Question

The two data sets in the table below are dependent random samples. The population of (x−y)(x-y) differences is approximately normally distributed. A claim is made that the mean difference (x−y)(x-y) is not equal to -23.5.

x	60	62	46	45	37	45	46	46
y	79	77	75	86	83	71	76	77

For each part below, enter only a numeric value in the answer box. For example, do not type "z =" or "t =" before your answers. Round each of your answers to 3 places after the decimal point.

(a) Calculate the value of the test statistic used in this test.

     Test statistic's value =

(b) Use your calculator to find the P-value of this test.

     P-value =

(c) Use your calculator to find the critical value(s) used to test this claim at the 0.04 significance level. If there are two critical values, then list them both with a comma between them.

     Critical value(s) =

(d) What is the correct conclusion of this hypothesis test at the 0.04 significance level?     

• There is not sufficient evidence to support the claim that the mean difference is not equal to -23.5
• There is sufficient evidence to warrant rejection the claim that the mean difference is not equal to -23.5
• There is not sufficient evidence to warrant rejection the claim that the mean difference is not equal to -23.5
• There is sufficient evidence to support the claim that the mean difference is not equal to -23.5

Answer 1

(a) Calculate the value of the test statistic used in this test.

Here, we have to use paired t test.

H₀: µ_d = -23.5 versus H_a: µ_d ≠ -23.5

Test statistic for paired t test is given as below:

t = (Dbar - µ_d)/[S_d/sqrt(n)]

From given data, we have

Dbar = -29.6250

S_d = 10.2809

n = 8

df = n – 1 = 7

α = 0.04

Critical t values = - 2.5168 and 2.5168

t = (Dbar - µ_d)/[S_d/sqrt(n)]

t = (-29.6250 – (-23.5))/[ 10.2809/sqrt(8)]

t = (-29.6250 + 23.5)/ 3.6348

t = -1.6851

Test statistic's value = -1.685

(b) Use your calculator to find the P-value of this test.

     

P-value = 0.136

(by using t-table)

(c) Use your calculator to find the critical value(s) used to test this claim at the 0.04 significance level. If there are two critical values, then list them both with a comma between them.

    

Critical value(s) = - 2.517, 2.517

(by using t-table)

(d) What is the correct conclusion of this hypothesis test at the 0.04 significance level?    

Here, we get

P-value = 0.136 > α = 0.04

So, we do not reject the null hypothesis

There is not sufficient evidence to support the claim that the mean difference is not equal to -23.5