# Statistics with R

## Student Resources

# Chapter 8: Confidence Interval Estimation

1. Find the 90% confidence interval estimate of μ if the sample mean x̅ = 72, the sample size is n = 64, and σ = 12.

- 71.21047 to 72.78953
- 70.20561 to 73.79439
- 69.53272 to 74.46728 X
- 68.71029 to 75.28971

**Solution:**

> MOE <- qnorm(0.05, lower.tail = FALSE) * (12 / sqrt(64))

> 72 + c(-MOE, MOE)

[1] 69.53272 to 74.46728

2. Find the 95% confidence interval estimate of m if the sample mean x̅ = -131, the sample size is n = 81, and σ = 18.

- -133.3364 to -128.6636
- -134.9199 to -127.0801 X
- -132.4803 to -129.5197
- -138.0559 to -123.9441

**Solution:**

> MOE <- qnorm(0.025, lower.tail = FALSE) * (18 / sqrt(81))

> -131 + c(-MOE, MOE)

[1] -134.9199 to -127.0801

3. Find the 99% confidence interval estimate of m if the sample mean x̅ = 0, the sample size is n = 121, and σ = 11.

- -2.575829 to 2.575829 X
- -4.722354 to 4.722354
- -1.888941 to 1.888941
- -1.133365 to 1.133365

**Solution:**

> MOE <- qnorm(0.005, lower.tail = FALSE) * (11 / sqrt(121))

> 0 + c(-MOE, MOE)

[1] -2.575829 to 2.575829

4. What sample size is required if we want a 95% confidence interval estimate with a margin of error of 8? Assume σ = 25.

- 44
- 32
- 38 X
- 46

**Solution:**

> (qnorm(0.025, lower.tail = FALSE) * 25 / 8) ^ 2

[1] 37.51425

5. What sample size is required if we want a 90% confidence interval estimate with a margin of error of 2? Assume σ = 8.

- 46
- 48
- 38
- 44 X

**Solution:**

> (qnorm(0.05, lower.tail = FALSE) * 8 / 2) ^ 2

[1] 43.2887

6. Suppose we draw a simple random sample of 500 from a large population and find that the sample proportion is = 0.55. What is the 95% confidence interval estimate of the population proportion p?

- 0.4937043 to 0.6062957
- 0.5063936 to 0.5936064 X
- 0.5174977 to 0.5825023
- 0.4524930 to 0.6475070

**Solution:**

> MOE <- qnorm(0.025, lower.tail = FALSE) * sqrt(0.55 * 0.45 / 500)

> 0.55 + c(-MOE, MOE)

[1] 0.5063936 to 0.5936064

7. Suppose we draw a simple random sample of 400 from a large population and find that the sample proportion is *** x̅ = 0.65. What is the 99% confidence interval estimate of the population proportion p?

- 0.5885704 to 0.7114296 X
- 0.6065627 to 0.6934373
- 0.6145336 to 0.6854664
- 0.5383098 to 0.7616902

**Solution:**

> MOE <- qnorm(0.005, lower.tail = FALSE) * sqrt(0.65 * 0.35 / 400)

> 0.65 + c(-MOE, MOE)

[1] 0.5885704 to 0.7114296

8. Suppose we draw a simple random sample of 840 from a large population and find that the sample proportion is = 0.40. What is the 90% confidence interval estimate of the population proportion p?

- 0.3479852 to 0.4520148
- 0.2848842 to 0.5151158
- 0.3721969 to 0.4278031 X
- 0.3649768 to 0.4350232

**Solution:**

> MOE <- qnorm(0.05, lower.tail = FALSE) * sqrt(0.60 * 0.40 / 840)

> 0.40 + c(-MOE, MOE)

[1] 0.3721969 to 0.4278031

9. What sample size is required if we want a 90% confidence interval estimate of p with a margin of error of 0.025? Use a planning value of 0.40.

- 998
- 924
- 1414
- 1039 X

**Solution:**

> ((qnorm(0.05, lower.tail = FALSE) / 0.025) ^ 2) * (0.40) * (0.60)

[1] 1038.929

10. What sample size is required if we want a 99% confidence interval estimate of p with a margin of error of 0.03? Assume that there are no data available for the purpose of estimating a planning value.

- 1620
- 1844 X
- 1557
- 1994

**Solution:**

> ((qnorm(0.005, lower.tail = FALSE) / 0.03) ^ 2) * (0.50) * (0.50)

[1] 1843.027