Question

6. Pedestrian Deaths A researcher wanted to determinse whether pedestrian deaths were uniformly distributed over the days of the week. She randomly selected 300 pedestrian deaths, recorded the day of the week on which the death occurred, and obtained the following results (the data are based on information obtained from the Insurance Institute for Highway Safety) Day of the Week Sunday Monday Tuesday Wednesday Frequency Day of Frequency the Week Thursday Friday Saturday 41 49 61 39 40 30 40 Test the belief that the day of the week on which a fatality happens involving a pedestrian occurs with equal frequency at the a -0.05 level of significance. 4. Unwed Women Having Children The Pew Research Group asked the following question of individuals who earned in excess of $100,000 per year and those who earned less than $100,000 per year:"Do you believe that it is morally wrong in women to have children? Of the 1205 individuals who earned in excess of $100,000 who earned less than $100,000 per year, 695 said yes. Construct a 95% confidence interval to determine if there is a difference in the proportion of individuals who believe it is morally wrong for unwed women to have children. for unwed per year, 710 said yes:of the 1310 individuals 3. Gas Mileage The Environmental Protection Agency (EPA) states that a 2013 Kia Optima should get 28 miles per ga report the miles per gallon that they get on their vehicle. Treat s there reason to believe that individuals are getting different 19.4 on average. The website www.fueleconomy.gov allows users to 13 Kia Optima the following data as a random sample of ten 20 owners The data represent the miles per gallon on their vehicle. gas mileage than the EPA states should be attained? 176 26.7 30.2 19.9 Source: www.fueleconomy.gov 30.3 275 16.7 20.2 23.3 2. Ca on fast-food when they use a credit card? The following data represent a random sample of credit-card and cash purchases sh or Credit? Do people tend to spend more money Credit 23.89 13.89 15.54 10.35 12.76 18.32 20.67 18.36 19.16 Cash 6.78 9.21 6.26 18.98 10.76 21.76 Source: Brian Ortiz, student at Joliet Junior College 11.36 8.90 15.64 13.78 Test whether the sample evidence suggests that people spend more when using a credit card. Use the 0.01 level of significance. Note: Normal probability plots indicate that each sample could come from a population that is normally distributed 7. Wet Suits Do wet suits allow a swimmer to swim faster? Researchers measured the speed (in meters per second) of swimmers both with and without a wetsuit. The results of the study are shown in the table. Conduct the appropriate test to determine whether the data suggest that wet suits allow a swimmer to swim faster. Use an -0.05 level af significance Swimmer 1 Without 1.49 137 1.35 127 1.12 164 With 157 1.47 1.42 1.35 1.22 1.75 7 8 910 11 12 Swimmer Without1.59 1.52 1.50 145 1.44 1.41 With 1.64 1.57 1.56 1.53 1.49 1.51 Source: Data from de Lucas R.D, Balildan, P. Neiva.C.M.,Greco, с.С. & Denadai, B.S (2000). The effects of wet suits on physiological and biomechanical indices during swimming. Journal of Science and Medicine in Sport 3 (1):1-8 S. Profile of Smokers The following data represent the smoking status from a random sample of 1054 U.S, residents 18 years or older by level of education. Smoking Status Number of Years of Education 512 12 13-15 16 or more Source: National Health Interview Survey Current 178 137 Former Never 208 143 69 25 51 34 Test whether smoking status and level of education are independent at the α 0.05 level of significance. For the following problem assume that the prerequisites have been met. Use and state the 6 steps to hypothesis testing: 1 -state the null and alternative hypotheses; 2 -state the level of significance; 3 - state the calculator command and the values entered; 4 state the value of the test statistic and p-value; 5 - state whether to reject or fail to reject the null hypothesis; 6-state the αnclusion in the context of the problem to test the stated claim. 1. Always Wear a Helmet The National Highway Traffic Safety Administration publishes reports about motorcycle fatalities and helmet use. The distribution shows the proportion of fatalities by location of injury for motorcycle accidents Location Multiple of injury Locations Head Neck Thorax Lumbar'Spine Proportion 0.57 Abdomen 0.31 0.03 0.06 0.03 The following data show the location of injury and number of fatalities for 2068 riders not wearing a helmet. Abdomen Location Multiple of injury Locations Head Neck Thorax Lumbar/Spine Number 1036 864 38 47 (a) Does the distribution of fatal injuries for riders not wearing a helmet follow the distribution for all riders? Use the α-0.05 level of significance. (b) Compare the observed and expected counts for each category. What does this information tell you?

Answer 1

6)

We want to test if the data comes from a uniform distribution. If the data follows a uniform distribution, then each day will have the same frequency of accidents.

We have 7 days, and a total of 300 accidents in this week, so 300/7=42.86 accidents per day.

The following table is obtained (fo-fe)2fe (39-42.9)2/42.9 0.355 (40-42.9)2142.9 0.196 (30-429)2/42.9 = 3.879 (40-42.9)142.90.196 (41-42.9)2142.9 0.084 (49-429)2/42.9 = 0.867 (61-42.9)2/42.9 7.637 Categories Sun Mon Tue Wed Thur Fri Sat Observed 39 40 30 40 Expected 300 0.143-42.9 300*0.143-42.9 300*0.143-42.9 300 0.143-42.9 300*0.143-42.9 300*0.143-42.9 300 0.143-42.9 49 61 300 13.214 Sum- 300

1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested Но : Pi-0.143, P2-0.143,Pa-0.143,P4-0.143,P5-0.143, P6-0.143,Pr-0.143 Ha: Some of the population proportions differ from the values stated in the null hypothesis This corresponds to a Chi-Square test for Goodness of Fit (2) Rejection Region Based on the information provided, the significance level is α 0.05, the number of degrees of freedom is df-7-1 = 6, so then the rejection region for this test is R = {x2 x2 > 12.592) (3)_Test Statistics The Chi-Squared statistic is computed as follows: 0.355+0.196+3.879+0.196+0.084+0.867 +7.637 13.214 Ei Since it is observed that X2-13.214 > X2-12.592, it is then concluded that the null hypothesis is rejected

(5) Conclusion It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that some of the population proportions differ from those stated in the null hypothesis, at the α- 0.05 significance level.

Since the hypothesis test is rejected, we conclude that the data does not follow a uniform distribution.