1 Problem 1: Old Faithful The most famous geyser in the world, Old Faithful in Yellowstone National Park, has mean time between eruptions of 85 minutes. Past research has shown that the population standard deviation is known to be 21.25 and that the distribution of time between eruptions is bimodal. a) What is the probability that a random sample of 35 time intervals between eruptions has a mean longer than 93 minutes? b) What is the probability that a random sample of size 15 time intervals between eruptions has a mean longer than 93 minutes? Problem 2: Red Light Cameras To combat red-light-running crashes, many states are installing red light cameras at dangerous intersections. These cameras photograph the license plates of vehicles that run red lights and automatically issue tickets. How effective are these photo enforcement programs? The Virginia Department of Transportation (VDOT) conducted a comprehensive study of its newly adopted program and published the results in a 2012 study. In one portion of the study, VDOT provided crash data both before and after installation of the cameras at several intersections. The data, measured as the number of crashes caused by red light running per intersection per year, for 13 randomly selected intersections in Fairfax County are given in the first columns in the StatCrunch file. a) What is (are) the parameter(s) we are conducting inference on? Choose one of the following symbols ( D (the mean difference from a matched pairs design) 2 (the mean difference from independent samples) and describe it in words. b) Depending on your answer to part (a), construct one or two histograms and one or two boxplots to visualize the distribution(s) of your sample data. Also, properly title and label your graphs. Copy and paste these graphs into your assignment. Below the graphs, answer the following questions. i. ii. iii. Are there any major deviations from normality? Are there any outliers present? Is it appropriate to conduct statistical inference procedures, why or why not? If the answer to part iii is no, do not complete the rest of #2. c) At the 0.01 significance level, test the claim that the installation of the cameras decreased the mean number of crashes at these particular intersections. i. State the null and alternative hypotheses. ii. State the significance level for this problem. iii. Calculate the test statistic (subtract after - before) iv. Calculate the P-value and include the probability notation statement. v. State whether you reject or do not reject the null hypothesis. 2 vi. State your conclusion in context of the problem (i.e. interpret your results). d) For the above situation, construct a 96.2% confidence interval for the above data. Interpret the confidence interval as we learned in class. Verify your interval using StatCrunch Note: For part d, to earn full credit, show how you obtained the critical value for the confidence interval in StatCrunch. Then, write out the confidence interval formula you would use, and the steps necessary to construct the confidence interval. Problem 3: Styles of Instruction A researcher wanted to know whether there was a difference in the level of understanding among students learning StatCrunch based on the style of instruction. In a previous semester of STAT 250, Section 1 was taught StatCrunch with video tutorials and Section 2 was taught StatCrunch with written instructions. One simple random sample of 28 was taken from each section and the students in each sample were given a StatCrunch quiz that tested basic procedures. The data provided in StatCrunch represents the quiz scores the students received. At the 0.01 significance level, can the researcher conclude from these data that there is a significant difference in quiz scores between the two methods of instruction? a) What is (are) the parameter(s) of interest? Choose one of the following symbols ( D (the mean difference from a matched pairs design) 2 (the mean difference from independent samples) and describe it in words. b) Depending on your answer to part (a), construct one or two histograms and one or two boxplots to visualize the distribution(s) of your sample data. Remember to properly title and label these graphs. Copy and paste these graphs into your assignment. Below the graphs, answer the following questions. i. Are there any major deviations from normality? ii. Are there any outliers present? iii. Is it appropriate to conduct statistical inference procedures, why or why not? If the answer to part iii is no, do not complete the rest of #3. c) At the 0.01 significance level, can the researcher conclude from these data that there is a significant difference in quiz scores between the two methods of instruction? i. State the null and alternative hypotheses. ii. State the significance level for this problem. iii. Calculate the test statistic. iv. Calculate the P-value and include the probability notation statement. v. State whether you reject or do not reject the null hypothesis. vi. State your conclusion in context of the problem (i.e. interpret your results). d) Construct a 95% confidence interval for the above data using our formulas and verify the interval using StatCrunch. Note: To earn full credit, explain how you obtained the critical value for the confidence interval, write out the formula you would use and the steps necessary to construct the confidence interval. 3 Problem 4: Monday Night Deals In an attempt to increase business on Monday nights, a restaurant offers a free dessert with every dinner order. Before the offer, the mean number of dinner customers on Monday was 150. The numbers of diners on a random sample of 12 days while the offer was in effect are selected. Can you conclude that the mean number of diners increased while the free dessert offer was in effect? a) What is (are) the parameter(s) we are conducting inference on? Choose one of the following symbols ( D (the mean difference from a matched pairs design) 2 (the mean difference from independent samples) and describe it in words. b) Construct a histogram and a boxplot to visualize the distribution of your sample data. Copy and paste these graphs into your assignment. Below the graphs, answer the following questions. i. Are there any major deviations from normality? ii. Are there any outliers present? iii. Is it appropriate to conduct statistical inference procedures, why or why not? If the answer to part iii is no, do not complete the rest of #4. c) At the 0.01 significance level, can you conclude that the mean number of diners increased from 150 while the free dessert offer was in effect? i. State the null and alternative hypotheses. ii. State the significance level for this problem. iii. Calculate the test statistic. iv. Calculate the P-value and include the probability notation statement. v. State whether you reject or do not reject the null hypothesis. vi. State your conclusion in context of the problem (i.e. interpret your results). d) Construct a 99% confidence interval for the above data. Interpret the confidence interval. Intersectio Before Cam After Camera 1 3.6 1.87 2 0.27 0 3 0.29 0 4 4.55 1.79 5 3.09 2.04 6 2.29 3.14 7 2.4 2.72 8 1.09 0.24 9 3.15 1.57 10 3.21 0.43 11 1.88 0.28 12 1.35 1.09 13 7.35 4.92 Section 1 Section 2 79.8 84.6 86.2 70.8 91.2 86.8 85.4 94.8 86.4 76.7 85.6 91.8 82.5 82.6 83.9 84.3 86.1 73.7 83.4 83.9 85.2 83.3 89.2 83.3 88.1 80.3 80.7 89.5 90.5 75.1 95 78.7 84.1 83.4 87.8 76 89.1 79.3 90 88.7 87.5 74.9 82.2 77.8 84.5 72 86.4 92.5 93 83.7 89.9 80.5 79.3 84.5 87.3 78.3 Diners 206 190 196 209 191 174 309 212 111 40 199 153