Exam 3: Randomness in Data

Suppose $80 \%$ of the incoming email messages for a college's computer system are spam. -Display your answer "Because you are considering a large sample size, the Central Limit Theorem predicts the sampling distribution of sample proportions of messages that are spam is approximately normal, centered at 80 with standard deviation $\sqrt{.8(.2) / 200}$ or .0283 . So the $z$ -score for a proportion of .75 equals $(.75-.8) / .0461$ or -1.77 . The probability lying to the right (exceeding .75) equals .9616." as a shaded area in a well-labeled sketch.

Suppose the number of French fries in a small bag at a particular fast-food restaurant follows a normal model with mean 49 fries and standard deviation 5 fries. -Use this model to estimate the proportion of the bags that contain at least 40 fries.

In general, which vary more: averages or individual observations?

If you bet one dollar on a color in a spin of a roulette wheel, then you will either win one dollar or lose one dollar. The probability is $20 / 38$ (about .526) that you will lose one dollar and 18/38 (about .474 ) that you will win one dollar. a. Explain what it means to say that the probability of winning one dollar is 474 , as if to someone with no formal knowledge of statistics. b. Consider losing one dollar to mean that your winnings are (_1) dollar. Determine the expected value of your winnings from betting one dollar on one spin of a roulette wheel. c. Write a sentence or two interpreting what this expected value means in this context.

According to the "quick facts" listed on Cal Poly's website in fall 2011, the proportion of Cal Poly undergraduates who are in the College of Business is. Suppose that you select a simple random sample of 150 Cal Poly undergraduates, and let $\hat{p}$ represent the sample proportion who are in the College of Business. -Describe the (approximate) sampling distribution of $\hat{p}$ . Be sure to refer to shape, center, and variability.

Give an intuitive explanation for "Averages based on many observations vary less than averages based on a few observations".

Suppose that a tire manufacturer believes that the lifetimes of its tires follow a normal distribution with mean 48,000 miles and standard deviation 5,000 miles. -Determine the probability that a randomly selected tire lasts for more than 55,000 miles.

Suppose that a tire manufacturer believes that the lifetimes of its tires follow a normal distribution with mean 48,000 miles and standard deviation 5,000 miles. -Produce a well-labeled sketch of this normal distribution.

According to the "quick facts" listed on Cal Poly's website in fall 2011, the proportion of Cal Poly undergraduates who are in the College of Business is. Suppose that you select a simple random sample of 150 Cal Poly undergraduates, and let $\hat{p}$ represent the sample proportion who are in the College of Business. -Check the conditions for whether the normal approximation (CLT) is reasonable to apply in this case.

Suppose that a family has five children, consisting of three girls and two boys. Call the girls Abigail, Brianna, and Courtney; call the boys Duane and Evan. Suppose further that two of these children are to be selected at random. -List the sample space of all possible pairs of these five children.

Suppose a phone company reports that the average duration of a cell phone call is 1.7 minutes with a standard deviation of 1.4 minutes. -Use the CLT to approximate the probability that in a random sample of 60 calls, the sample average duration would be less than 2 minutes.

Reconsider the activities involving the sampling of Reese's Pieces candies. Continue to suppose that $45 \%$ of all Reese's Pieces are orange. -Does sampling variability refer to a statistic, a parameter, or both?

Scores on the verbal ability portion of the Graduate Record Examination (GRE) follow a normal distribution with mean 500 and standard deviation 115 . a. Between what two values do the middle $95 \%$ of scores fall? b. If your score is 650 , then you did better than what percentage of exam takers? c. Suppose you study a random sample of 100 GRE verbal exams. Determine the probability that the average (mean) score in this sample will be between 485 and 515.

Suppose that a phone company reports that the average duration of a cell phone call is 1.7 minutes, with a standard deviation of 1.4 minutes. -Suppose you want to examine a random sample of 60 cell phone calls. Do you think it would be reasonable to use the Central Limit Theorem to describe the sampling distribution of the sample mean call duration? Explain.

Suppose that a tire manufacturer believes that the lifetimes of its tires follow a normal distribution with mean 48,000 miles and standard deviation 5,000 miles. -Determine the $z$ -score corresponding to 55,000 miles.

Suppose that two employees have to be fired from a group of seven: - Amy, female, age 35 - Bob, male, age 41 - Chad, male, age 43 - Diana, female, age 44 - Elvis, male, age 49 - Frank, male, age 61 - Ginger, female, age 62 Suppose for now that the company selects the two employees to be fired at random, so all possible pairs of people have the same probability of being fired. a. List all outcomes in the sample space. In other words, list all possible pairs of people that can be chosen from this group of seven people. (To save time and space, just use initials. For example, write AB to represent the (Amy, Bob) pair.) b. Determine the probability (both as a fraction and a decimal) that both employees selected are female. c. Determine the probability (both as a fraction and a decimal) that the average age of the employees selected is older than 50. d. Determine the probability that both employees selected are older than 60 . Suppose that Frank and Ginger are the employees selected to be fired. They proceed to file a lawsuit, alleging that the company is guilty of age discrimination for choosing the two oldest employees to fire. The company responds that it did not discriminate; it simply happened to choose these two by random chance. e. Does your probability analysis provide evidence to support the allegation of age discrimination? Explain (with 1-3 sentences) the reasoning process behind your answer.

Reconsider the activities involving the sampling of Reese's Pieces candies. Continue to suppose that $45 \%$ of all Reese's Pieces are orange. -Are you more likely to get between $35 \%$ and $55 \%$ orange candies if you take a random sample of 40 candies or a random sample of 400 candies? Explain briefly.

Suppose that $85 \%$ of the email messages arriving to a college's computer system are spam. You want to take a random sample of incoming email messages and perform probability calculations involving the sample proportion that are spam. How large must the sample size be so you can reasonably apply the Central Limit Theorem?

Give brief (one-sentence) explanations for your answers to the following. -In which case is it more likely for the proportion of heads to be close to one-half: if you flip a fair coin twice or if you flip a fair coin 200 times?

Suppose that $15 \%$ of all students at your school ride a bicycle to class each day. Suppose further that you take a random sample of 80 students at your school. a. Is $15 \%$ number a parameter or a statistic? Explain briefly. b. Are you guaranteed that $15 \%$ of your sample ride a bicycle to class each day? Explain briefly. c. Describe the sampling distribution of the sample proportion of these 80 students who ride a bicycle to class each day. (Hint: Be sure to refer to the shape, center, and spread of this sampling distribution.) Also include a well-labeled sketch of this distribution.