Question

The mean and standard deviation of a random sample of n measurements are equal to and ​, respectively. a. Find a ​% confidence interval for if n. b. Find a ​% confidence interval for if n. c. Find the widths of the confidence intervals found in parts a and b. What is the effect on the width of a confidence interval of quadrupling the sample size while holding the confidence coefficient​ fixed?

Answer 1

Complete Question

The complete question is shown on the first uploaded image

Answer:

a

`33.55 <  \mu < 35.5`

b

`34.03 <  \mu < 34.969`

c

Generally the width at n = 49 is mathematically represented as

`w =  2 *  E`

`w =  2 *  0.952`

`w =  1.904`

Generally the width at n = 196 is mathematically represented as

`w =  2 *  E`

`w =  2 *  0.4687`

`w =  0.9374`

d

The correct option is E

Explanation:

From the question we are told that

The sample mean is
`\= x  =  34.5`

The standard deviation is
`s =  3.4`

Generally given that the confidence level is 95% then the level of significance is

`\alpha = (100 -  95)\%`

=>
`\alpha = 0.05`

Generally from the normal distribution table the critical value of
`(\alpha )/(2)` is

`Z_{(\alpha )/(2) } =  1.96`

Considering question a

From the question n = 49

Generally the margin of error is mathematically represented as

`E = Z_{(\alpha )/(2) } *  (s )/(√(n) )`

=>
`E = 1.96*  ( 3.4 )/(√(49) )`

=>
`E =  0.952`

Generally 95% confidence interval is mathematically represented as

`\= x -E <  p <  \=x  +E`

`34.5 -0.952 <  p < 34.5 + 0.952`

=>
`33.55 <  \mu < 35.5`

Considering question b

From the question n = 196

Generally the margin of error is mathematically represented as

`E = Z_{(\alpha )/(2) } *  (s )/(√(n) )`

=>
`E = 1.96*  ( 3.4 )/(√(196) )`

=>
`E =  0.4687`

Generally 95% confidence interval is mathematically represented as

`\= x -E <  p <  \=x  +E`

`34.5 -0.4687 <  p < 34.5 +0.4687`

=>
`34.03 <  \mu < 34.969`

Considering question c

Generally the width at n = 49 is mathematically represented as

`w =  2 *  E`

`w =  2 *  0.952`

`w =  1.904`

Generally the width at n = 196 is mathematically represented as

`w =  2 *  E`

`w =  2 *  0.4687`

`w =  0.9374`

Now when the sample size is quadrupled i.e from n = 49 to n = 196

The width of the confidence interval decrease by 2 from 1.904 to 0.9374