Question

The data for a random sample of 10 paired observations are shown in the following table.

Pair	Population1	Population 2
1	19	24
2	25	27
3	31	36
4	52	53
5	49	55
6	34	34
7	59	66
8	47	51
9	17	20
10	51	55

1. If you wish to test whether these data are sufficient to indicate that the mean for population 2 is larger than that for population 1, what are the appropriate null and alternative hypotheses? Define any symbols you use.

2. Conduct the test from part a, using α=.05. What is your decision?

3. Find a 95% confidence interval for μd. Interpret this interval.

4. What assumptions are necessary to ensure the validity of the preceding analysis?

Answer 1

let us consider = population 1 mean

= population 2 mean

to test whether these data are sufficient to indicate that the mean for population 2 is larger than that for population 1

let =

H0 :

Ha:

using minitab > paired sample t test > we have

Paired T-Test and CI: Population1, Population 2

Paired T for Population1 - Population 2

N Mean StDev SE Mean
Population1 10 38.40 15.06 4.76
Population 2 10 42.10 15.81 5.00
Difference 10 -3.700 2.214 0.700

95% upper bound for mean difference: -2.417
T-Test of mean difference = 0 (vs < 0): T-Value = -5.29 P-Value = 0.000

since p value is less than 0.05 so we reject h0 and conclude that the mean for population 2 is larger than that for population

95% CI for mean difference: (-5.284, -2.116)

since the confidence interval of difference of mean of two populations contains only negative values so we can be 95 % confident   that population mean difference lies in between this interval .

the assumption required are the sample taken should be random and on same observations. and the sample should be taken from normally distributed population.