Q: A researcher wants to estimate the proportion of defective products in a factory. She takes a random sample of 400 products and finds 32 are defective. What is the 95% confidence interval for the proportion of defective products? (Use $$ z = 1.96 $$ for 95% confidence.)

$$ 0.08 \pm 1.96 \sqrt{ \dfrac{ 0.08 \times 0.92 }{ 400 } } $$

Q: A sample of 500 people is surveyed, and 150 say they prefer a new product. What is the estimated standard error of the sample proportion?

$$ \sqrt{ \dfrac{ 0.3 \times 0.7 }{ 500 } } $$

Question 1

A random sample of 200 voters is taken from a city, and 120 say they support a new public transportation initiative. Construct a 95% confidence interval for the proportion of all voters in the city who support the initiative. Which of the following is the correct formula for the confidence interval?

Accepted Answer

$$ \hat{p} \pm 1.96 \sqrt{ \dfrac{ \hat{p}(1-\hat{p}) }{ n } } $$

Question 2

Which of the following conditions must be met for the sampling distribution of the sample proportion $$ \hat{p}  to be $$approximately normal?

Accepted Answer

$$ np \u003e 10 $$ and $$ n(1-p) \u003e 10 $$

Question 3

A store owner wants to estimate the proportion of customers who shoplift. In a random sample of 120 customers, 6 are detected as shoplifters. What is the point estimate $$ \hat{p} $$ for the proportion of shoplifters?

Accepted Answer

$$ \dfrac{6}{120} = 0.05 $$

Question 4

If you increase the confidence level from 90% to 99% for a confidence interval for a population proportion, what happens to the width of the interval (assuming the same sample and data)?

Accepted Answer

The interval becomes wider.

Question 5

A sample of 600 crime files shows that 380 involved the use of a firearm. What is the sample proportion $$ \hat{p} $$?

Accepted Answer

$$ \dfrac{380}{600} = 0.633 $$

Question 6

What is the formula for the margin of error (ME) in a confidence interval for a population proportion?

Accepted Answer

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

Question 7

Suppose a 90% confidence interval for the proportion of heads when spinning a quarter 250 times is (0.51, 0.61). What does this interval mean?

Accepted Answer

We are 90% confident that the true proportion of heads for this quarter is between 0.51 and 0.61.

Question 8

Which of the following best describes the standard error of the sample proportion $$ \hat{p} $$?

Accepted Answer

$$ \sqrt{ \dfrac{ \hat{p}(1-\hat{p}) }{ n } } $$

Question 9

A researcher wants to estimate the proportion of defective products in a factory. She takes a random sample of 400 products and finds 32 are defective. What is the 95% confidence interval for the proportion of defective products? (Use $$ z = 1.96 $$ for 95% confidence.)

Accepted Answer

$$ 0.08 \pm 1.96 \sqrt{ \dfrac{ 0.08 	imes 0.92 }{ 400 } } $$

Question 10

If a sample size increases while keeping the confidence level the same, what happens to the width of the confidence interval for a population proportion?

Accepted Answer

The interval becomes narrower.

Question 11

Which of the following statements about confidence intervals for proportions is correct?

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.

Question 12

A 99% confidence interval for a population proportion is calculated as (0.60, 0.70). Which of the following is true if the confidence level is reduced to 90% using the same data?

Accepted Answer

The new interval will be narrower than (0.60, 0.70).

Question 13

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

Accepted Answer

The population must be normally distributed.

Question 14

A sample of 500 people is surveyed, and 150 say they prefer a new product. What is the estimated standard error of the sample proportion?

Accepted Answer

$$ \sqrt{ \dfrac{ 0.3 	imes 0.7 }{ 500 } } $$

Question 15

Which of the following best explains why we use $$ z -$$values in constructing confidence intervals for proportions?

Accepted Answer

Because the sampling distribution of $$ \hat{p}  is $$approximately normal for large samples.

Question 16

A researcher calculates a 95% confidence interval for a population proportion and finds it to be (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 17

If a sample proportion $$ \hat{p} = 0.4  is $$obtained from a sample of $$ n = 100 $$, what is the margin of error for a 90% confidence interval? (Use $$ z = 1.645 $$ for 90% confidence.)

Accepted Answer

$$ 1.645 \sqrt{ \dfrac{ 0.4 	imes 0.6 }{ 100 } } $$

Question 18

A 95% confidence interval for a population proportion is calculated as (0.52, 0.68). If the true population proportion is 0.65, what can we conclude?

Accepted Answer

The interval contains the true population proportion.

Question 19

Which of the following would result in a narrower confidence interval for a population proportion, all else being equal?

Accepted Answer

Increasing the sample size.

Confidence Intervals for Proportions: Concepts, Formulas, and Applications

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