# 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

#### Solution Preview

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Please show all work

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 Zα 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 ІzІ> Zα 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.