Question

Use the sample information x¯ ⎯ x¯ = 40, σ = 7, n = 13 to calculate the following confidence intervals for μ assuming the sample is from a normal population.

(a) 90 percent confidence. (Round your answers to 4 decimal places.)
  
The 90% confidence interval is from __to__

(b) 95 percent confidence. (Round your answers to 4 decimal places.)
  
The 95% confidence interval is from __to__

(c) 99 percent confidence. (Round your answers to 4 decimal places.)
  
The 99% confidence interval is from __to__

(d) Describe how the intervals change as you increase the confidence level.
  

A- The interval gets narrower as the confidence level increases.

B- The interval gets wider as the confidence level decreases.

C- The interval gets wider as the confidence level increases.

D- The interval stays the same as the confidence level increases.

Answer 1

Solution :

Given that,

= 40

= 7

n = 13

a ) At 90% confidence level the z is ,

= 1 - 90% = 1 - 0.90 = 0.10

/ 2 = 0.10 / 2 = 0.05

Z_/2 = Z_0.05 = 1.645

Margin of error = E = Z_/2* (/n)

= 1.645 * ( 7 / 13)

= 3.19

At 90% confidence interval estimate of the population mean is,

- E < < + E

40 - 3.19 < < 40 + 3.19

36.81< < 43.19

b) At 95% confidence level the z is ,

  = 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

Z_/2 = Z_0.025 = 1.960

Margin of error = E = Z_/2* (/n)

= 1.960 * ( 7 / 13)

= 3.80

At 95% confidence interval estimate of the population mean is,

- E < < + E

40 - 3.80 < < 40 + 3.80

36.20< < 43.80

c ) At 99% confidence level the z is ,

= 1 - 99% = 1 - 0.99 = 0.01

/ 2 = 0.01 / 2 = 0.005

Z_/2 = Z_0.005 = 2.576

Margin of error = E = Z_/2* (/n)

= 2.576 * ( 7 / 13)

= 5.00

At 99% confidence interval estimate of the population mean is,

- E < < + E

40 - 5.00< < 40 + 5.00

35.00< < 45.00

d ) Option c ) is correct.

The interval gets wider as the confidence level increases.