Question

A simple random sample with n=56 provided a sample mean of 22.5 and a sample standard deviation of 4.4

a.Develop a 90% confidence interval for the populationmean.

b.Develop a 95% confidence interval for the populationmean.

c.Develop a 99% confidence interval for the populationmean.

d.What happens to the margin of error and the confidenceinterval as the confidence level is increased?

Answer 1

SOLUTION:

From given data,

A simple random sample with n=56 provided a sample mean of 22.5 and a sample standard deviation of 4.4

sample of size n = 56

sample mean = =  22.5

standard deviation = σ = 4.4

Formula for for the population mean µ

   Z_/2 ( / )

(a).Develop a 90% confidence interval for the population mean.

Confidence interval is 90%

90% = 90/100 = 0.90

= 1 - Confidence interval = 1-0.90 = 0.10

/2 = 0.10 / 2

= 0.05

Z_/2 = Z_0.05 = 1.64

for the population mean µ

margin of error = E = Z_/2 ( / )

= Z_0.05 ( / )

= 1.64 (4.4 / )

= 1.64* 0.58797473

= 0.964278

The confidence interval

   Z_/2 ( / )

   E

22.5 0.964278

(22.5 -0.964278, 22.5 +0.964278)

(21.535722, 23.464278)

(21.536, 23.464)

(b).Develop a 95% confidence interval for the population mean.

Confidence interval is 95%

95% = 95/100 = 0.95

= 1 - Confidence interval = 1-0.95 = 0.05

/2 = 0.05 / 2

= 0.025

Z_/2 = Z_0.025 = 1.96

for the population mean µ

margin of error = E = Z_/2 ( / )

= Z_0.025 ( / )

= 1.96 (4.4 / )

= 1.96* 0.58797473

= 1.1524305

The confidence interval

   Z_/2 ( / )

   E

22.5 1.1524305

(22.5 -1.1524305, 22.5 +1.1524305)

(21.3475695, 23.6524305)

(21.348, 23.652)

(c).Develop a 99% confidence interval for the population mean.

Confidence interval is 99%

99% = 99/100 = 0.99

= 1 - Confidence interval = 1-0.99 = 0.01

/2 = 0.01 / 2

= 0.005

Z_/2 = Z_0.005  = 2.58

margin of error = E = Z_/2 ( / )

= Z_0.005 ( / )

= 2.58 (4.4 / )

= 2.58* 0.58797473

= 1.516975

for the population mean µ

   Z_/2 ( / )

   E

22.5 1.516975

(22.5 -1.516975, 22.5 +1.516975)

(20.983025 , 24.016975)

(20.983, 24.017)

d.What happens to the margin of error and the confidence interval as the confidence level is increased?

If the confidence interval as the confidence level is increased Then the margin of error also increased.