Q: Suppose a survey is conducted to estimate the proportion of Canadian adults who have at least one tattoo. The sample proportion is $$\hat{p} = 0.22$$ from a sample of $$n = 1000. $$What is the standard deviation of $$\hat{p}$$?

$$\sqrt{\dfrac{0.22 \times 0.78}{1000}}$$

Q: A simple random sample of $$n = 400 is $$taken from a population with $$p = 0.4. $$What is the mean and standard deviation of the sampling distribution of $$\hat{p}$$?

Mean $$= 0.4$$, Standard deviation $$= \sqrt{\dfrac{0.4 \times 0.6}{400}}$$

Question 1

Which of the following best describes the difference between a population parameter and a sample statistic?

Accepted Answer

A population parameter is a fixed value that describes a characteristic of the entire population, while a sample statistic is a value calculated from a sample and used to estimate the parameter.

Question 2

Which of the following is an example of measurement bias in a survey?

Accepted Answer

Asking a question in a way that leads respondents toward a particular answer.

Question 3

A pollster wants to estimate the proportion of voters who support a new policy. Which sampling method is most likely to produce an unbiased estimate?

Accepted Answer

Selecting a simple random sample of voters from the entire population.

Question 4

Suppose a survey is conducted to estimate the proportion of Canadian adults who have at least one tattoo. The sample proportion is $$\hat{p} = 0.22$$ from a sample of $$n = 1000. $$What is the standard deviation of $$\hat{p}$$?

Accepted Answer

$$\sqrt{\dfrac{0.22 	imes 0.78}{1000}}$$

Question 5

Which of the following conditions must be met for the Central Limit Theorem (CLT) for sample proportions to apply?

Accepted Answer

$$np \geq 10$$ and $$n(1-p) \geq 10$$

Question 6

A researcher wants to estimate the proportion of students who own a car with a margin of error of $$0.03 at a$$ $$95\%$$ confidence level. What formula should be used to determine the minimum required sample size?

Accepted Answer

$$n = \dfrac{z$$^{2}$$ p$$^{*}$$(1-p^{*}$$)}{$$E^{2}$$}$

Question 7

If a $$95\%$$ confidence interval for a population proportion is $$(0.45, 0.55)$$, which of the following is a correct interpretation?

Accepted Answer

We are $$95\%$$ confident that the true population proportion is between $$0.45$$ and $$0.55.$$

Question 8

Which of the following statements about increasing the confidence level from $$90\% to$$ $$99\% is $$correct?

Accepted Answer

The confidence interval becomes wider.

Question 9

A simple random sample of $$n = 400 is $$taken from a population with $$p = 0.4. $$What is the mean and standard deviation of the sampling distribution of $$\hat{p}$$?

Accepted Answer

Mean $$= 0.4$$, Standard deviation $$= \sqrt{\dfrac{0.4 	imes 0.6}{400}}$$

Question 10

Which of the following is true about bias in an estimator?

Accepted Answer

An estimator is unbiased if its expected value equals the true parameter.

Question 11

A survey uses the question: 'Don't you agree that recycling is important?' Which type of bias is most likely present?

Accepted Answer

Measurement bias

Question 12

Suppose you want to estimate the proportion of adults who exercise regularly. You take a random sample of $$n = 900$$ and find $$p = 0.3. $$What is the $$95\%$$ confidence interval for the population proportion? (Use $$z^{*} = 1.96$)

Accepted Answer

$$0.3 \pm 1.96 	imes \sqrt{\dfrac{0.3 	imes 0.7}{900}}$$

Question 13

Which of the following best describes the effect of increasing the sample size on the margin of error for a fixed confidence level?

Accepted Answer

The margin of error decreases.

Question 14

A news organization wants to plan a poll with a margin of error of $$0.04 at a$$ $$95\%$$ confidence level. What is the minimum sample size required if no prior estimate of $$p is $$available? (Use $$z^{*} = 1.96$)

Accepted Answer

$$n = \left(\dfrac{1.96}{0.04}\$$right)^{2}$$ 	imes 0.25$$

Question 15

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

Accepted Answer

The population must be normally distributed.

Question 16

A survey of 800 randomly selected adults found that 320 support a new law. Construct a $$95\%$$ confidence interval for the proportion of all adults who support the law. (Use $$z^{*} = 1.96$)

Accepted Answer

$$0.4 \pm 1.96 	imes \sqrt{\dfrac{0.4 	imes 0.6}{800}}$$

Question 17

Which of the following best describes the sampling distribution of $$\hat{p} as $$the sample size increases?

Accepted Answer

It becomes more closely approximated by a normal distribution.

Question 18

A poll reports a $$95\%$$ confidence interval for the proportion of voters supporting a candidate as $$(0.48, 0.54). $$What does this mean?

Accepted Answer

We are $$95\%$$ confident that the true proportion of all voters who support the candidate is between $$0.48$$ and $$0.54.$$

Chapter 7

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