Question

Construct a 98​% confidence interval to estimate the population mean with x overbar equals 57 and sigma equals 11 for the following sample sizes.

​a) n equals 38 ​b) n equals 41 ​c) n equals 66

Answer 1

Solution :

Given that,

Point estimate = sample mean = = 57

Population standard deviation = = 11

At 98% confidence level the z is ,

= 1 - 98% = 1 - 0.98 = 0.02

/ 2 = 0.02 / 2 = 0.01

Z_/2 = Z_0.01 = 2.326

(a)

Sample size = n = 38

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

= 2.326 * (11 / 38)

= 4.15

At 98% confidence interval estimate of the population mean is,

- E < < + E

57 - 4.15 < < 57 + 4.15

52.85 < < 61.15

(52.85 , 61.15 )

(b)

Sample size = n = 41

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

= 2.326 * (11 / 41)

= 4.00

At 98% confidence interval estimate of the population mean is,

- E < < + E

57 - 4.00 < < 57 + 4.00

53.00< < 61.00

(53.00 , 61.00 )

(c)

Sample size = n = 66

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

= 2.326 * (11 / 66)

= 3.15

At 98% confidence interval estimate of the population mean is,

- E < < + E

57 - 3.15 < < 57 + 3.15

53.85 < < 60.15

(53.85 , 60.15 )