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Hypothesis testing, ANOVA, statistics and probability

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10.30
Note: For tests on two proportions, two means, or two variances it is a good idea to check your work by using MINITAB, MegaStat, or the LearningStats two-sample calculators in Unit 10.

In Dallas, some fire trucks were painted yellow (instead of red) to heighten their visibility. During a test period, the fleet of red fire trucks made 153,348 runs and had 20 accidents, while the fleet of yellow fire trucks made 135,035 runs and had 4 accidents. At α = .01, did the yellow fire trucks have a significantly lower accident rate? (a) State the hypotheses. (b) State the decision rule and sketch it. (c) Find the sample proportions and z test statistic. (d) Make a decision. (e) Find the p-value and interpret it. (f) If statistically significant, do you think the difference is large enough to be important? If so, to whom, and why? (g) Is the normality assumption fulfilled? Explain.
Accident Rate for Dallas Fire Trucks
Statistic Red Fire Trucks Yellow Fire Trucks
Number of accidents x1 = 20 accidents x2 = 4 accidents
Number of fire runs n1 = 153,348 runs n2 = 135,035 runs

10.46
Note: For tests on two proportions, two means, or two variances it is a good idea to check your work by using MINITAB, MegaStat, or the LearningStats two-sample calculators in Unit 10.

To test the hypothesis that students who finish an exam first get better grades, Professor Hardtack kept track of the order in which papers were handed in. The first 25 papers showed a mean score of 77.1 with a standard deviation of 19.6, while the last 24 papers handed in showed a mean score of 69.3 with a standard deviation of 24.9. Is this a significant difference at α = .05? (a) State the hypotheses for a right-tailed test. (b) Obtain a test statistic and p-value assuming equal variances. Interpret these results. (c) Is the difference in mean scores large enough to be important? (d) Is it reasonable to assume equal variances? (e) Carry out a formal test for equal variances at α = .05, showing all steps clearly.

10.64
Note: For tests on two proportions, two means, or two variances it is a good idea to check your work by using MINITAB, MegaStat, or the LearningStats two-sample calculators in Unit 10.

A cognitive retraining clinic assists outpatient victims of head injury, anoxia, or other conditions that result in cognitive impairment. Each incoming patient is evaluated to establish an appropriate treatment program and estimated length of stay. To see if the evaluation teams are consistent, 12 randomly chosen patients are separately evaluated by two expert teams (A and B) as shown. At the .10 level of significance, are the evaluator teams consistent in their estimates? State your hypotheses and show all steps clearly. LengthStay
Estimated Length of Stay in Weeks
Patient
Team 1 2 3 4 5 6 7 8 9 10 11 12
A 24 24 52 30 40 30 18 30 18 40 24 12
B 24 20 52 36 36 36 24 36 16 52 24 16

11.24
1. Choose an appropriate ANOVA model. State the hypotheses to be tested.
2. Display the data visually (e.g., dot plots or MegaStat's line plots). What do the displays show?
3. Do the ANOVA calculations using the computer.
4. State the decision rule for α = .05 and make the decision. Interpret the p-value.
5. In your judgment, are the observed differences in treatment means (if any) large enough to be of practical importance?
6. Do you think the sample size is sufficient? Explain. Could it be increased? Given the nature of the data, would more data collection be costly?
7. Perform Tukey multiple comparison tests and discuss the results.
8. *Perform a test for homogeneity of variances. Explain fully.
In a bumper test, three types of autos were deliberately crashed into a barrier at 5 mph, and the resulting damage (in dollars) was estimated. Five test vehicles of each type were crashed, with the results shown below. Research question: Are the mean crash damages the same for these three vehicles? Crash1
Crash Damage ($)
Goliath Varmint Weasel
1,600 1,290 1,090
760 1,400 2,100
880 1,390 1,830
1,950 1,850 1,250
1,220 950 1,920

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10.30
Note: For tests on two proportions, two means, or two variances it is a good idea to check your work by using MINITAB, MegaStat, or the LearningStats two-sample calculators in Unit 10.

In Dallas, some fire trucks were painted yellow (instead of red) to heighten their visibility. During a test period, the fleet of red fire trucks made 153,348 runs and had 20 accidents, while the fleet of yellow fire trucks made 135,035 runs and had 4 accidents. At α = .01, did the yellow fire trucks have a significantly lower accident rate? (a) State the hypotheses. (b) State the decision rule and sketch it. (c) Find the sample proportions and z test statistic. (d) Make a decision. (e) Find the p-value and interpret it. (f) If statistically significant, do you think the difference is large enough to be important? If so, to whom, and why? (g) Is the normality assumption fulfilled? Explain.
Accident Rate for Dallas Fire Trucks
Statistic Red Fire Trucks Yellow Fire Trucks
Number of accidents x1 = 20 accidents x2 = 4 accidents
Number of fire runs n1 = 153,348 runs n2 = 135,035 runs

SOLUTION:

State the null and alternate hypotheses

Null hypothesis: H0: P1<P2
Alternative hypothesis: H1: P1 > P2
Select a level of significance

The Z&#945; critical value with 0.01 significance level is given by 2.33

Identify the test statistic

Sample size n1 =153348 and had accident samples = X1 = 20

Sample size n2 =135035 and had accident samples = X2 = 4

The first sample proportion =0.00013 and 0.99987

The second sample proportion =0.00003 and =0.99997

The standard error =

Test Statistic:

3.057449

Formulate a decision rule

Since &#1030;z&#1030;> Z&#945; Therefore there is no evidence to accept the hypothesis

Make a decision

Thus the yellow fire truck does not have a lower significant rate

Compute p-value and interpret it

p value = P[Z>3.057]
= 1-P [Z<3.057]
= 1-9989
=0 .0011

Is the normality assumption fulfilled? Explain
The normality holds good since the sample size is large

10.46
Note: For tests on two proportions, two means, or two variances it is a good idea to check your work by ...

Solution Summary

The solution examines hypothesis testing, ANOVA, and probability. The expert uses Minitab.

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1.
a. The score on the entrance exam at a well-known, exclusive law school are normally distributed with a mean score of 200 and a standard deviation equal to 50. At what value should the lowest passing score be set if the school wishes only 2.5 percent of those taking the test to pass?

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2.
Independent random samples were taken from normal distributions of the yearly production of ships built by the International Ship Building Company under (1) a fixed-position layout and (2) a project layout. The results are given in the accompanying table. Test the null hypothesis that the mean for the fixed-position layout is equal to that for the project layout. Use .05 for the level of significance.

Fixed-position layout 8 0 1 5 1 5 1
Project layout 6 1 1 1 4 4 4

3.
The accompanying table shows the grades for three randomly selected samples of students in economics, statistics, and accounting classes. Test the hypothesis that the mean grades are the same for all three classes. Use .05 for the level of significance.

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80 100 95
90 90 90
70 90 88
100 75 82
60 95 85
100 60
80
60

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