Question

(a) Find the margin of error for the given values of​ c, σ​, and n. c = 0.90​, σ = 3.8​, n = 100

E= _ (Round to three decimal places as​ needed.)

(b) Construct the confidence interval for the population mean μ.

c = 0.90 ​, x=9.1​, σ = 0.3 ​, and n = 47

A 90​% confidence interval for μ is _, _ (Round to two decimal places as​ needed.)

(c)  Construct the confidence interval for the population mean μ.

c=0.95 , x=16.2, σ =2.0, and n =35

A 95​% confidence interval for μ is _, _ (Round to two decimal places as​ needed.)

Answer 1

Solution :

Given that,

a) Population standard deviation =    = 3.8

Sample size = n = 100

At 90% confidence level

= 1 - 90%  

= 1 - 0.90 =0.10

/2 = 0.05

Z/2 = Z_{0.05 = 1.645}

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

E = 1.645 * ( 3.8 /  100 )

E = 0.625

b) Point estimate = sample mean = = 9.1

Population standard deviation =    = 0.3

Sample size = n =47

At 90% confidence level

= 1 - 90%  

= 1 - 0.90 =0.10

/2 = 0.05

Z/2 = Z_{0.05 = 1.645}

Margin of error = E = Z/2 * ( /n)
= 1.645 * ( 0.3 /  47 )
= 0.07
At 90% confidence interval estimate of the population mean is,

  ± E

= 9.1 ± 0.07   

( 9.03, 9.17)  

c) Point estimate = sample mean = = 16.2

Population standard deviation =    = 2.0

Sample size = n =35

At 95% confidence level

= 1 - 95%  

= 1 - 0.95 =0.05

/2 = 0.025

Z/2 = Z_{0.025 = 1.96}

Margin of error = E = Z/2 * ( /n)
= 1.96 * ( 2.0 /  35 )
= 0.66
At 90% confidence interval estimate of the population mean is,

  ± E

= 16.2 ± 0.66

( 15.54, 16.86)