# 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.

1. 71.21047  to  72.78953
2. 70.20561  to  73.79439
3. 69.53272  to  74.46728 X
4. 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.

1. -133.3364  to  -128.6636
2. -134.9199  to  -127.0801 X
3. -132.4803  to  -129.5197
4. -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.

1. -2.575829  to  2.575829 X
2. -4.722354  to  4.722354
3. -1.888941  to  1.888941
4. -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.

1. 44
2. 32
3. 38 X
4. 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.

1. 46
2. 48
3. 38
4. 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?

1. 0.4937043  to  0.6062957
2. 0.5063936  to  0.5936064 X
3. 0.5174977  to  0.5825023
4. 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?

1. 0.5885704  to  0.7114296 X
2. 0.6065627  to  0.6934373
3. 0.6145336  to  0.6854664
4. 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?

1. 0.3479852  to  0.4520148
2. 0.2848842  to  0.5151158
3. 0.3721969  to  0.4278031 X
4. 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.

1. 998
2. 924
3. 1414
4. 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.

1. 1620
2. 1844 X
3. 1557
4. 1994

Solution:

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

[1] 1843.027