an element of a test of hypothesis?

Accepted Answer

Calculation of the sample mean.

Question 11

A Type II error in hypothesis testing is best described as:

Accepted Answer

Failing to reject the null hypothesis when it is false.

Question 12

In a hypothesis test about a population proportion, which of the following conditions must be met to use the $$z-$$test?

Accepted Answer

The sample is large and random, and the data are qualitative.

Question 13

A company claims that at least 55% of voters favor candidate A. Which of the following is the correct set of hypotheses to test if the true proportion is less than claimed?

Accepted Answer

$$H_0$$: p = 0.55$$ vs H_a$$: p \u003c 0.55$$

Question 14

If a hypothesis test is conducted at $$\alpha = 0.01$$ instead of $$\alpha = 0.05$$, what is the effect on the probability of a Type I error?

Accepted Answer

The probability of a Type I error decreases.

Question 15

A sample of 80 graduate students finds that 42 have made a purchase online. Construct a 95% confidence interval for the proportion of graduate students who have made a purchase online. Which formula should be used?

Accepted Answer

$$\hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Question 16

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

Accepted Answer

The probability of making a Type I error.

Question 17

A psychologist tests whether the proportion of psychosomatic illnesses has changed from 30%. In a sample of 80, 36 have such illnesses. What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{36}{80} = 0.45$$

Question 18

Which of the following best describes the alternative hypothesis $$H_a$$?

Accepted Answer

It gives an opposing conjecture to the one given by the null hypothesis.

Question 19

A test of hypothesis about a population proportion is conducted and the p-value is found to be 0.03. If $$\alpha = 0.05$$, what is the correct conclusion?

Accepted Answer

Reject the null hypothesis.

Hypothesis Testing for Population Proportions and Means

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