Question

Construct a 95​% confidence interval to estimate the population mean with x overbar equals 104 and sigma equals 28 for the following sample sizes. ​

a) n equals 30 ​b) n equals 48 ​c) n equals 66

Answer 1

Solution :

Given that,

Point estimate = sample mean = = 104

Population standard deviation = = 28

a)

Sample size = n = 30

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 * (28 / 30)

= 10.02

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

- E < < + E

104 - 10.02 < < 104 + 10.02

93.98 < < 114.02

(93.98 , 114.02)

b)

Sample size = n = 48

  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 * (28 /48)

= 7.92

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

- E < < + E

104 - 7.92 < < 104 + 7.92

96.08 < < 111.92

(96.08 , 111.92)

c)

Sample size = n = 66

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 * (28 / 66)

= 6.76

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

- E < < + E

104 - 6.76 < < 104 + 6.76

97.24 < < 110.76

(97.24 , 110.76 )