Question 1

Which formula gives the point estimate for a population proportion based on a sample of size $$n$$ with $$x$$ successes?

Accepted Answer

$$\frac{x}{n}$$

Question 2

A 95% confidence interval for a population proportion $$p is $$constructed as $$\hat{p} \pm 1.96 \sigma_{\hat{p}}. $$What does the value 1.96 represent?

Accepted Answer

The critical value $$z_{0.025}$$ for a 95% confidence level

Question 3

If a confidence interval for a population proportion does NOT contain the true proportion, what is the most likely cause?

Accepted Answer

The sample statistic was in the tail of the sampling distribution

Question 4

Which formula is used to calculate the margin of error $$E$$ for a $$(1-\alpha)\cdot100\%$$ confidence interval for a population proportion?

Accepted Answer

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

Question 5

A social worker wants to estimate the proportion of people providing eldercare with a margin of error of 0.03 and 95% confidence. If no prior estimate is available, what value should be used for $$\hat{p} in $$the sample size formula?

Accepted Answer

$$\hat{p} = 0.50$$

Question 6

Which of the following is NOT a property of the Student’s t-distribution?

Accepted Answer

It is identical to the normal distribution for all sample sizes

Question 7

When constructing a confidence interval for a population mean with unknown population standard deviation, which distribution should be used?

Accepted Answer

Student’s t-distribution

Question 8

What is the formula for the lower bound of a $$(1-\alpha)\cdot100\%$$ confidence interval for a population mean $$\mu$$ using Student’s t-distribution?

Accepted Answer

$$\bar{x} - t_{\alpha/2} \frac{s}{\sqrt{n}}$$

Question 9

Which condition must be checked before using Student’s t-distribution for confidence intervals when $$n \u003c 30$$?

Accepted Answer

The sample data must be approximately normal with no outliers

Question 10

What is the formula for the sample size needed to estimate a population mean $$\mu$$ within margin of error $$E at$$ $$(1-\alpha)\cdot100\%$$ confidence?

Accepted Answer

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

Question 11

Which distribution is used to construct confidence intervals for the population variance $$\$$sigma^2$$?

Accepted Answer

Chi-square distribution

Question 12

What is the formula for the lower bound of a $$(1-\alpha)\cdot100\%$$ confidence interval for the population variance $$\$$sigma^2$$?

Accepted Answer

$$\frac{(n-1)s^2$}{\$$chi^2$$_{\alpha/2}}$$

Question 13

Which of the following is a characteristic of the chi-square distribution?

Accepted Answer

It is not symmetric

Question 14

In bootstrapping, what is the main requirement for the bootstrap distribution to be valid for confidence interval estimation?

Accepted Answer

The bootstrap distribution must be symmetric and centered near the original sample statistic

Question 15

Which method should be used to construct a confidence interval for a population mean when the sample size is small and the data are not approximately normal?

Accepted Answer

Nonparametric methods or bootstrapping

Question 16

If the sample size increases, what happens to the margin of error for a confidence interval for a population proportion?

Accepted Answer

The margin of error decreases

Question 17

Which formula is used to estimate the sample size needed for a population proportion when no prior estimate is available?

Accepted Answer

$$n = \left( \frac{z_{\alpha/2}}{E} \$$right)^2$$ \cdot 0.25$$

Question 18

Which of the following best describes the relationship between the t-distribution and the normal distribution as sample size increases?

Accepted Answer

The t-distribution approaches the normal distribution

Question 19

A confidence interval for a population mean is reported as ($$21.09$$, $$29.59). $$What is the margin of error?

Accepted Answer

$$\frac{29.59 - 21.09}{2}$$

Chapter 9: Estimating the Value of a Parameter: Confidence Intervals for Proportions, Means, and Variances

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