Question

It has been claimed that the mean age of all FHSU Virtual College statistics students is less than thirty-one years old. Using the "Age" sample data collected from this year's classes, test this hypothesis--that is does the collected data statistically support or contradict this claim? Justify your answer through a formal hypothesis testing procedure with a 0.05 (5%) level of significance. Again, necessary claims, calculations, and values must be shown below and to the right. Give your proper/final conclusion in sentence form below as related to the context/situation. (Hint: Since the population s.d. is not known, make sure to realize the need to use the t-distribution for testing purposes.)

H₀: H₁: Sample's Mean (x-bar): Sample's S.D. (s): Critical Value: Sample's Test Statistic: P-value: CONCLUSION:

Individual ID# Gender Foot Length Height Age Armspan Number in Family Hair Color

1 Male 29.0 179.5 32 181.0 3 Brown

2 Male 29.5 185.5 27 187.5 5 Red

3 Male 25.0 175.5 26 176.0 9 Brown

4 Female 20.5 160.0 27 160.5 1 Blonde

5 Female 25.5 172.0 33 137.0 6 Blonde

6 Female 24.0 163.0 32 165.0 4 Brown

7 Female 24.0 167.5 33 172.0 4 Blonde

8 Male 25.0 175.5 23 170.0 4 Red

9 Female 24.5 165.0 28 155.5 7 Blonde

10 Male 25.5 175.0 37 176.5 4 Brown

11 Female 25.5 167.5 24 183.0 2 Blonde

12 Female 23.0 162.5 30 162.5 6 Brown

13 Female 25.0 161.5 26 155.0 8 Blonde

14 Male 21.5 176.0 55 179.0 5 Brown

15 Female 24.0 157.5 41 163.0 2 Brown

16 Female 26.5 177.5 29 175.0 2 Red

17 Female 25.0 175.0 47 186.0 4 Brown

18 Male 24.0 162.5 21 164.0 4 Brown

19 Female 24.5 157.5 42 155.0 8 Black

20 Male 27.0 182.0 38 172.0 3 Black

21 Female 28.0 169.0 41 165.0 3 Brown

22 Male 30.0 185.0 23 190.5 8 Brown

23 Female 21.5 158.0 41 147.5 3 Brown

24 Female 24.0 167.5 21 173.0 5 Blonde

25 Female 24.5 165.0 52 162.0 3 Blonde

26 Female 25.0 160.0 23 154.0 4 Brown

27 Female 24.0 159.0 33 163.0 6 Brown

28 Female 23.5 171.0 17 169.5 4 Black

29 Female 27.0 170.0 38 169.0 4 Black

30 Male 21.5 160.0 21 165.0 3 Black

31 Male 30.5 193.0 24 203.0 11 Black

32 Male 27.0 191.0 38 188.0 9 Black

33 Female 24.0 163.0 41 176.0 3 Blonde

34 Female 25.5 166.0 28 165.0 4 Brown

35 Female 22.0 171.0 31 154.5 4 Brown

36 Female 22.0 173.0 22 171.0 4 Brown

37 Female 22.0 163.0 20 167.0 4 Brown

38 Female 24.0 170.0 23 172.0 5 Brown

39 Male 29.0 193.0 28 208.5 7 Brown

40 Male 25.0 188.0 20 194.0 3 Brown

41 Female 25.0 170.0 21 165.0 3 Blonde

42 Male 25.5 193.0 31 190.5 4 Blonde

43 Male 29.0 189.0 29 193.0 9 Blonde

44 Female 25.0 157.5 57 152.0 6 Brown

45 Male 30.0 193.0 18 195.0 4 Red

46 Female 24.0 167.5 24 149.0 6 Brown

47 Female 24.0 162.0 36 161.0 6 Brown

48 Female 24.0 168.0 19 160.0 4 Blonde

49 Male 25.5 180.0 22 176.5 4 Brown

50 Male 27.0 185.5 31 190.0 2 Brown

51 Male 30.5 177.5 13 182.5 6 Red

52 Female 24.0 170.0 28 175.0 5 Blonde

53 Female 26.5 167.5 26 166.0 5 Brown

54 Male 29.5 190.5 22 196.0 3 Brown

55 Female 25.0 160.0 46 167.0 10 Blonde

56 Male 29.5 183.0 22 181.0 3 Brown

57 Female 21.5 162.5 18 167.5 3 Brown

58 Female 21.5 153.0 34 160.5 7 Black

59 Female 24.0 160.0 21 165.0 3 Brown

60 Female 22.0 160.5 21 157.5 4 Brown

61 Male 24.0 175.5 18 183.0 3 Brown

62 Female 21.5 164.5 23 150.0 1 Brown

63 Female 22.0 157.0 34 170.0 2 Brown

64 Male 30.0 156.0 36 183.0 3 Brown

65 Male 28.0 177.5 37 188.0 6 Black

66 Female 20.5 157.5 54 160.0 4 Blonde

67 Female 24.0 163.0 26 157.5 3 blonde

68 Male 26.0 185.5 38 192.0 7 Brown

69 Male 23.0 167.5 22 172.5 4 Brown

70 Male 29.5 172.0 30 181.0 4 Brown

71 Female 23.0 150.0 42 152.0 4 Brown

72 Male 22.0 174.0 44 175.0 4 Blonde

73 Female 25.0 165.0 20 165.0 7 Blonde

74 Male 26.0 180.0 42 175.5 2 Black

75 Male 26.5 180.0 22 182.0 2 Brown

Answer 1

Objective: To test whether the mean age of all FHSU Virtual College statistics students is less than thirty-one years old.Let denote the population mean age of all FHSU Virtual College statistics students. We have to test:

Vs

As mentioned, since, the population standard deviation is unknown, the appropriate test to test the above hypothesis would be a one sample t test:  

But before running this test, we must ensure that the data satisfies the assumptions of this test:

- The data is continuous - The observations are independent - The data is normally distributed - There are no outliers   

Assuming that all the assumptions are satisfied:

The test statistic is given by:

with critical region given by: for left tailed test.

From the given data,

Sample Mean

Sample Standard Deviation

where, xi , i = 1,2,3 ......,75 denote the observations.

= 9.863

Substituting the values obtained in the test statistic:

= -0.726

Comparing this observed test statistic value with the critical value

Using excel,

..............(since, it is a one tailed test and excel function is only for two tailed test; hence 2*0.05 = 0.10; Here, since this is a left tailed test, the t value would have a negative sign)

We get

Since, t = -0.726 > -1.666, does not lie in the critical region, we fail to reject H0 at 5% level.

Also, the p-value of the test is obtained as:

We get p-value = 0.235 > 0.05. We fail to reject H0. We may conclude that the data does not provide sufficient evidence to support the claim that the mean age of all FHSU Virtual College statistics students is less than thirty-one years.