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Chapter 6: Q65E (page 363)

Suppose you wish to estimate the mean of a normal population

using a 95% confidence interval, and you know from prior information thatσ2≈1

a. To see the effect of the sample size on the width of the confidence interval, calculate the width of the confidence interval for n= 16, 25, 49, 100, and 400.

b. Plot the width as a function of sample size non graph paper. Connect the points by a smooth curve and note how the width decreases as nincreases.

Short Answer

Expert verified

a. The width of the confidence interval for n=16 is 0.98, The width of the confidence interval for n=25 is 0.784, The width of the confidence interval for n=49 is 0.56, The width of the confidence interval for n=100 is 0.392 The width of the confidence interval for n=400 is 0.196

b.From the drawn graph, It is observed that the width of the confidence interval decrease as the sample size increase.

Step by step solution

01

Given information

Know from prior information thatσ2≈1

02

Finding the width of the confidence interval

a.

The width of the confidence interval is

Width=2×zα/2×σn

The critical value for a 95% confidence interval iszα/2=z0.025=1.96

For sample size n=16

Width=2×zα/2×σn=2×1.96×116=0.98

For sample size n=25

Width=2×zα/2×σn=2×1.96×125=0.784

For sample size n=49

Width=2×zα/2×σn=2×1.96×149=0.56

For sample size n=100

Width=2×zα/2×σn=2×1.96×1100=0.392

For sample size n=400

Width=2×zα/2×σn=2×1.96×1400=0.196

03

Plotting the width of the confidence interval

b.

Here it is observed that the width of confidence interval decrease as the sample size increase.

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