Question

Given two dependent random samples with the following results:

Population 1	70	60	72	55	69	50	55	74
Population 2	72	56	81	50	79	60	50	78

Can it be concluded, from this data, that there is a significant difference between the two population means?

Let d=(Population 1 entry)−(Population 2 entry). Use a significance level of α=0.1for the test. Assume that both populations are normally distributed.

State the null and alternative hypotheses

Find the value of the standard deviation of the paired differences

Find the value of the test statistic

Determine the decision rule for rejecting the null hypothesis (ie. reject if t / |t| is greater than or less than .....)

Make the decision

Answer 1

Solution:

From given data we calculate following:

Difference = Sample 1 - Sample 2 Sample 2 Sample 1 -2 72 70 4 56 60 -9 81 72 5 50 55 -10 79 69 -10 60 50 50 55 -4 78 74 2.625 65.75 63.125 Average 6.675 13.21 9.203 St. Dev st O2 LO LC

From the sample data, it is found that the corresponding sample means are:

Sample mean (x̄1) = 63.125 , x̄2 = 65.75

Also, the provided sample standard deviations are:

s1​=9.203 and s2​=13.21

So, for differences

$\dpi{100} \bar{D}$ = −2.625 ,

standard deviation of the paired differences= S_D ​= 6.675

The following null and alternative hypotheses need to be tested:

Но: др %3D 0 На: др # 0

Test statistic:

-2.625 =-1.112 t = 6.675//8 sD/Vn

P-Value: Based on the information provided, the significance level is α=0.1, and the degrees of freedom are df = 7.

Using excel , =TDIST(1.112,7,2)

P-value = 0.3028

Using the P-value approach: The p-value is p = 0.3027, and since p = 0.3027 ≥0.1, it is concluded that the null hypothesis is not rejected.

Hence, it is concluded, from this data, that there is a no significant difference between the two population means

Done

tc = critical t =