Statistics - CSW - Meinke Problem Set 1 Name__________________________ Problem Set Covers Lessons 6 to 9 Sections 3.1, 3.2 3.3, 3.4, 3.5, 4.1, 4.2, 4.3, 5.1, 5.2, 5.4, 6.1, and 6.2 in Openstax Introduction to Statistics This assignment is graded for correctness and will be included in the Test category of your grade. 45 points Instructions: Show all your work. Indicate clearly the methods you use, because you will be graded on the correctness of your methods as well as on the accuracy and completeness of your results and explanations. Your numerical answers must be written in context of the question with the appropriate units. When required, show the calculation of z-scores and draw and shade Normal Curve where appropriate. If you are using a function on your calculator or in Desmos to complete the calculation. Write the function and the parameters used. For example : If I use normalcdf on my calculator I will write normalcdf(lower:30, upper:50, : 32, : 4.2) = 0.68302 µ σ If i use normaldistr in Desmos I will write normaldist(32,4.2) min: 30, max: 50 = 0.68302 You may work with your classmates. You may ask for help from me during class or during ofʨice hours. You may not get help from anyone other than members of this class or me. 1. In some courses (but certainly not in an Introduction to Statistics course!), students are graded on a “Normal curve.” For example, students between 0 and 0.5 standard deviations above the mean receive a C+; between 0.5 and 1.0 standard deviations above the mean receive a B –; between 1.0 and 1.5 standard deviations above the mean receive a B; between 1.5 and 2.0 standard deviations above the mean receive a B+, etc. The class average on an exam was 60 with a standard deviation of 10. a. What is the lowest and highest grade that will earn a B. b. What is the proportion of students who will receive a B if the marks are actually Normally distributed? Sketch and shade a normal distribution curve.

2. From past experience, the owner of Paddy’s Sports Store finds that the mean number of Celtics hats sold in a sales campaign is 845, the standard deviation is 15, and a histogram of the demand is approximately Normal. The owner is willing to accept a 2.5% chance that the Celtics hat will be sold out. About how many Celtics hats should the owner order for an upcoming sales campaign? Sketch and shade a normal distribution curve. 3. Lamar is shopping for a used car, and he’s interested in determining the typical mileage on cars that are three or four years old. He looks at an online car-buying site and compares the number of miles on 30 cars that are three years old to 30 cars that are four years old. His results are summarized by Minitab below. All values are in thousands of miles. Descriptive Statistics: Mileage on Four year old cars in thousands of miles Sample size n=30 cars Mean 56.68 Standard deviation 17.82 Minimum 23.60 Q1 47.80 Median 54.70 Q3 64.50 Maximum 100.30 Descriptive Statistics: Mileage on Three year old cars in thousands of miles Sample size n=30 cars Mean 33.33 Standard deviation 12.70 Minimum 14.10 Q1 22.33 Median 32.10 Q3 39.23 Maximum 66.40 Both distributions are approximately Normally distributed. a. One car that Lamar is interested in is four years old and has been driven 40 thousand miles. Another one is three years old and has 30 thousand miles on it. How does the number of miles on these cars compare, relative to other cars of the same age? Provide appropriate statistical calculations to support your answer. (provide calculation work and 2 to 4 sentences)

5. The two-way table below gives information on seniors and juniors at a high school and the way they get to school. Car Bus Walk Total Juniors 146 106 48 300 Seniors 146 64 40 250 Total 292 170 88 550 a. You select one student from this group at random. What is the probability that this student typically takes a bus to school? Show your work. b. Given that a student is a junior, what is the probability that they walk to school? Show your work. c. You select one student from this group at random. Let event W = Student Typically walks to school, and J = Student is a Junior i. Explain why or why not the events W and J are mutually exclusive. ii. Explain why or why not the events W and J are independent. Support your answer with numerical evidence. 6. An airline estimates that the probability that a random call to their reservation phone line results in a reservation being made is 0.31. This can be expressed as P(call results in reservation) = 0.31. Assume each call is independent of other calls. Describe what the Law of Large Numbers says in the context of this situation..

7. Wile E. Coyote is pursuing the Road Runner. The Road Runner chooses his route randomly. The probability that he takes the high road is 0.8. The probability that he takes the low rode is 0.2. If he takes the high road, the probability that Wile E. catches him is 0.01. If he takes the low road, the probability that Wile E. catches him is 0.05. Find the probability that the Road Runner is caught. Draw a tree diagram or a two way table. 8. At a grocery store the probability that a randomly-chosen shopper buys apples is 0.21, that the shopper buys potato chips is 0.36, and that the shopper buys both apples and potato chips is 0.09. (a) Let A = Randomly-chosen shopper buys apples, C = Randomly-chosen shopper buys potato chips. Draw a Venn diagram or create a two-way table that summarizes the probabilities above. (b) What is the probability that a randomly-selected shopper does not buy apples and does not buy potato chips. Show how you arrived at your answer.

11. Binomial Setting? A binomial distribution will be the correct distribution for one of these settings and not for the other. Briefly discuss both settings and why or why not they are binomial. (B. I. N. S.) a. According to New Jersey Transit, the 8:00 A.M. weekday train from Princeton to New York City has a 90% chance of arriving on time on a randomly selected day. Suppose this claim is true. Choose 6 days at random. Let W = the number of days on which the train arrives late. b. Exactly 10% of the students in a school with 1500 students are left-handed. Select students at random from the school, one at a time, until you find one who is left-handed. Let V = number of students chosen 12. A company that rents DVDs from vending machines in grocery stores has developed the following probability distribution function for the random variable X = the number of DVDs a customer rents per visit to a machine. (a) Find and interpret with a sentence the mean (expected value) of X. Show you work. (b) Find and interpret with a sentence the standard deviation of X. Show your work.

13. Over the course of the last five seasons, a professional soccer player has succeeded in scoring a goal on 84% of his penalty kicks. Assume that the success of each kick is independent. Assume that each penalty kick has the same probability of success. (a) What is the probability that he scores exactly 7 goals in the next 10 penalty kicks? Start by defining a discrete random variable. Show your work. (b) What is the probability that he scores 5 or fewer of his next 10 penalty kicks? Show your work. (c) Suppose that our soccer player is out of action with an injury for several weeks. When he returns, he only scores on 5 of his next 10 penalty kicks. Is this evidence that his success rate is now less than 84%? Explain.