Question

A random sample of 13 items is drawn from a population whose standard deviation is unknown. The sample mean is x¯ = 950 and the sample standard deviation is s = 10. Use Appendix D to find the values of Student’s t.

1. Construct an interval estimate of mu with 99% confidence. (Round your answers to 3 decimal places.)
The 99% confidence interval is from_____ to ______ .
2. Construct an interval estimate of mu with 99% confidence, assuming that s = 20. (Round your answers to 3 decimal places.)
The 99% confidence interval is from_____ to ______ .
3. Construct an interval estimate of mu with 99% confidence, assuming that s = 40. (Round your answers to 3 decimal places.)
The 99% confidence interval is from_____ to ______ .

Answer 1

1. The 99% confidence interval is from 941.527 to 958.473

2. The 99% confidence interval is from 933.054 to 966.946

3. The 99% confidence interval is from 916.108 to 983.892

Explanation:

The confidence interval is given by

`\text {confidence interval} = \bar{x} \pm MoE\\\\`

Where
`\bar{x}` is the sample mean and Margin of error is given by

`$ MoE = t_(\alpha/2)((s)/(√(n) ) ) $ \\\\`

Where n is the sample size,

s is the sample standard deviation,

`t_{\alpha/2` is the t-score corresponding to some confidence level

The t-score corresponding to 99% confidence level is

Significance level = α = 1 - 0.99 = 0.01/2 = 0.005

Degree of freedom = n - 1 = 13 - 1 = 12

From the t-table at α = 0.005 and DoF = 12

t-score = 3.055

1. 99% Confidence Interval when s = 10

The margin of error is

`MoE = t_(\alpha/2)((s)/(√(n) ) ) \\\\MoE = 3.055\cdot (10)/(√(13) ) \\\\MoE = 3.055\cdot 2.7735\\\\MoE = 8.473\\\\`

So the required 99% confidence interval is

`\text {confidence interval} = \bar{x} \pm MoE\\\\\text {confidence interval} = 950 \pm 8.473\\\\\text {confidence interval} = 950 - 8.473, \: 950 + 8.473\\\\\text {confidence interval} = (941.527, \: 958.473)\\\\`

The 99% confidence interval is from 941.527 to 958.473

2. 99% Confidence Interval when s = 20

The margin of error is

`MoE = t_(\alpha/2)((s)/(√(n) ) ) \\\\MoE = 3.055\cdot (20)/(√(13) ) \\\\MoE = 3.055\cdot 5.547\\\\MoE = 16.946\\\\`

So the required 99% confidence interval is

`\text {confidence interval} = \bar{x} \pm MoE\\\\\text {confidence interval} = 950 \pm 16.946\\\\\text {confidence interval} = 950 - 16.946, \: 950 + 16.946\\\\\text {confidence interval} = (933.054, \: 966.946)\\\\`

The 99% confidence interval is from 933.054 to 966.946

3. 99% Confidence Interval when s = 40

The margin of error is

`MoE = t_(\alpha/2)((s)/(√(n) ) ) \\\\MoE = 3.055\cdot (40)/(√(13) ) \\\\MoE = 3.055\cdot 11.094\\\\MoE = 33.892\\\\`

So the required 99% confidence interval is

`\text {confidence interval} = \bar{x} \pm MoE\\\\\text {confidence interval} = 950 \pm 33.892\\\\\text {confidence interval} = 950 - 33.892, \: 950 + 33.892\\\\\text {confidence interval} = (916.108, \: 983.892)\\\\`

The 99% confidence interval is from 916.108 to 983.892

As the sample standard deviation increases, the range of confidence interval also increases.