Question

Independent random samples from two normal distributions returned the values:
Sample 1: 45, 78, 121, 45, 67, 89, 95, 99, 79, 74
Sample 2: 67, 32, 67, 45, 21, 67, 89, 78, 77, 96, 94, 99
Find a 95% confidence interval for the difference in population means.

Answer 1

Answer:

The 95% confidence interval for the difference in population means is = ( -12.0663 , 31.79972 )

= ( -12.1 , 31.8 ) ... rounded to 1 decimal place

Explanation:

Here by considering that both the samples having equal variance, we use the independent sample t test to find  95% confidence interval for the difference in population means. ( We can check equality of variances of two samples by usinf F- test)

The 95% confidence interval for the difference in population means can be given by:

where   and   are sample means, n1 and n2 are sample sizes, Sp is pooled standard deviation and t is table value with (n1 +n2 -2) df

Sp is given as :

where s1^2 and s2^2 are sample variances

Now the summary statistics of two samples from excel ( we need sample means and sample variance that can also be calculared manually)

sample1		sample2
Mean	79.2	Mean	69.33333
Standard Error	7.454752	Standard Error	7.312643
Median	78.5	Median	72
Mode	45	Mode	67
Standard Deviation	23.574	Standard Deviation	25.33174
Sample Variance	555.7333	Sample Variance	641.697
Kurtosis	-0.07293	Kurtosis	-0.37501
Skewness	0.039448	Skewness	-0.74227
Range	76	Range	78
Minimum	45	Minimum	21
Maximum	121	Maximum	99
Sum	792	Sum	832
Count	10	Count	12

Here S1^2 = 555.7333 and S2^2 = 641.679

hence

Sp = 24.556329

Now the 95% confidence interval is

Here = 79.2 and   = 69.3333 and t = 2.086 ( from table with alpha=0.05 and df = 20)

= ( -12.0663 , 31.79972 ) = ( -12.1 , 31.8 ) ... rounded to 1 decimal place

MINITAB output for reference:

Two-Sample T-Test and CI: sample1, sample2

Two-sample T for sample1 vs sample2

N Mean StDev SE Mean
sample1 10 79.2 23.6 7.5
sample2 12 69.3 25.3 7.3

Difference = μ (sample1) - μ (sample2)
Estimate for difference: 9.9
95% CI for difference: (-12.1, 31.8) # Our Answer
T-Test of difference = 0 (vs ≠): T-Value = 0.94 P-Value = 0.359 DF = 20
Both use Pooled StDev = 24.5563