Question 1

A researcher wants to estimate the proportion of students at a university who own a laptop. In a random sample of 500 students, 350 report owning a laptop. What is the point estimate for the population proportion?

Accepted Answer

$$\frac{350}{500} = 0.70$$

Question 2

Which of the following best describes the margin of error in a confidence interval for a population proportion?

Accepted Answer

It measures the accuracy of the point estimate.

Question 3

Suppose you want to estimate the proportion of voters who support a new policy with a margin of error of $$0.04$$ and $$95\%$$ confidence. If no prior estimate is available, what value should you use for $$\hat{p} to $$calculate the required sample size?

Accepted Answer

$$0.5$$

Question 4

A 90% confidence interval for a population proportion is calculated as $$0.63 \pm 0.02. $$What does this interval mean?

Accepted Answer

We are 90% confident that the true proportion is between $$0.61$$ and $$0.65.$$

Question 5

Which formula is used to determine the sample size needed to estimate a population proportion $$p$$ with margin of error $$E$$ and confidence level $$(1-\alpha)\cdot100\%$$ when a prior estimate $$\hat{p} is $$available?

Accepted Answer

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

Question 6

Which property distinguishes the Student’s t-distribution from the standard normal distribution as sample size increases?

Accepted Answer

The t-distribution approaches the standard normal distribution as sample size increases.

Question 7

A sample of 17 pennies minted after 1982 has a mean weight of $$2.464$$ grams and a standard deviation of $$0.02$$ grams. What is the 99% confidence interval for the population mean weight, using $$t_{\alpha/2} = 2.575$$?

Accepted Answer

$$2.464 \pm 2.575 \cdot \frac{0.02}{\sqrt{17}}$$

Question 8

Which formula gives the margin of error for a confidence interval for a population mean when the population standard deviation is unknown?

Accepted Answer

$$E = t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$$

Question 9

A sociologist wants to estimate the proportion of residents who only speak English at home with a margin of error of $$0.03$$ and 90% confidence. If a prior estimate is $$0.824$$ and $$z_{\alpha/2} = 1.645$$, what is the required sample size?

Accepted Answer

$$n = 0.824(1-0.824) \left( \frac{1.645}{0.03} \$$right)^2$$

Question 10

Which formula is used to construct a $$(1-\alpha)\cdot100\%$$ confidence interval for a population mean $$\mu$$ when the population standard deviation is unknown?

Accepted Answer

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

Question 11

Which of the following statements about the chi-square distribution is correct?

Accepted Answer

The shape of the chi-square distribution depends on the degrees of freedom.

Question 12

Given a sample variance $$s^2$$ from a normal population, which formula is used to construct a $$(1-\alpha)\cdot100\%$$ confidence interval for the population variance $$\$$sigma^2$$?

Accepted Answer

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

Question 13

A stock analyst computes the sample standard deviation of weekly returns for Microsoft as $$s = 4.6974$$ for $$n = 15$$ weeks. For a 90% confidence interval, the chi-square critical values are $$\$$chi^2$$_{0.95} = 6.571$$ and $$\$$chi^2$$_{0.05} = 23.685. $$What is the lower bound for the population standard deviation?

Accepted Answer

$$\sqrt{\frac{14 \cdot 22.0659}{23.685}}$$

Question 14

Which of the following is NOT a requirement for constructing a confidence interval for a population proportion?

Accepted Answer

The sample size must be greater than 5% of the population.

Question 15

If the area under the t-distribution to the right of $$t_{0.20} is$$ $$0.20$$ with $$10$$ degrees of freedom, what is the value of $$t_{0.20}$$?

Accepted Answer

$$0.8791$$

Question 16

Which formula is used to determine the sample size needed to estimate a population mean $$\mu$$ with margin of error $$E$$ and confidence level $$(1-\alpha)\cdot100\%$$?

Accepted Answer

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

Question 17

Which procedure should you use to estimate the population mean if the data come from a population that is approximately normal with no outliers and $$n \geq 30$$?

Accepted Answer

Compute a z-interval.

Estimating Population Parameters: Proportion, Mean, and Standard Deviation

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