Instruction: Please read each questions carefully. Please show all work for full credit. You may type or hand-write your answers.
1. Aresearcher is interested in whether individuals with a smart phone (population 1) or without a smart phone (population 2) spend more time on social networking sites. Assume that both populations are normally distributed. A Simple Random Sample (SRS) of 26 smart phone users and 21 non-smart phone users were drawn from their respective populations. The following statistics (in hours per week) were calculated from each sample. Use the following information to test the hypothesis that the means are different using α = 0.05.
Smart phone users: n=26 x¯ = 23.5 s1 = 3.8
Non-users: n = 21 x¯ = 18.5 s2 = 7.9
1) Hypotheses:
2) Full equality of variances test:
3) Decision:
4) Test statistic for means tests:
5) P-value statement and value:
6) Conclusions and interpretation (i.e. make a decision using the two methods above and answer the question being posed in one to two sentences)
7) Rejection regions (Draw images in boxes and label test statistics on image)
2. We are interested in comparing the average supermarket prices of two leading colas. Our sample was taken by randomly selecting eight supermarket and recording the price of a six-pack of each brand of cola at each supermarket. Conduct a full hypothesis test to determine if a difference exists between mean prices at α = 0.10. The data are shown in the following table:
|
Supermarket
|
Brand 1 price
|
Brand 2 price
|
|
1
|
2.25
|
2.30
|
|
2
|
2.47
|
2.45
|
|
3
|
2.38
|
2.44
|
|
4
|
2.27
|
2.29
|
|
5
|
2.15
|
2.25
|
|
6
|
2.25
|
2.25
|
|
7
|
2.36
|
2.42
|
|
8
|
2.37
|
2.40
|
1) Hypothesesand significance level:
2) Test statistic:
3) P-value statement and value:
4) Conclusions and interpretation (i.e. make a decision using the two methods above and answer the question being posed in one to two sentences)
5) Rejection region (draw picture below and label test statistic on image)
3. The amount of money spent on health care is an importance issue for workers because many companies provide health care that only partially covers many medical procedures. The director of employee benefits at a midsize company wants to determine the amount spent on health care by the typical hourly worker in the company. A random sample of 25 workers is elected and the amount they spent on their families' health care needs during the past year is given below.
|
400
|
201
|
218
|
207
|
531
|
|
143
|
290
|
273
|
208
|
211
|
|
345
|
314
|
197
|
225
|
172
|
|
254
|
398
|
326
|
223
|
108
|
|
248
|
219
|
342
|
123
|
432
|
|
Column
|
N
|
Mean
|
Std. err.
|
Std. dev
|
|
Costs $
|
25
|
264.32
|
20.35
|
101.73
|
1) Discuss all assumptions and conditions for inference using this data set.
2) Construct a 99% confidence interval for the amount the typical worker spends on health care. Complete this step even if you have concerns regarding some of the conditions from part (1). Show all work and the formula used.
3) Interpret your confidence interval from part (2) in the context of the question. In addition, discuss the accuracy of this interpretation specifically in this case.
4. Insurance adjusters are concerned at the high estimates they are receiving for auto repairs from garage I compared garage II. To verify their suspicions, each of 15 cars recently involved in an accident was taken to both garages for separate estimates of repair costs. The estimates from the two garages are given in the table below.
|
Car
|
Garage I
|
Garage II
|
Difference
|
|
1
|
17.6
|
17.3
|
0.3
|
|
2
|
20.2
|
19.1
|
1.1
|
|
3
|
19.5
|
18.4
|
1.1
|
|
4
|
11.3
|
11.5
|
-0.2
|
|
5
|
13
|
12.7
|
0.3
|
|
6
|
16.3
|
15.8
|
0.5
|
|
7
|
15.3
|
14.9
|
0.4
|
|
8
|
16.2
|
15.3
|
0.9
|
|
9
|
12.2
|
12
|
0.2
|
|
10
|
14.8
|
14.2
|
0.6
|
|
11
|
21.3
|
21
|
0.3
|
|
12
|
22.1
|
21
|
1.1
|
|
13
|
16.9
|
16.1
|
0.8
|
|
14
|
17.6
|
16.47
|
0.9
|
|
15
|
18.4
|
17.5
|
0.9
|
|
Garage I
|
Garage II
|
Difference (Garage I - II)
|
|
Mean = 16.85
|
Mean = 16.23
|
Mean = 0.62
|
|
Std. dev = 3.20
|
Std. dev = 0.94
|
Std. dev = 0.394
|
Hypothesis test result #1 (μ1: mean of garage 1,μ2: mean of garage 2)
H0:μ1-μ2 = 0
HA:μ1-μ2> 0
|
Sample diff.
|
Std. Err
|
Degree of freedom
|
T-stat
|
P-value
|
|
0.62
|
1.12
|
27.8
|
0.56
|
0.29
|
Hypothesis test result #2
H0:μD = 0
HA:μD> 0
|
Sample diff.
|
Std. Err
|
Degree of freedom
|
T-stat
|
P-value
|
|
0.62
|
0.11
|
14
|
5.93
|
<0.0001
|
1) Which of the two results shown above, Test Result #1 or #2, should the researchers use to test their hypothesis? Completely explain your reasons for selecting this methods of analysis.
2) Using the output that your chose for part (1), what can the researchers conclude concerning the repair costs at the two garages?