Solver ... Hypothesis Testing for Two or more Samples
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20160524174937unit_4_practice_problems_solution_layout.docxUnit 4
Practice Problems
Two Sample Hypothesis Test for the Mean (Independent Samples)
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Level of Significance |
0.05 |
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No Pool |
Pool |
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Sample Mean |
202763.158 |
225258.065 |
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Sample Size |
38 |
62 |
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Sample Standard Deviation |
33716.973 |
46371.959 |
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Standard Error (Computed) |
8037.41 |
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Test Statistic (Computed) |
-2.80 |
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This is a two-tailed test since we are testing if the mean is different for the houses with vs. without a pool.
The p-value of .0062 is less than .05, and so we reject the null hypothesis and conclude that the mean sales price for homes is different for houses with vs. without a pool. The fact that the sample mean is greater for houses with a pool indicates that houses with a pool likely have a higher mean price in the population.
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Lower |
Upper |
p-Value |
Decision |
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Crit Value |
Crit Value |
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Two-Tailed Test |
-1.9600 |
1.9600 |
0.0062 |
Reject the null hypothesis |
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Practice Problem Two
Two Sample Hypothesis Test for the Mean (Paired Samples)
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Level of Significance |
0.05 |
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Standard Error (Computed) |
3545.08 |
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Test Statistic (Computed) |
0.25 |
This is a two-tailed test since we are testing whether the mean appraisals given by the two agents are different from one another. The p-value of .8001 is greater than .05, and so we fail to reject the null hypothesis; there is insufficient evidence to conclude the mean appraisals are different for the two agents. As a result we conclude that the two agents’ appraisals are not significantly different from one another.
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Lower |
Upper |
p-Value |
Decision |
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Crit Value |
Crit Value |
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Two-Tailed Test |
-1.9842 |
1.9842 |
0.8001 |
Do not reject the null hypothesis |
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d |
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Practice Problem Three
Two Sample Hypothesis Test for the Proportion
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Level of Significance |
0.05 |
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Brick |
Wood or Stucco |
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Number of Successes |
30 |
32 |
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Sample Size |
40 |
60 |
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Proportion (Computed) |
0.75 |
0.53 |
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Z Test Statistic (Computed) |
2.19 |
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This is an upper-tailed test since we are testing whether the proportion for brick homes with a pool is greater than that of wood & stucco homes with a pool. The p-value of .0144 is less than .05, and so we reject the null hypothesis. As a result, we conclude that brick homes are more likely to have a swimming pool than wood & stucco homes.
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Lower Crit Value |
Upper Crit Value |
p-Value |
Decision |
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Upper-Tail Test |
n/a |
1.6449 |
0.0144 |
Reject the null hypothesis |
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H1: (P1 > P2 or P2 < P1) |
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Practice Problem Four
Analysis of Variance
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SUMMARY |
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Groups |
Count |
Sum |
Average |
Variance |
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2 BR |
24 |
4807000.000 |
200291.667 |
1904911231.884 |
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3 BR |
26 |
5574000.000 |
214384.615 |
1205046153.846 |
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4 BR |
26 |
5552000.000 |
213538.462 |
1457058461.538 |
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5 BR |
24 |
5738000.000 |
239083.333 |
2430688405.797 |
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ANOVA |
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Source of Variation |
SS |
df |
MS |
F |
F crit |
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Treatment |
18885182948.718 |
3 |
6295060982.906 |
3.635 |
2.699 |
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Error |
166271407051.282 |
96 |
1731993823.451 |
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Total |
185156590000.000 |
99 |
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The p-value of .0156 is less than .05, and so we reject the null hypothesis. Thus we conclude that a difference exists in the mean sales price based on number of bedrooms that a home has. Looking at the Post Hoc tests, we see that homes with 5 bedrooms are worth significantly more than homes with 2, 3 or 4 bedrooms. There doesn't appear to be a significant difference in mean prices for homes with 2, 3 or 4 bedrooms.
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Null hypothesis: |
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Alternate hypothesis: |
Not all the means are equal |
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Significance level: |
0.05 |
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p-Value |
0.0156 |
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Decision: |
Reject the null hypothesis |
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Mean Differences Post-Hoc Tests |
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Difference |
Lower |
Upper |
Difference in Treatment Means??? |
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Comparison |
of Means |
Conf. Limit |
Conf. Limit |
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2 BR to 3 BR |
14092.949 |
-9291.274 |
37477.172 |
- no difference - |
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2 BR to 4 BR |
13246.795 |
-10137.428 |
36631.018 |
- no difference - |
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3 BR to 4 BR |
846.154 |
-22065.612 |
23757.920 |
- no difference - |
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2 BR to 5 BR |
38791.667 |
14944.345 |
62638.989 |
***Means are different*** |
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3 BR to 5 BR |
24698.718 |
1314.495 |
48082.941 |
***Means are different*** |
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4 BR to 5 BR |
25544.872 |
2160.649 |
48929.095 |
***Means are different*** |
