Question

A random sample of n measurements was selected from a population with standard deviation σ=13.6and unknown mean μ. Calculate a 90 % confidence interval for μ for each of the following situations:

(a) n=35, x=78.5

(b) n=50, x¯=78.5

(c)n=70, x¯=78.5

Answer 1

Solution :

Given that,

= 78.5

= 13.6

n = 35

At 90% confidence level the z is ,

= 1 - 90% = 1 - 0.90 = 0.10

/ 2 = 0.10 / 2 = 0.05

Z_/2 = Z_0.05 = 1.645

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

= 1.645 * ( 13.6/ 35)

= 3.7816

At 90% confidence interval estimate of the population mean is,

- E < < + E

78.5 - 3.7816 < < 78.5 + 3.7816

74.7184< < 82.2816

(74.7184 ,  82.2816 )

Solution :

Given that,

= 78.5

= 13.6

n = 50

At 90% confidence level the z is ,

= 1 - 90% = 1 - 0.90 = 0.10

/ 2 = 0.10 / 2 = 0.05

Z_/2 = Z_0.05 = 1.645

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

= 1.645 * ( 13.6/ 50)

= 3.1639

At 90% confidence interval estimate of the population mean is,

- E < < + E

78.5 - 3.1639 < < 78.5 + 3.1639

75.3361< < 81.6639

(75.3361 ,  81.6639 )

Solution :

Given that,

= 78.5

= 13.6

n = 70

At 90% confidence level the z is ,

= 1 - 90% = 1 - 0.90 = 0.10

/ 2 = 0.10 / 2 = 0.05

Z_/2 = Z_0.05 = 1.645

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

= 1.645 * ( 13.6/ 70)

= 2.6740

At 90% confidence interval estimate of the population mean is,

- E < < + E

78.5 - 2.6740 < < 78.5 + 2.6740

75.8260< < 81.1740

(75.8260 , 81.1740 )