Fran is training for her first marathon, and she wants to know if there is a significant difference between the mean number of miles run each week by group runners and individual runners who are training for marathons. She interviews 50 randomly selected people who train in groups, and finds that they run a mean of 45.5 miles per week. Assume that the population standard deviation for group runners is known to be 2.6 miles per week. She also interviews a random sample of 47 people who train on their own and finds that they run a mean of 44.4 miles per week. Assume that the population standard deviation for people who run by themselves is 1.1 miles per week. Test the claim at the 0.02 level of significance. Let group runners training for marathons be Population 1 and let individual runners training for marathons be Population 2. Step 3 of 3: Draw a conclusion and interpret the decision.

Gary has discovered a new painting tool to help him in his work. If he can prove to himself that the painting tool reduces the amount of time it takes to paint a room, he has decided to invest in a tool for each of his helpers as well. From records of recent painting jobs that he completed before he got the new tool, Gary collected data for a random sample of 6 medium-sized rooms. He determined that the mean amount of time that it took him to paint each room was 4.6 hours with a standard deviation of 0.3 hours. For a random sample of 9 medium-sized rooms that he painted using the new tool, he found that it took him a mean of 4.1 hours to paint each room with a standard deviation of 0.4 hours. At the 0.10 level, can Gary conclude that his mean time for painting a medium-sized room without using the tool was greater than his mean time when using the tool? Assume that both populations are approximately normal and that the population variances are equal. Let painting times without using the tool be Population 1 and let painting times when using the tool be Population 2. Step 3 of 3 : Draw a conclusion and interpret the decision.

1. Suppose you own a hat shop and decide to order hats in only one size for the coming season. To decide which size to order, you look at last year's sales figures, which are itemized according to size. Should you find the mean, median, or mode for the data? Why? 2. A movie chain conducts a popcorn poll in which each person entering a theater and buying a box of popcorn is asked a yes-no question. Which "average" do you think is used to report the result and why? 3. When a government agency reports the average rainfall for a state for a year, which "average" do you think it uses and why? 4. Karl had scores of \( 90,95,85 \), and 90 on his first four tests. a. Find the median, mean, and mode. b. Karl scored a 20 on his fifth exam. Which of the three averages would Karl want the instructor to use to compute his grade? Why? c. Which measure is affected most by an extreme score?

5. The mean of the five numbers given is 50 : \[ 20 \quad 35 \quad 5060 \quad 85 \] a. Add four numbers to the list so that the mean of the nine numbers is still 50 . b. Explain how you could choose the four numbers to add to the list so that the mean does not change. c. How does the mean of the four numbers you added to the list compare to the original mean of 50 ? 6. Lea drives 5 mi at 30 mph and then 5 mi at 50 mph. Is the mean speed for the trip 40 mph? \$. If you why not? theater, what type of data would you use and why? Explain the types of measures of center and spread that you would report for your data.

9. The owners of a restaurant want to address their concerns of declining customer satisfaction. Examine the following sampling methods for bias. Which would provide good feedback? Why? i. A postcard questionnaire is placed with every bill in hopes that patrons fill it out at home and mail it back. ii. Servers are instructed to encourage patrons to visit the web site, the address of which is printed on the bottom of the receipt, hosting a short survey. iii. Patrons can fill out a questionnaire at the bar and drop it in a locked box, thereby entering a drawing for a free dinner for two. iv. A pollster is hired to stand outside the restaurant and ask every other exiting patron some questions.

Company X is known to release some employees and hire new ones every year without changing the total number of employees in the company. A statistical analysis was done, and the following was observed: in year 2020, there were 150 male employees who constituted \( 75 \% \) of the total number of employees. In year \( 2021,60 \% \) of the employees were females. What is the change in the number of female employees from year 2020 to 2021 ?