Question

A randomly selected sample of college basketball players has the following heights in inches. See Attached Excel for Data. Compute a 95% confidence interval for the population mean height of college basketball players based on this sample and fill in the blanks appropriately. < μ < (round to 3 decimal places)

Answer 1

Complete Question

The complete question is shown on the first uploaded image

Answer:

The confidence interval is
`64.86<\mu<67`

Explanation:

From the question we are given the following data

The following heights are

66, 65, 67, 62, 62, 65, 61, 70, 66, 66, 71, 63, 69, 65, 71, 66, 66, 69, 68, 62, 65, 67, 65, 71, 65, 70, 62, 62, 63, 64, 67, 67

The sample size is n =32

The confidence level is
`k = 95`% = 0.95

The mean is evaluated as

`\= x = 66+ 65+ 67+ 62+ 62+ 65+ 61+ 70+ 66+ 66+ 71+63+ 69+ 65+ 71+ 66+ 66+ 69+ 68+ 62+ 65+ 67,+\\65+ 71+ 65+ 70+ 62+ 62+ 63+ 64+ 67+ 67 / 32`

=>
`\= x = (2108)/(32)`

=>
`\= x = 65.875`

The standard deviation is evaluated as

`\sigma = √( v)`

Now

`v = ( 66-65.875 )^2+(65-65.875)^2+( 67-65.875)^2+ (62-65.875)^2+ (62-65.875)^2+ (65-65.875)^2+( 61-65.875)^2+ (70-65.875)^2+ (66-65.875)^2+ (66-65.875)^2+ (71+63-65.875)^2+ (69-65.875)^2+ (65-65.875)^2+ (71-65.875)^2+( 66-65.875)^2+ (66-65.875)^2+ (69-65.875)^2+ (68-65.875)^2+ (62-65.875)^2+ (65-65.875)^2+ (67-65.875)^2,+\\(65-65.875)^2+ (71-65.875)^2+ (65-65.875)^2+ (70-65.875)^2+( 62-65.875)^2+( 62-65.875)^2+ (63-65.875)^2+ (64-65.875)^2+ (67-65.875)^2+ (67-65.875)^2 / 32`

=>
`v= 8.567329`

=>
`\sigma = √(8.567329)`

=>
`\sigma = 2.927`

The level of significance is evaluated as

`\alpha = 1 - 0.95`

`\alpha = 0.05`

The degree of freedom is evaluated as

`Df = n- 1 \equiv Df = 32 -1 = 31`

The critical values for the level of significance is obtained from the z -table as

`t_c = t_(\alpha/2 ) , Df = t _(0.05/2), 31 =\pm 1.96`

The confidence interval is evaluated as

`\mu = \= x \pm t_c * (\sigma )/(√(n) )`

substituting values

`\mu =65.875 \pm 1.96* (2.927)/(√(32) )`

`\mu =65.875 \pm 1.01415`

=>
`64.86<\mu<67`