Question

A random sample of 36 individuals were selected from the batch of products to estimate a feature of interest. Suppose that the sample mean is 125 and the standard deviation for this batch is assumed to be 24.

a) construct a 95% confidence interval for the mean value for this feature of the batch.

b) if we want to make the sampling error +/- 7, how many more individuals should be selected to achieve a confidence level 95%?

c) based on the current sample, how confident can you claim that the true value for the mean is between 118 and 132?

Answer 1

Solution :

Given that,

(a)Point estimate = sample mean = = 125

Population standard deviation =    = 24

Sample size = n =36

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 = 1.96

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

= 1.96* ( 24 /  36 )

= 7.84

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

- E < < + E

125- 7.84 <   < 125 + 7.84

117.16 <   < 132.84

( 117.16 ,  132.84 ) (rounded)

(118 and 132)

(b)Margin of error = E = +/ - = 7

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 = 1.96

sample size = n = [Z_/2* / E] ²

n = ( 1.96* 24 /7 )²

n =45

Sample size = n =45

(c)  95% confidence interval for the mean value is fall between lower bound 118 and upper bound = 132 than is claim that it is true value of mean