# Hypotheses testing problems

1. 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.

2. Does lovastatin (a cholesterol-lowering drug) reduce the risk of heart attack? In a Texas study, researchers gave lovastatin to 2,325 people and an inactive substitute to 2,081 people (average age 58). After 5 years, 57 of the lovastatin group had suffered a heart attack, compared with 97 for the inactive pill. (a) State the appropriate hypotheses. (b) Obtain a test statistic and p-value. Interpret the results at α = .01. (c) Is normality assured? (d) Is the difference large enough to be important?

(e) What else would medical researchers need to know before prescribing this drug widely? (Data are from Science News 153 [May 30, 1998], p. 343.)

3. 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.

4. Sample of 25 concession stand purchases at the October 22 matinee of Bride of Chucky showed a mean purchase of $5.29 with a standard deviation of $3.02. For the October 26 evening showing of the same movie, for a sample of 25 purchases the mean was $5.12 with a standard deviation of $2.14. The means appear to be very close, but not the variances. At α = .05, is there a difference in variances? Show all steps clearly, including an illustration of the decision rule. (Data are from a project by statistics students Kim Dyer, Amy Pease, and Lyndsey Smith.)

5. 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?

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 1920

****All work must be shown and completed using megastat add on for Excel. Can you also show which where the numbers get plugged into megastat?****

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Please see the attached files for fully formatted explanation.

Statistics Problems 2

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.

(a) State the hypotheses.

H0: There is no significant difference in the accidents rate of red fire trucks and yellow fire trucks

H1: The yellow fire trucks have a significantly lower accident rate

(b) State the decision rule and sketch it.

Reject the null hypothesis if the calculated value of test statistic is less than the critical value of Z

(c) Find the sample proportions and z test statistic.

Hypothesis test for two independent proportions

p1 p2 pc

0.0001 0.00003 8.33878E-05 p (as decimal)

0.0001 0.00003 8.33878E-05 p (as fraction)

19.997 4.05105 24.0476292 X

153348 135035 288383 n

0.0001 difference

0 hypothesized difference

3.41E-05 std. error

2.946306 z

(d) Make a decision.

Reject the null hypothesis. The sample provides enough evidence to support the claim that the yellow fire trucks have a significantly lower accident rate.

(e) Find the p-value and interpret it.

P value = P(Z>2.960988745) =0.001533266.

A p-value is a measure of how much evidence we have against the null hypothesis. The p-value measures consistency by calculating the probability of observing the results from your sample of data or a sample with results more extreme, assuming the null hypothesis is true. The smaller the p-value, the greater the inconsistency.

Here there is a greater in consistence against the null hypothesis.

(f) If statistically significant, do you think the difference is large enough to be important? If so, to whom, and why?

The test Statistic used is given by where

Thus if the difference in the proportion is large enough the value of Z will be high. A high of Z indicate that the null hypothesis is significant.

(g) Is the normality assumption fulfilled? Explain.

Normal assumption can be used ...

#### Solution Summary

The solution provides step by step method for the calculation of test statistic . Formula for the calculation and Interpretations of the results are also included. Interactive excel sheet is included. The user can edit the inputs and obtain the complete results for a new set of data.

The question is from 'Applied Statistics in Business and Economics by David P Doane and Lori E. Seward, McGraw Hill; 1st Edition (2006) '