Additional practice with these ideas can be found in the problems for Tintle Section P.1.1

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1 Psych 10 / Stats 60, Practice Problem Set 3 (Week 3 Material) Part 1: Decide if each variable below is quantitative, ordinal, or categorical. If the variable is categorical, also decide whether or not it is binary. a. Telephone number b. Height, measured in inches c. Star rating of a hotel (e.g., one star, two stars, etc.) d. College year (freshman, sophomore, junior, or senior) e. Whether or not a student is an engineering major f. Number of questions asked during a lecture g. Whether or not a person left a tip h. State where a person was born Additional practice with these ideas can be found in the problems for Tintle Section P.1.1 Part 2: Describing distributions 1. Below are two distributions of favorite primary color for Group A and Group B. Compare the central tendency and the variability of the two distributions. Color Group A: Frequency Group B: Frequency red yellow blue Imagine you d like to determine a representative value in the distribution of a quantitative variable. Describe one reason why you might prefer to use the mean, one reason why you might prefer to use the median, and one reason why you might prefer to use the mode. 3. Suppose we have a distribution of times for how long it takes each student to get from their residence to our classroom ( commute times ). For each question, determine whether it is best answered by looking at a measure of central tendency, a measure of variability, or a measure of shape. a. How long does it take a typical student to get from their residence to our classroom? b. Do students tend to have similar or different commute times from each other? c. Do we have multiple subgroups (types) of commutes within our class? d. If I select a random student, how well will I be able to predict their commute time? e. Are commute times normally distributed?

2 4. Distributions of income are usually positively skewed. Describe why this might be the case. 5. Distributions of GPA are usually negative skewed. Describe why this might be the case. 6. Without doing any math, which set of scores has the smallest standard deviation? a. 11, 17, 31, 53 b. 5, 11, 42, 22 c. 145, 143, 145, 147 d. 27, 105, 10, For each sample of values below, calculate the mean, median, mode (if there is one), range, sum of squares, variance, and standard deviation without using an R functions (although vector operations might be very useful). Then, double check your answers using R functions. [A hint for R: an easy mistake to make is to forget to create a vector using c() before using any functions like mean() or sd(); mean(1,2,3) does not produce the same output as mean(c(1,2,3))] a. 5, 1, 5, 5, 4 b. 81, 86, 81, 82, 81 c. 9, 4, 0, 6, 1, 4 8. For each pair of bets below, calculate the expected value and variance of each bet, and explain which one you would prefer to take and why. Pair A: Bet #1: We take a Jack, Queen, King, and Ace and shuffle them, and you pick a card at random. If you select the Jack you lose $10, if you select the Queen you lose $5, and if you select the King or Ace you win $10. Bet #2: We take a Jack, Queen, and King and shuffle them, and you pick a card at random. If you select the Jack you lose $10, if you select the Queen you lose $1, and if you select the King you win $8. Pair B: Bet #1: You roll a six-sided die. If you roll a 6 then you win $100, but if you roll anything else you lose $20. Bet #1: You roll a six-sided die. If you roll a 5 or 6 then you win $10, if you roll a 3 or 4 you get nothing, and if you roll a 1 or 2 you lose $10.

3 9. Apple sells three-year insurance for new computers for $249; this insurance provides free repairs in case the computer breaks. Suppose the price to repair a broken computer is always exactly $300, and that a computer will break 0 or 1 times in the first three years of owning it. How likely would it need to be that the computer will break for it to be a good idea, in the long-run, to buy the insurance? Describing locations in distributions 10. Below is a frequency table describing a distribution of n = 200 scores on a 5-point exam. a. Just by looking at the frequency column, describe the shape of the distribution. b. Fill in the rest of the table to include relative frequency and cumulative frequency for each score. c. What is the quantile corresponding to a cumulative frequency of.19? Score Frequency (f) Relative frequency (rf) Cumulative frequency (cf) We started looking at the cumulative distribution function in lecture. The plot to the right shows the cumulative distribution functions for two distributions (A in solid blue lines, B in dashed orange lines). Based on this plot, can you determine which distribution has more variability? (Hint: one place to start would be to try to estimate the interquartile range for each distribution, i.e., find the quantiles that correspond to cumulative frequencies of.25 and.75). cdf x distribution A B 12. Convert each value to a z-score if we have a distribution with a mean of 10 and a standard deviation of 1.5. Values: 5, 6, 8, 10, 13, Convert each value to a z-score if we have a distribution with a mean of 10 and a standard deviation of 3. Values: 5, 6, 8, 10, 13, Convert each value to a z-score if we have a distribution with a mean of -4 and a standard deviation of 7. Values: -22, -8, -4, 1, 16

4 Transformation of variables 15. Suppose that a set of exam scores has a mean of 80 with a standard deviation of 6. Calculate the mean and standard deviation of the new, transformed distributions of exam scores if each scenario below happens. You also might want to think about what will happen to the shape of the distribution. a. Each score is increased by 12 b. Each score is increased by 20% c. Each score is reduced so it is only 75% of what it used to be d. Each score is decreased by 10 e. Each score is increased by 20% and then 5 extra points are added to each score 16. Diana runs a mean of 5 kilometers a day, with a standard deviation of 1 kilometer. What is the mean number of miles she runs per day, and the standard deviation of miles run each day. (There are.62 miles in a kilometer). 17. Imagine that students bring a mean of 4 framed photographs to campus, with a standard deviation of 2 photographs. If students are randomly assigned to live in quad-style dorm rooms (four students per room), what is the mean and standard deviation of the distribution of total number of framed photographs per dorm room? 18. Imagine that instead, in the scenario above, we calculated the mean number of photographs per student in each dorm room, and examined the distribution of these means. What is the mean and standard deviation of this distribution of mean number of photographs per student in each dorm room? 19. Suppose bags of M&Ms have a mean number of 2 green candies per fun-sized (mini) bag, with a standard deviation of 3. If these fun-sized bags are randomly grouped and sold in bulk packages of 20, what is the mean and standard deviation of the distribution of total number of green candies in a bulk package? 20. Suppose that instead, in the scenario above, we calculated the mean number of green candies per fun-sized bag for each bulk package (in other words, for each bulk package, we count the total number of green candies per fun pack, and then take the mean of these values), and examined the distribution of these means. What is the mean and standard deviation of this distribution of mean number of green candies per fun-sized bag in each bulk package?

5 Extra challenge problems (these types of problems are more challenging than what will appear on the quiz, but thinking about them will help strengthen your understanding of the material): 21. We have a room full of people and the distribution of heights for these people has a mean of 68 inches and a standard deviation of 4 inches. We also have a group of hats, and the distribution of heights for these hats has a mean of 12 inches with a standard deviation of 5 inches. If each person randomly selects a hat and puts it on and we measure the height of each person + hat pair, what is the mean and standard deviation of this distribution of heights? 22. On any given day, the probability that Roman will eat breakfast at the dining hall is.20, the probability that he will eat lunch at the dining hall is.80, and the probability that he will eat dinner at the dining hall is.60. We ll assume that these outcomes are independent, i.e., knowing that he ate breakfast at the dining hall one day doesn t give us any extra information about whether he ate lunch at the dining hall that same day. We re interested in the distribution of number of meals eaten at the dining hall per day. What is the mean and standard deviation of this distribution? (Hint: calculate the mean and variance for the three distributions number of meals per day separately for breakfast, lunch, and dinner (0 or 1), and then think about the mean and variance of the sum of these variables).