Question: Prepare a written response to the following questions.
Part -1:
1. List the four steps of hypothesis testing, and explain the procedure and logic of each.
2. Based on the information given for the following studies, decide whether to reject the null hypothesis. Assume that all populations are normally distributed. For each, give:
a. The Z-score cutoff (or cutoffs) on the comparison distribution at which the null hypothesis should be rejected.
b. The Z-score on the comparison distribution for the sample score.
c. Your conclusion.
|
Population |
|
|
|
| Study |
µ |
σ
|
Sample Score |
p |
Tails of Tests |
| A |
5 |
1 |
7 |
0.05 |
1 (high predicted) |
| B |
5 |
1 |
7 |
0.05 |
2 |
| C |
5 |
1 |
7 |
0.01 |
1 (High predicted) |
| D |
5 |
1 |
7 |
0.01 |
2 |
3. A researcher predicts that listening to music while solving math problems will make a particular brain area more active. To test this, a research participant has her brain scanned while listening to music and solving math problems, and the brain area of interest has a percentage signal change of 58. From many previous studies with this same math problem's procedure (but not listening to music), it is known that the signal change in this brain is normally distributed with a mean of 35 and a standard deviation of 10.
a. Using the .01 level, what should the researcher conclude? Solve this problem explicity using all five steps of hypothesis testing, and illustrate your answer with a sketch showing the comparison distribution, the cutoff (or cutoffs), and the score of the sample on this distribution.
Explain your answer to someone who has never had a course in statistics (but who is familiar with mean, standard deviation, and Z scores).
Part -2:
1. Two boats, the Prada (Italy) and the Oracle (USA), are competing for a spot in the upcoming America's Cup race. They race over a part of the course several times. The sample times in minutes for the Prada were: 12.9, 12.5, 11.0, 13.3, 11.2, 11.4, 11.6, 12.3, 14.2, and 11.3. The sample times in minutes for the Oracle were: 14.1, 14.1, 14.2, 17.4, 15.8, 16.7, 16.1, 13.3, 13.4, 13.6, 10.8, and 19.0. For data analysis, the appropriate test is the t-Test: Two-Sample Assuming Unequal Variances.
The next table shows the results of this independent t-test. At the .05 significance level, can we conclude that there is a difference in their mean times? Explain these results to a person who knows about the t test for a single sample but is unfamiliar with the t test for independent means.
|
Hypothesis Test: Independent Groups (t-test, unequal variance)
|
|
Prada
|
Oracle
|
|
12.170
|
14.875
|
mean
|
|
1.056
|
2.208
|
std. dev.
|
|
10
|
12
|
n
|
|
16
|
df
|
|
-2.7050
|
difference (Prada - Oracle)
|
|
0.7196
|
standard error of difference
|
|
0
|
hypothesized difference
|
|
-3.76
|
t
|
|
.0017
|
p-value (two-tailed)
|
|
-4.2304
|
confidence interval 95.% lower
|
|
-1.1796
|
confidence interval 95.% upper
|
|
1.5254
|
margin of error
|
2. The Willow Run Outlet Mall has two Haggar Outlet Stores, one located on Peach Street and the other on Plum Street. The two stores are laid out differently, but both store managers claim their layout maximizes the amounts customers will purchase on impulse. A sample of ten customers at the Peach Street store revealed they spent the following amounts more than planned: $17.58, $19.73, $12.61, $17.79, $16.22, $15.82, $15.40, $15.86, $11.82, $15.85. A sample of fourteen customers at the Plum Street store revealed they spent the following amounts more than they planned when they entered the store: $18.19, $20.22, $17.38, $17.96, $23.92, $15.87, $16.47, $15.96, $16.79, $16.74, $21.40, $20.57, $19.79, $14.83. For Data Analysis, a t-Test: Two-Sample Assuming Unequal Variances was used.
At the .01 significance level is there a difference in the mean amount purchased on an impulse at the two stores? Explain these results to a person who knows about the t test for a single sample but is unfamiliar with the t test for independent means.
Hypothesis Test: Independent Groups (t-test, unequal variance)
Peach Street Plum Street
15.8680 18.2921 mean
2.3306 2.5527 std. dev.
10 14 n
20 df
-2.42414 difference (Peach Street - Plum Street)
1.00431 standard error of difference
0 hypothesized difference
-2.41 t
.0255 p-value (two-tailed)
-5.28173 confidence interval 99.% lower
0.43345 confidence interval 99.% upper
2.85759 margin of error
3. Fry Brothers heating and Air Conditioning, Inc. employs Larry Clark and George Murnen to make service calls to repair furnaces and air conditioning units in homes. Tom Fry, the owner, would like to know whether there is a difference in the mean number of service calls they make per day. Assume the population standard deviation for Larry Clark is 1.05 calls per day and 1.23 calls per day for George Murnen. A random sample of 40 days last year showed that Larry Clark made an average of 4.77 calls per day. For a sample of 50 days George Murnen made an average of 5.02 calls per day. At the .05 significance level, is there a difference in the mean number of calls per day between the two employees? What is the p-value?
Hypothesis Test: Independent Groups (t-test, pooled variance)
Larry George
4.77 5.02 mean
1.05 1.23 std. dev.
40 50 n
88 df
-0.25000 difference (Larry - George)
1.33102 pooled variance
1.15370 pooled std. dev.
0.24474 standard error of difference
0 hypothesized difference
-1.02 t
.3098 p-value (two-tailed)
-0.73636 confidence interval 95.% lower
0.23636 confidence interval 95.% upper
0.48636 margin of error
Part -3:
1. A consumer organization wants to know if there is a difference in the price of a particular toy at three different types of stores. The price of the toy was checked in a sample of five discount toy stores, five variety stores, and five department stores. The results are shown below.
Discount toy Variety Department
$12 15 19
13 17 17
14 14 16
12 18 20
15 17 19
An ANOVA was run and the results are shown below. At the .05 significance level, is there a difference in the mean prices between the three stores? What is the p-value? Explain why an ANOVA was used to analyze this problem.
One factor ANOVA
Mean n Std. Dev
13.2 5 1.30 Discount Toys
16.2 5 1.64 Variety
18.2 5 1.64 Department
15.9 15 2.56 Total
ANOVA table
Source SS df MS F p-value
Treatment 63.33 2 31.667 13.38 .0009
Error 28.40 12 2.367
Total 91.73 14
2. A physician who specializes in weight control has three different diets she recommends. As an experiment, she randomly selected 15 patients and then assigned 5 to each diet. After three weeks the following weight losses, in pounds, were noted. At the .05 significance level, can she conclude that there is a difference in the mean amount of weight loss among the three diets?
Plan A Plan B Plan C
5 6 7
7 7 8
4 7 9
5 5 8
4 6 9
An ANOVA was run and the results are shown below. At the .01 significance level, is there a difference in the weight loss between the three plans? What is the p-value? What can you do to determine exactly where the difference is?
One factor ANOVA
Mean n Std. Dev
5.0 5 1.22 Plan A
6.2 5 0.84 Plan B
8.2 5 0.84 Plan C
6.5 15 1.64 Total
ANOVA table
Source SS df MS F p-value
Treatment 26.13 2 13.067 13.52 .0008
Error 11.60 12 0.967
Total 37.73 14