, and the standard deviation of the paired differences, s. Use these values to test the hypotheses:Math 37 - Lecture 20
Examples (6.3)
Example 1 A survey of 40,000 American households in 1987 found that 30.5% of the households in the sample owned a cat (or vice versa).
(a) Is this sample proportion evidence that statistically significantly less than one-third of households own a cat at the a=.001 significance level?
(b) Form a 99% confidence interval to estimate the proportion of all American households that owned a pet cat in 1987.
(c) Comment on whether the sample data suggest that the proportion of cat owners among all American households is much less than one-third in a practical sense.
Example 2
|
Planet |
Distance (mill miles) |
Diameter (miles) |
Rev (days) |
|
Mercury |
36 |
3,030 |
88 |
|
Venus |
67 |
7,520 |
225 |
|
Earth |
93 |
7,826 |
365 |
|
Mars |
142 |
4,217 |
687 |
|
Jupiter |
484 |
88,838 |
4,332 |
|
Saturn |
887 |
74,896 |
10,760 |
|
Uranus |
1,765 |
31,762 |
30,684 |
|
Neptune |
2,791 |
30,774 |
60,188 |
|
Pluto |
3,654 |
1,428 |
90,467 |
Does it make sense to construct a 95% confidence interval for the mean diameter? Explain.
Example 3 - A researcher looking for evidence of extrasensory perception (ESP) tests 500 subjects. Four of these subjects do significantly better (p-value <.01) than random guessing.
(a) How often do we expect someone to get lucky and test significantly?
(b) Do you think you have conclusive evidence that these four people have ESP and aren't just guessing?
Example 4 - To decide whether a newly developed gasoline additive increases gas mileage you will compare the gas mileage for cars with and without the additive.
(a) Outline a completely randomized design using 20 cars to test this.
(b) A recent study randomly selected a single group of 10 cars and had each of the 10 cars driven both with and without the additive.
- Does this design differ from the one you gave in (a)?
- How many data values are obtained for each method?
- Explain any advantages this design has over yours
- Where is randomization used in this design?
- How many samples of cars do you have?
(c) Below is the data for the study done:
|
Car |
With additive |
Without additive |
Difference |
|
1 |
25.7 |
24.9 |
|
|
2 |
20.0 |
18.8 |
|
|
3 |
28.4 |
27.7 |
|
|
4 |
13.7 |
13.0 |
|
|
5 |
18.8 |
17.8 |
|
|
6 |
12.5 |
11.3 |
|
|
7 |
28.4 |
27.8 |
|
|
8 |
8.1 |
8.2 |
|
|
9 |
23.1 |
23.1 |
|
|
10 |
10.4 |
9.9 |
To analyze these data compute the paired difference in mileage for each car. Now compute the average paired difference, , and the standard deviation of the paired differences, s. Use these values to test the hypotheses:
H0: no difference in the gas mileage (m=0)
Ha: the additive increases the gas mileage (m>0)
where m is the true average difference in mileage.
(Substitute s in for s in your test statistic.)
(d) What's your conclusion?
(e) What assumptions does this procedure require?