Question

a. Estimate, with 95 percent confidence, the mean height for all students enrolled in BMGT 230, lastsemester.b. How would the width of the interval in Part a change if the standard deviation equaled 3.33 inches, holding all other values/ideas constant?c. How would the width of the interval in Part a change if the mean equaled 71 inches, holding all other values/ideas constant?

Answer 1

Complete Question

The complete question is shown on the first uploaded image

Answer:

a

The 95% confidence interval is
`228.5<  \mu <  231.5`

b

The width will reduce by one

c

The width will remain the same

Explanation:

Considering question a

From the question we are told that

The sample size is n = 36

The standard deviation is
`\sigma =4.44 \ inches`

The mean is
`\mu = 230 \ inches`

From the question we are told the confidence level is 95% , hence 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`

Generally the margin of error is mathematically represented as

`E_1 = Z_{(\alpha )/(2) } *  (\sigma )/(√(n) )`

=>
`E_1 = 1.96*  (4.44)/(√(36) )`

=>
`E_1 = 1.4504`

Generally the width of the confidence interval is

`W_1 = 2 * E_1`

=>
`W_1 = 2 * 1.4504`

=> =>
`W_1 = 3`

Generally 95% confidence interval is mathematically represented as

`\= x -E <  \mu <  \=x  +E`

=>
`230 -1.4504 <  \mu < 230 + 1.4504`

=>
`228.5<  \mu <  231.5`

Considering question b

when
`\sigma_1 = 3.33 \ inches`

Generally the margin of error is mathematically represented as

`E_2 = Z_{(\alpha )/(2) } *  (\sigma_1 )/(√(n) )`

=>
`E_2 = 1.96*  (3.33)/(√(36) )`

=>
`E_2 = 1.0878`

Generally the width of the confidence interval is

`W_2 = 2 * E_2`

=>
`W_2 = 2 * 1.0878`

=>
`W_2 = 2`

So comparing
`W_1 \ and \ W_2` we see that the width will decrease by 1

Considering question b

When
`\= x = 71`

Generally 95% confidence interval is mathematically represented as

`\= x -E <  \mu <  \=x  +E`

=>
`69.55<  \mu <  72.45`

Generally the width is mathematically represented as

`W_3 = 72.45 - 69.55`

=>
`W_3 = 3`

Comparing
`W_2 \ and \ W_1` we see that the width of the confidence interval remain the same