Q: A manufacturer tests 300 batteries and finds 10 defective. Test $$H_0$$: p = 0.05$$ vs H_a$$: p \u003c 0.05$$ at $$\alpha = 0.01$$. What is the value of the test statistic z$?

$$z = \frac{0.0333 - 0.05}{\sqrt{0.05 \times 0.95 / 300}}$$

Question 1

Which of the following best describes the difference between the population proportion $$p$$ and the sample proportion $$\hat{p}$$?

Accepted Answer

$$p is $$the proportion of the entire population with the attribute, while $$\hat{p} is $$the proportion of the sample with the attribute.

Question 2

A poll of 1010 employees found that 202 responded 'yes' to playing hooky. What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{202}{1010} = 0.20$$

Question 3

Which formula gives the margin of error $$E$$ for a one-proportion $$z-$$interval?

Accepted Answer

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

Question 4

Suppose you want a 95% confidence interval for a population proportion. What is the appropriate $$z_{\alpha/2}$$ value to use?

Accepted Answer

$$z_{\alpha/2} = 1.960$$

Question 5

A manufacturer tests 300 batteries and finds 10 defective. Test $$H_0$$: p = 0.05$$ vs H_a$$: p \u003c 0.05$$ at $$\alpha = 0.01$$. What is the value of the test statistic z$?

Accepted Answer

$$z = \frac{0.0333 - 0.05}{\sqrt{0.05 	imes 0.95 / 300}}$$

Question 6

Which of the following is an assumption for using the one-proportion $$z-$$interval procedure?

Accepted Answer

Both $$x$$ and $$n-x$$ must be at least 5.

Question 7

If you want to estimate a population proportion with a margin of error no greater than $$E at $$confidence level $$1-\alpha$$, and you have no prior estimate for $$\hat{p}$$, what sample size $$n$$ should you use?

Accepted Answer

$$n = \frac{1}{4} \left( \frac{z_{\alpha/2}}{E} \$$right)^2$$

Question 8

Which scenario best describes when to use a left-tailed $$z-$$test for a population proportion?

Accepted Answer

When testing if the population proportion is less than a specified value.

Question 9

A 98% confidence interval for a population proportion is $$(0.895, 0.925). At$$ $$\alpha = 0.02$$, what can you conclude about $$H_0$$: p = 0.90$$ vs H_a$$: p 
eq 0.90$$?

Accepted Answer

Fail to reject $$H_0$$ because $$0.90 is $$inside the interval.

Question 10

Which of the following best describes the sampling distribution of $$\hat{p}$$ for large $$n$$?

Accepted Answer

$$\hat{p} is $$approximately normal with mean $$p$$ and variance $$\frac{p(1-p)}{n}.$$

Question 11

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

Accepted Answer

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

Question 12

A sample of 2000 adults finds 1820 prefer mobile banking. Construct a 95% confidence interval for the population proportion. What is the correct formula for the endpoints?

Accepted Answer

$$\hat{p} \pm 1.96 \sqrt{\frac{\hat{p}(1-\hat{p})}{2000}}$$

Question 13

Which value of $$\hat{p}$$ maximizes the product $$\hat{p}(1-\hat{p})$$?

Accepted Answer

$$\hat{p} = 0.5$$

Question 14

If the P-value for a right-tailed $$z-$$test is 0.03 and $$\alpha = 0.05$$, what is the correct conclusion?

Accepted Answer

Reject $$H_0.$$

Question 15

Which of the following is NOT a step in the one-proportion $$z-$$test procedure?

Accepted Answer

Compute the sample mean.

Question 16

A researcher wants to estimate the proportion of people who support a new policy with a margin of error of 0.02 at 99% confidence. What $$z_{\alpha/2}$$ value should be used?

Accepted Answer

$$z_{\alpha/2} = 2.576$$

Question 17

Which of the following statements about confidence intervals for a population proportion is TRUE?

Accepted Answer

A 95% confidence interval means that if we repeated the sampling process many times, about 95% of the intervals would contain the true population proportion.

Inference for a Population Proportion: Confidence Intervals and Hypothesis Testing

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