Question

2.) Find the 90% confidence intervals for population mean for the following a.) sample mean is 53 and  = 7.1 for n = 90 b.) sample mean is 285 and  = 7.1 for n = 28 c.) sample mean is 149.7 and s = 23.8 for n = 20

Answer 1

(A)

Here, we have given that,

n= Number of observation =90

= sample mean =53

S= sample standard deviation=7.1

Now, we want to find the 90% confidence interval for population mean

Formula is as follows,

Where

E=Margin of error =

Now,

Degrees of freedom = n-1 = 90-1=89

c=confidence level =0.90

=level of significance=1-c=1-0.90=0.10

and we know that confidence interval is always two tailed

t-critical = 1.662 Using Excel=TINV(prob=0.10, D.F=89)

Now,

=1.244

We get the 90% confidence interval for the population mean

Lower limit =51.76

Upper limit =54.24

Interpretation:

This is the 90% CI which shows that we have 90% confidence that this population mean will fall within this interval.

(B)

Here, we have given that,

n= Number of observation =28

= sample mean =285

S= sample standard deviation=7.1

Now, we want to find the 90% confidence interval for population mean

Formula is as follows,

Where

E=Margin of error =

Now,

Degrees of freedom = n-1 = 28-1=27

c=confidence level =0.90

=level of significance=1-c=1-0.90=0.10

and we know that confidence interval is always two tailed

t-critical = 1.703 Using Excel=TINV(prob=0.10, D.F=27)

Now,

  

=2.285

We get the 90% confidence interval for the population mean

Lower limit =282.71

Upper limit =287.29

Interpretation:

This is the 90% CI which shows that we have 90% confidence that this population mean will fall within this interval.

(C)

Here, we have given that,

n= Number of observation =20

= sample mean =149.7

S= sample standard deviation=23.8

Now, we want to find the 90% confidence interval for population mean

Formula is as follows,

Where

E=Margin of error =

Now,

Degrees of freedom = n-1 = 20-1=19

c=confidence level =0.90

=level of significance=1-c=1-0.90=0.10

and we know that confidence interval is always two tailed

t-critical = 1.729 Using Excel=TINV(prob=0.10, D.F=19) also we find it using t table

Now,

  

=9.202

We get the 90% confidence interval for the population mean

Lower limit =140.50

Upper limit =158.90

Interpretation:

This is the 90% CI which shows that we have 90% confidence that this population mean will fall within this interval.