Question

5. (a) Construct a 99% confidence interval for the mean height of the entire female/male SCC student body.

(b) What is the width of this interval?

(c) Write a sentence interpreting your confidence interval.

(d) Review your 93% and 99% confidence intervals above. Which is wider and why?

male Student # Gender Height Shoe Age Hand

1 M 67 10 19 R

2 M 74 12 17 R

3 M 72 11.5 19 R

4 M 69 10 35 R

5 M 66 9 18 R

6 M 71 10.5 17 R

7 M 72 10.5 17 R

8 M 66 10 20 R

9 M 67 10 18 R

10 M 71 10.5 24 R

11 M 66 10 21 R

12 M 71 10.5 18 R

13 M 69 10 22 R

14 M 66 9.5 18 L

15 M 76 14 18 R

16 M 69 11 22 R

17 M 68 9 19 R

18 M 70 12 30 R

19 M 67 10 24 R

20 M 70 11 21 R

21 M 70 10 52 R

22 M 63 9 27 R

23 M 69 11 22 R

24 M 72 10 22 R

25 M 76 11.5 20 L

26 M 75 11 17 R

27 M 72 11 50 L

28 M 69 11 20 R

29 M 70 12 20 R

30 M 69 11.5 23 R

31 M 70 11 18 R

32 M 67 10 21 R

33 M 68 11 44 R

34 M 76 13 48 R

35 M 62 8 23 L

36 M 69 9 19 R

37 M 72 10 60 R

38 M 73 11.5 41 R

39 M 70 9.5 39 R

40 M 78 15 24 R

41 M 65 8.5 23 R

42 M 68 9.5 20 R

Answer 1

5 a) Sample mean of the average height of the entire student body is

Sample standard deviation of the average height of the student body

Since the population standard deviation is unknown, hence we will use the t-statistics to find the 99% confidence interval for the population means.

The confidence interval for mean is given by

Degree of freedom = n- 1 = 42-1 = 41

The 99% confidence interval is

b) Width of the interval is

Width = 71.233 - 68.287 = 2.946

c) Interpretation: There is a 95% chance that the true average height will lie in the interval calculated above.

d) t-critical for 93% confidence is

t-critical = 1.86

93% confidence interval for the population mean is

Width of 93% confidence interval = 70.774 - 68.746 = 2.028

As we can see, the 99% confidence interval is wider.

Reason - A larger confidence is nothing but a larger probability of finding the true mean in a given interval. A larger probability means a larger favorable space. That is why for 99% surety, the width is wider whereas, for 93% surety, the width is narrower. More intuitively, we are more likely to correctly guess the true mean if we have a larger range. As we get more and more precise (narrow width), the chances that the guess will be correct will decrease.

Hence, the 99% confidence interval is wider than the 93% confidence interval.

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