Question

You wish to test the following claim (H1H1) at a significance level of α=0.002α=0.002.

      Ho:μ=75.1Ho:μ=75.1
      H1:μ>75.1H1:μ>75.1

You believe the population is normally distributed, but you do not know the standard deviation. You obtain the following sample of data:

data
77
87.3
81.7
96.8
113.2
82.2
81.2
84.6

4a. What is the critical value for this test? (Report answer accurate to three decimal places.)
critical value =

4b. What is the test statistic for this sample? (Report answer accurate to three decimal places.)
test statistic =

4c. The test statistic is...

A.) in the critical region

or

B.) not in the critical region

4d. This test statistic leads to a decision to...

A.) reject the null

B.) accept the null

C.) fail to reject the null

4e. As such, the final conclusion is that...

A.) There is sufficient evidence to warrant rejection of the claim that the population mean is greater than 75.1.

B.) There is not sufficient evidence to warrant rejection of the claim that the population mean is greater than 75.1.

C.) The sample data support the claim that the population mean is greater than 75.1.

D.) There is not sufficient sample evidence to support the claim that the population mean is greater than 75.1.

Answer 1

Given that the hypotheses are:

Ho:μ=75.1

H1:μ>75.1

and the sample as 77, 81.2, 81.7, 82.2, 84.6, 87.3, 96.8, 113.2, also the distribution is normal but since the population standard deviation is unknown hence T-distribution is applicable for hypothesis testing. Based on the hypothesis it will be a right-tailed test.

For the given sample the mean is calculated as:

Mean = (77 + 81.2 + 81.7 + 82.2 + 84.6 + 87.3 + 96.8 + 113.2)/8
= 704/8
Mean = 88

and the Sample standard deviation as:

Standard Deviation s = √(1/8 - 1) x ((77 - 88)2 + ( 81.2 - 88)2 + ( 81.7 - 88)2 + ( 82.2 - 88)2 + ( 84.6 - 88)2 + ( 87.3 - 88)2 + ( 96.8 - 88)2 + ( 113.2 - 88)2)
= √(1/7) x ((-11)2 + (-6.8)2 + (-6.3)2 + (-5.8)2 + (-3.4)2 + (-0.7)2 + (8.8)2 + (25.2)2)
= √(0.1429) x ((121) + (46.24) + (39.69) + (33.64) + (11.56) + (0.49) + (77.44) + (635.04))
= √(0.1429) x (965.1)
= √(137.91279)
= 11.7419

4 a) Rejection region:

Based on the fegree of freedom, df n-1= 8-1=7 and given significance level and the type of hypothesis the critical value for t -test is computed using excel formula for normal distribution which is =T.INV(1-0.02,7), thus tc computed as 2.517.

So, reject Ho if t >tc.

4b) Test statistic is calculated as:

4c) So, as we can see that t >tc hence it falls in the critical region.

A.) in the critical region

4d) Decision:

Since t>tc and the test statistic is in the critical region hence we reject t the null hypothesis.

A.) reject the null

4e) Conclusion:

Since t> tc hence the conclusion made is:

C.) The sample data support the claim that the population mean is greater than 75.1.