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Name:_____________________ Date: 7/01/2011 STAT 50 Exam 1 Please show ALL your work on the problems below. No more than 1 point will be given to problems if you only provide the correct answer and insufficient work. Some formulas you may need: x x n PA 1 P A x x s n 1 n x 2 x 2 2 P A B P A PB P A B n(n 1) P A B P A PB | A 1. (15 points) In order to figure out the percentage of students at PCC who smoke, Greg asked the students in his stats class and found that 8 out his 33 students smoke. a) Describe the population b) Describe the sample c) Describe (in words) the population parameter d) Describe (in words) the sample statistic e) What is the value of the sample statistic? 2. (27 points) Data: 41 35 77 54 41 21 62 For this data, find the a) mean b) median d) midrange e) range f) standard deviation c) mode g) variance h) In the context of this problem, are the numbers 35 and 54 far apart? Why or why not? 3. (6 points) Define a random sample and a simple random sample. 4. (3, 5, 5 points) Consider the procedure where you draw a single card from a standard poker deck. Let A be the event that you draw an Ace. a) Find P( A) b) What does your answer in part (a) mean? c) If you were to repeat this procedure a total of 26,000 times, how many times will an Ace be drawn? 5. (6 points) A meteorologist has determined that there is a 72% chance that it will be sunny tomorrow, a 45% chance that it will be windy tomorrow, and a 38% chance that it will be both sunny and windy tomorrow. Find a) The probability that it will be windy or sunny tomorrow b) The probability that it will not be sunny tomorrow 6. (15 points) Suppose you have data that has a normal distribution with mean 52 and standard deviation 4 . Fill in the blanks below: 68% of the data lies between _____ and _____ _____% of the data lies between 44 and 60 99.7% of the data lies between _____ and _____ 7. (39 points) A box contains 12 colored balls with numbers on them as shown below. Consider the procedure where you draw one ball out of the box. a) Find the sample space. b) Let B denote the event that you draw a blue ball from the box, let R denote the event that you draw a red ball from the box, let G denote the event that you draw a green ball from the box, let Y denote the event that you draw a yellow ball from the box and let E denote the event that you draw an even numbered ball from the box. Find the following: R E B Y E G P (G ) P(R ) P( E R) PE G P( E | G ) P(S ) c) Are the events E and G disjoint? Why or why not? d) Are the events E and G independent? Why or why not? Name:_____________________ Date: 7/08/2011 STAT 50 Exam 2 Please show ALL your work on the problems below. No more than 1 point will be given to problems if you only provide the correct answer and insufficient work. 1. (24 points) In this problem we are going to analyze the “3rd 12” bet in roulette. Locate the box labeled “3rd 12” in the picture above. If you place a bet in this box, you will win if any of the numbers between 25 and 36 (inclusive) come up and you will lose if anything else comes up. If you win, you will win double your bet and if you lose, you will lose the amount of your bet. Suppose you decide to bet $25 on the “3rd 12” box. Let X be the amount of money you will win when making this bet once. a) Find the probability distribution of X c) Find the standard deviation of X d) Explain the meaning of your answer in part (b) b) Find the expected value of X 2. (2, rest 6 points) Lucy hates her job and as a result often comes home in a bad mood because of it. It is estimated that on any given day, the probability that Lucy comes home 4 in a bad mood because of her job is . Let X be the number of times Lucy comes home 5 in a bad mood because of her job this work week (Monday through Friday). a) What is the name for the distribution of the random variable X? b) List the 6 things you should always list when working with this kind of random variable c) What is the probability that Lucy does not come home in a bad mood because of her job at all this week? d) What is the probability that Lucy comes home in a bad mood because of her job exactly 3 times this week? e) What is the probability that Lucy comes home in a bad mood because of her job at least once this week? (this is a continuation of problem 2) f) How many times do you expect Lucy to come home this week? g) Explain what your answer in part (d) means h) Explain what your answer in part (f) means i) What is the probability that Lucy comes home in a bad mood because of her job at least 3 times this week? (Hint: Rewrite the phrase “at least 3 times” using and, or, or not) 3. (10 points) Consider the procedure where you flip 3 coins one at a time. a) What is the sample space? b) Define your own random variable on this procedure 4. (12 points) Back in the ‘80s, the California Lottery was a game where 6 numbers are drawn out of a group of 49 (the numbers were 1-49). To play this game, you would buy a ticket where you would pick 6 numbers of your own and if your numbers matched the 6 numbers that were drawn, then you hit the jackpot. a) How many such tickets are possible? b) If you buy 20 different tickets, what is your probability of winning the jackpot? 5. (6 points) Suppose you have just rented 4 movies and you are excited to watch them, but can’t decide which order to watch them in. How many different orders are possible? 6. (18 points) At John’s sandwich shop, sandwiches are ordered by making 3 decisions: First the customer decides what kind of bread he wants, then he decides the kind of meat and finally he decides the kind of cheese. At John’s sandwich shop, there are 4 kinds of bread, 8 kinds of meat and 5 kinds of cheese. a) How many different sandwiches can be made? b) How many of these sandwiches contain cheddar cheese (cheddar cheese is one of the cheese options)? c) If you were to pick a sandwich at random, what is the probability that it will have cheddar cheese? Some formulas you may need: PA 1 P A P A B P A PB P A B P A B P A PB | A P A B P A PB P A B P A PB P(at least one) 1 P(none) n Pr EV xp(x) EV np n! (n r )! Var n x 2 Cr n! (n r )! r! p( x) 2 2 npq x npq 2 p ( x) 2 Name:_____________________ Date: 7/15/2011 STAT 50 Exam 3 Please show ALL your work on the problems below. No more than 1 point will be given to problems if you only provide the correct answer and insufficient work. 1. (24 points) The average number of cars you see stalled along side the road as you drive from Los Angeles to Las Vegas is 3.7. a) What is the probability that on your next trip to Vegas you will see 7 cars stalled along side the road? b) What is the probability that on your next trip to Vegas you will see at least one car stalled along side the road? c) What is the probability that on your next trip to Vegas you will see between 4 and 6 cars (inclusive) stalled along side the road? d) What is the probability that on your next trip to Vegas you will see at least 2 cars stalled along side the road? 2. (6, 4, 6 points) Suppose the random variable X has a uniform distribution on the interval [9, 14]. a) Find the value of c that makes this a probability distribution b) Find P ( X 11) c) Find P(10 X 12) 3. (12 points) Suppose X is a random variable whose density curve is given below. a) What are the possible values of X? b) Find P(35 X 8) 4. (6, 10 points) The lengths of pregnancies are normally distributed with a mean of 268 days and a standard deviation of 15 days. a) If one pregnant woman is randomly selected, find the probability that her length of pregnancy is less than 260 days. b) If 25 pregnant women are randomly selected, find the probability that their lengths of pregnancies have a mean between 260 days and 270 days. 5. (10, 6 points) There is an 80% chance that a prospective employer will check the educational background of a job applicant. For 100 randomly selected job applicants, a) find the probability that between 80 and 92 (inclusive) of these applicants have their educational background checked b) find the probability that exactly 85 of these applicants have their educational backgrounds checked 6. (4, 4, 4, 6, 10, 8 points) In order to figure out the percentage of registered voters who are planning to vote for President Obama in the next election, a statistics student polls everyone he sees at the Cheesecake Factory and finds that 47 of the 72 registered voters at Cheesecake will vote for Obama in the next election. a) What is the population? b) What is the sample? c) What is the population parameter? (in words and symbol) d) What is the best point estimate for the population parameter we are trying to estimate? e) Find a 97% confidence interval for the percentage of registered voters who plan on voting for Obama in the next election. f) What does the 97% in a 97% confidence interval mean? Some formulas you may need: PA 1 P A P A B P A PB P A B P A B P A PB P(at least one) 1 P(none) P( X x) n C x p x q n x P( X x) X X e x x! X npq 2 npq np Z X n X E z / 2 pˆ qˆ n Name:_____________________ Date: 7/21/2011 STAT 50 Exam 4 Please show ALL your work on the problems below. No more than 1 point will be given to problems if you only provide the correct answer and insufficient work. 1. (20, 4 points) Stella wants to sell expensive jewelry at a booth at PCC. However, before she sets up a booth, she wants to know if students at PCC are too broke to buy her jewelry. So Stella decides she wants to figure out the average amount of money PCC students bring with them to school. She asked 25 PCC students how much money they had on them and after collecting all the data, she determined that the average amount of money these 25 students had on them was $17.34 with a standard deviation of $4.21. a) Find a 90% confidence interval for the average amount of money all PCC students bring with them to school. b) In the context of this problem, explain what the 90% means in a 90% confidence interval. 2. (24 points) The weights of 100 different M&M’s were measured and have a mean weight of 0.85 grams with a standard deviation of 0.0518 grams. Construct a 95% confidence interval for the variance of the weights of all M&M’s. 3. (24 points) Adults were randomly selected for a Newsweek poll. They were asked if they “favor or oppose using federal tax dollars to fund medical research using stem cells obtained from human embryos.” Of those polled, 481 were in favor and 401 were opposed. A politician claims that people don’t really understand the stem cell issue and their responses to such questions are random responses equivalent to a coin toss. Use a 0.01 significance level to test the claim that the proportion of subjects who respond in favor is equal to 0.5. 4. (24 points) The Coca-Cola bottling company has developed a new machine that they claim is much more consistent when it fills cans of Coke. The standard deviation for the amount of Coke filled in their cans by their older machines is estimated to be about 0.15 ounces. To determine if this new machine is better, the new machine was used to fill 100 cans of Coke. The standard deviation of the amount of Coke in the cans from this sample is 0.12 ounces. Use a 0.05 significance level to test the claim that the standard deviation of the amount of Coke filled in their cans by this new machine is less than 0.15 ounces. 5. (24 points) A scientist developed a pill that is supposed to help students become better at math. To test the pill, a sample of 350 total students who failed the math placement exam at PCC were selected and divided into 2 groups, a treatment group and a placebo group. Each of the 350 students is supposed to take one pill everyday and take PreAlgebra. The treatment group consists of 192 people and 134 of them passed PreAlgebra. Of the 158 people in the placebo group, 97 passed Pre-Algebra. Use a 0.05 significance level to determine if people who take this pill have a better chance of passing Pre-Algebra than those who don’t take the pill. Some formulas you may need: E z / 2 E t / 2 z n 0.25 / 2 E pˆ qˆ n E z / 2 n 1s 2 n 1s 2 2 2 s Z 2 R n pˆ p pq n z Z t n pˆ 1 pˆ 2 p1 p 2 pq pq n1 n2 n n 1s 2 L X z n /2 E R2 X s 2 2 n p x1 x2 n1 n2 n 1s 2 (n 1) s 2 2 q 1 p 2 L2 Name:_____________________ Date: 7/28/2011 STAT 50 Exam 5 Please show ALL your work on the problems below. No more than 1 point will be given to problems if you only provide the correct answer and insufficient work. 1. (24 points) A new weight loss pill has come on the market the company that manufactures it claims that you can eat anything you want all day long as long as you take one of these pills a day right before you go to sleep. The company claims that you will see the loss in 30 days. 700 people volunteered to do a double blind experiment to test if the drug really works or not. The subjects were randomly selected to be in either the treatment group or in the placebo group. Over the course of 30 days, the 382 members of the treatment group had an average weight loss of 14.8 lbs with a standard deviation of 6.4 lbs, while the 318 members of the placebo group had an average weight loss of 4.9 lbs with a standard deviation of 5.9 lbs. Use a hypothesis test at the 0.05 significance level to test the claim that people who take this diet pill have a mean weight loss higher than those who don’t and thus that the pill actually works. 2. (24 points) Some people claim that Diet Coke contains a chemical that makes people have to pee much quicker than if they drink regular Coke. To test this claim, Greg had some of his students drink a can of Diet Coke after class and timed how long it took them to have to use the restroom. This experiment was repeated on another day with the same students using Coke instead of Diet Coke. The results are summarized in the table below. Use a 0.05 significance level to test the claim that Diet Coke makes people have to pee sooner than regular Coke. Student Name Time it took to use the restroom (Diet Coke) Time it took to use the restroom (Coke) Karen 7 min 15 min Upa 18 min 21 min Laura 3 min 2 min Camille 9 min 12 min Doug 16 min 16 min 3. (24 points) These days it seems that most Armenian women get married around 18 years old whereas Armenian men get married at all sorts of ages. In a simple random sample of 30 Armenian women, the average age at which they got married (for the first time) was 19.3 years old with a standard deviation of 1.1 years old. In another simple random sample of Armenian men, the average age at which they got married (for the first time) was 26.9 years old with a standard deviation of 4.8 years old. Use a 0.05 significance level to test the claim that the ages at which Armenian women get married (for the first time) vary less that the ages at which Armenian men get married (for the first time). 4. (6, 10, 10, 6, 6, 10 points) In order to study the relationship between how much people smoke and how long it takes to develop lung cancer, 10 smokers who already have lung cancer were asked how many cigarettes they smoke in a day and how old they were when they first developed lung cancer. The data is summarized below. No, of cigarettes smoked per day (x) Age at which they first developed lung cancer (y) 3 68 10 61 16 48 24 44 5 68 18 49 8 59 10 59 In order to facilitate the rest of the calculations, here are some of the calculations already done for you: 2 x 118 x 1,742 y 562 y 2 32,192 xy 6,194 a) Find r b) Use a 0.05 significance level to test the claim that there is a linear relationship between x and y. 12 54 12 52 c) Find the equation of the least squares regression line for this data d) Find the best point estimate for the age at which a person who smokes 20 cigarettes per day is likely to develop lung cancer. e) Repeat part (d) if your conclusion in part (b) was false. f) Find a 95% prediction interval for the age at which a person who smokes 20 cigarettes per day is likely to develop lung cancer. Some formulas you may need: z x1 x2 1 2 12 n1 t x1 x2 1 2 s 2p n1 s 2p 22 12 n1 22 n2 n2 s 2p n1 1s12 n2 1s22 n1 1 n2 1 df n1 n2 2 n2 E t / 2 t E z / 2 x1 x2 1 2 s12 s 22 n1 n2 s 2p n1 s 2p n2 df is the smaller of n1 1 and n2 1 E t / 2 s12 s 22 n1 n 2 t d d sd E t / 2 sd df n 1 n n F s12 s 22 df1 n1 1 df 2 n2 1 n x 2 x 2 s r b1 n xy x y n x x 2 2 se 2 n2 2 2 y x x xy n x x 2 b0 n x 2 x 2 b0 y b1 xy yˆ b1 x b0 n y y n xy x y t y nn 1 r 1 r2 n2 2 2 df n 2 n x 0 x 1 n n x 2 x 2 2 E t / 2 s e 1 Name:___________________ Date: 6/28/2011 STAT 50 Quiz 1 1. (5 points) In order to determine the percentage of people in California who drink energy drinks, a Statistics student goes to lunch at Panda Express and asks the first 100 people he sees there if they drink energy drinks or not. Of the people surveyed, 34 of them said that they do drink energy drinks. a) Describe the population b) Describe the sample c) Describe (in words) the population parameter d) Describe (in words) the sample statistic e) What is the value of the sample statistic? 2. (2 points) Explain the difference between stratified sampling and cluster sampling. 3. (3 points) The final exam scores for my last statistics class are below (out of 250). Make a relative frequency table and relative frequency histogram for the data using classes of length 25 whose first class starts with 100. 193 238 120 183 218 108 173 244 240 180 162 189 151 141 196 196 205 195 180 159 Name:___________________ Date: 6/29/2011 STAT 50 Quiz 2 Some formulas you may need: x x 1. (8 points) Here is some data: 29 s 41 22 n 1 31 29 57 n x 2 x 2 2 n 26 x x nn 1 41 For this data, find the a) mean b) median d) midrange e) range f) standard deviation c) mode g) variance 2. (1 point) Suppose a set of data has a standard deviation of s 0.1 . With that in mind, are the data points 52 and 55 considered close together, or far apart? Explain why! 3. (1 point) Suppose you know that your data has a normal distribution with a mean 73 and a standard deviation of 8 . Fill in the blank below: ______ % of the data is between the values of 57 and 89 Name:___________________ Date: 6/30/2011 STAT 50 Quiz 3 1. (6 points) A bag contains 8 scrabble letters. The letters in the bag are: E1 K5 J8 Q10 P3 C3 G2 I1 The procedure is to draw a single letter from the bag without looking in the bag. The number next to each letter tells you how many points each letter is worth. Let A be the event that a vowel is drawn, and let B be the event that a letter worth 5 or more points. a) A b) B d) P( A) e) P(B) c) S f) P(S ) 2. (2 points) In order for the weatherman to determine if it is going to rain today, he looked at past records and noticed that out of 1287 days in the past that had all the same conditions as today has, it has rained on 821 of those days. What is the probability that it rains today? 3. (2 points) Consider the procedure where you draw a single card from a standard deck 1 of 52 poker cards. The probability of drawing a spade is (or 25%). 4 a) Explain as clearly as possible what this probability means. b) If you drew a card from the deck, then replaced the card back into the deck, shuffled and drew a card again and continued this 40,000 times, how many times would you draw a spade? Name:___________________ Date: 7/05/2011 Some formulas you may need: P( A B) P( A) P( B) STAT 50 Quiz 4 P( A B) P( A) P( B | A) P(at least one) 1 P(none) 1. (1, 1, 2, 1, 1, 1 points) An unprepared student is about to take a 5 question multiple choice exam in his stats class. Each of the 5 questions contains 3 choices and only one of the choices is the correct answer. Since the student has not studied, he is going to guess the answer to each question at random. a) What is the probability that the student answers the first question correctly b) What is the probability that the student answers the first question incorrectly c) Let I 1 be the event that the student answers the first question incorrectly and let I 2 be the event that the student answers the second question incorrectly. Are the events I 1 and I 2 independent? Why or why not? d) What is the notation for the probability that the student gets all 5 questions on the test incorrect (use the same notation from part (c) ( I 1 , I 2 , I 3 and so forth)). e) What is the probability that the student gets all 5 of the questions on the test incorrect? f) What is the probability that the student gets at least on question correct? 2. (2, 1 points) Consider the procedure where you draw 7 cards from a standard poker deck one at a time without replacement. a) What is the probability that you do not draw a single spade? b) What is the probability that you draw at least one spade? Name:___________________ Date: 7/06/2011 STAT 50 Quiz 5 Some formulas you may need: n Pr n! (n r )! n Cr n! (n r )! r! 1. (2, 1 points) a) How many samples of size 5 can be constructed out of a population consisting of 43 members? b) Suppose you want to select a SIMPLE RANDOM SAMPLE of size 5 from a population of 43 people. What is the probability that a given sample will be selected? 2. (2, 1 points) In horse racing, betting a trifecta means picking which horse will come in first, which will come in second, and which will come in third in the correct order. a) How many different trifecta bets are possible in a 9 horse race? b) If you make 7 trifecta bets in a 9 horse race, what is the probability that you will win one of these bets? 3. (2 points) How many 7 character license plates are there where the 2nd, 3rd and 4th characters are letters (A-Z), the rest of the characters are numbers (0-9), and no letter can be used more than once (but the numbers can repeat). 4. (2 points) A husband, wife, their baby boy and their dog go to a picture place to have their family photo taken. The photographer insists on lining the 4 up in a straight line, but is having trouble arranging them in a way that will give the best looking photo. So he takes one picture for every possible arrangement to make sure he hasn’t overlooked a good arrangement. How many photos did the photographer take? Name:___________________ Date: 7/07/2011 STAT 50 Quiz 6 Some formulas you may need: EV xp(x) Var x 2 p( x) 2 x 2 p ( x) 2 1. (6 points) In this problem, we are going to analyze the field bet in craps. The game of craps is played by rolling a pair of dice and observing the total of the two numbers on the dice. Locate the section in the picture above labeled FIELD. When you bet on the field, the die are rolled once and you will win if any of the totals 2, 3, 4, 9, 10, 11, or 12 are rolled and you will lose if any other total is rolled. More specifically, you will win the amount of your wager (you will win $5 if you bet $5) if you roll a 3, 4, 9, 10, or 11. Notice that the numbers 2 and 12 are circled. That’s because you win more if those totals are rolled. If you roll total of 2, you will win double your wager and if you roll a 12, you will triple your wager. Suppose you bet $100 on the field and let the random variable X be the amount of money you will win on a single roll of the dice. a) Find the probability distribution for X. (Hint: When finding P( X 100) , there are lots of ways for X to equal 100. Make sure to count them all) (problem 1 continued) b) Find the expected value of X. c) Find the standard deviation of X. e) Explain in words the meaning of your answer in part (b). f) Is this a good bet to make? Why or why not? 2. (2 points) Consider the procedure of drawing a single card from a poker deck. Define a random variable X on this procedure. 3. (2 points) What are the requirements of a probability distribution? Name:___________________ Date: 7/11/2011 STAT 50 Quiz 7 Some formulas you may need: P A B P A PB e x P( X x) x! P A B P A PB P A B 2 1. (1, 1, 1, 1, 1, 2, 3 points) The average number of earthquakes in California in one year that are over magnitude 3.0 is 78. Let X be the number of earthquakes in California in one year that are over 3.0 in magnitude. a) What is the name for the distribution of X? b) What are the possible values of X? c) What is the mean of X? d) What is the variance of X? e) What is the standard deviation of X? f) What is the probability that there will be 80 earthquakes in California over magnitude 3.0 next year? g) What is the probability that there will be between 77 and 80 (inclusive) earthquakes in California over magnitude next year? Name:___________________ Date: 7/12/2011 STAT 50 Quiz 8A,B 1. (1, 1, 1, 2, 2, 2, 1 points) Suppose Z has a standard normal distribution. a) What are the possible values for Z? b) What is the mean of Z? c) What is the standard deviation of Z? d) What is P( Z 1.52) ? e) What is P ( Z 0.42) ? f) What is P(1.71 Z 0.88) ? g) What is P ( Z 0.63) ? 2. (2 points) State the requirements for a density curve. 3. (2 points) Suppose X is a random variable with the density curve drawn below. a) What are the possible values of X? b) What is P(1 X 7) ? 4. (2 points) Suppose X is uniformly distributed over the interval [3,13]. a) Find c that makes this a probability density. b) Find P(0 X 7) 5. (4 points) Suppose the heights of 40 year old males is normally distributed with a mean of 70 inches and a standard deviation of 6 inches. What is the probability that a randomly selected 40 year old man’s height is between 64 inches and 76 inches? (Hint: Let X denote the height of a 40 year old male) Name:___________________ Date: 7/13/2011 STAT 50 Quiz 9 1. (1, 1, 2 points) a) What kind of object is the population parameter ? b) What kind of object is the sample statistic x ? c) Fill in the blanks: Since EV (x ) , x is an _______________ _______________ of . 2. (4 points) State the Central Limit Theorem 3. (2 points) Consider the procedure where you roll a pair of dice and let X denote the average of the two numbers rolled on the dice. Find P( X 4.5) Name:___________________ Date: 7/14/2011 Some formulas you may need: X X STAT 50 Quiz 10 X X n np npq 1. (2, 2, 1 points) The Coca-Cola company uses machines to fill their coke cans. The machines are set to fill the cans with 12 ounces of Coke but for various reasons, some cans get less than 12 ounces and some get more than 12 ounces. In fact, the amount of Coke in a can is normally distributed with a mean of 12 ounces and a standard deviation of 0.2 ounces. a) If one can of Coke is randomly selected, what is the probability that the amount of Coke in the can is less than 11.9 ounces? b) If 64 cans of Coke are randomly selected, what is the probability that the mean amount of Coke in the 64 cans is less than 11.9 ounces? c) If the amount of Coke in one can did not have a normal distribution, could part (b) be done in the same way? Explain as clearly as possible why or why not. 2. (2, 3 points) A gambler is about to play the game of Roulette 300 times, each time betting on red. a) Find the probability that the gambler wins at least 120 times b) Find the probability that the gambles wins between 135 times and 161 times. Name:___________________ Date: 7/18/2011 Some formulas you may need: STAT 50 Quiz 11 E z / 2 n z n /2 E 2 1. (2, 5, 3 points) A simple random sample of 50 adults (including males and females) is obtained, and each person’s red blood cell count (in cells per microliter) is measured. The sample mean is 4.63. The population standard deviation for red blood cell counts is 0.54. a) Find the best point estimate of the mean red blood cell count of all adults b) Construct a 99% confidence interval estimate of the mean red blood cell count of adults. c) What is the minimum sample size necessary so that the margin of error for a 99% confidence interval is 0.1? Name:___________________ Date: 7/19/2011 STAT 50 Quiz 12 Some formulas you may need: E t / 2 s n n 1s 2 n 1s 2 2 2 R L n 1s 2 R2 2 n 1s 2 L2 1. (1, 4, 1, 4 points) In a test of the Atkins weight loss program, 40 individuals participated in a randomized trial with overweight adults. After 12 months, the mean weight loss was found to be 2.1 lb, with a standard deviation of 4.8 lb. a) What is the best point estimate of the mean weight loss of all overweight adults who follow the Atkins program? b) Construct a 99% confidence interval estimate of the mean weight loss for all such subjects (this is a continuation of problem 1) c) What is the best point estimate of the standard deviation of the amount of weight loss of all overweight adults who follow the Atkins program? d) Construct a 95% confidence interval estimate of the standard deviation of the amount of weight loss for all such subjects Name:___________________ Date: 7/20/2011 A formula you may need: STAT 50 Quiz 13 z pˆ p pq n 1. (5 points) Cheating Gas Pumps When testing gas pumps in Michigan for accuracy, fuel-quality enforcement specialists tested pumps and found that 1299 of them were not pumping accurately (accurately means within 3.3 oz when 5 gallons are pumped), and 5686 pumps were accurate. Use a 0.01 significance level to test the claim of an industry representative that less than 20% of Michigan gas pumps are inaccurate. 2. (5 points) Driving and Cell Phones In a survey, 1640 out of 2246 randomly selected adults in the United States said that they use cell phones while driving. Use a 0.05 significance level to test the claim that the proportion of adults who use cell phones while driving is equal to 75%. Name:___________________ Date: 7/21/2011 Some formulas you may need: STAT 50 Quiz 14 z x n x t s 2 n 1s 2 2 n 1. (3 points) Weights of Pennies The U.S. mint has a specification that pennies have a mean weight of 2.5 g. A simple random sample of 37 pennies manufactured after 1983 was taken and those pennies have a mean weight of 2.49910 g and a standard deviation of 0.01648 g. Use a 0.05 significance level to test the claim that this sample is from a population with a mean weight equal to 2.5 g. 2. (3 points) The Doritos company claims that the average number of chips in their small bags of chips is 20 chips. A disgruntled consumer claims that Doritos is cheating their customers and are putting less chips in their bags than they should. To this end, a sample of 49 bags of chips is obtained and the average number of chips in this sample is 18.3 chips. Suppose that the standard deviation of the number of chips in all Doritos bags is 3.5 chips. Use a 0.01 significance level to test the claim that the mean number of chips in small Doritos bags is less than 20 chips. 3. (4 points) A can of Coke is supposed to contain 12 oz’s of soda. However, the amount of soda in each can will vary somewhat. The Coca-Cola company claims that the standard deviation of the amount of soda in their cans is 0.15 oz’s. To make sure that the amount of soda in each can is consistent and that their machines used to fill the cans are working properly, an inspector measures the amount of soda in 49 cans and finds that the standard deviation of the sample is 0.21 ounces. Test the claim that the standard deviation of the amount of soda in a can of Coke is larger than 0.15 oz’s at the 0.05 significance level. Is the Coca-Cola company lying? Do their filling machines need upgrading? Name:___________________ Date: 7/26/2011 STAT 50 Quiz 15 1. (2, 6, 2 points) A simple random sample of 13 four-cylinder cars is obtained, and the braking distances are measured. The mean braking distance is 137.5 ft and the standard deviation is 5.8 ft. A simple random sample of 12 six-cylinder cars is obtained and the braking distances have a mean of 136.3 ft with a standard deviation of 9.7 ft. a) Describe the two populations b) Construct a 90% confidence interval estimate of the difference between the mean braking distance of four-cylinder cars and the mean breaking distance of six-cylinder cars c) Use the confidence interval from part (b) to test the claim that the mean braking distance of four-cylinder cars is greater than the mean breaking distance of six-cylinder cars at the 0.05 significance level Some formulas you may need: z x1 x2 1 2 12 n1 t x1 x2 1 2 s 2p n1 s 2p 22 12 n1 22 n2 n2 s 2p n1 1s12 n2 1s22 n1 1 n2 1 df n1 n2 2 n2 E t / 2 t E z / 2 x1 x2 1 2 s12 s 22 n1 n2 s 2p n1 s 2p n2 df is the smaller of n1 1 and n2 1 E t / 2 s12 s 22 n1 n 2 Name:___________________ Date: 7/27/2011 STAT 50 Quiz 16 1. (5 points) Is Friday the 13th Unlucky? Researchers collected data on the number of hospital admissions resulting from motor vehicle crashes, and results are given below for Fridays on the 6th of a month and Fridays on the following 13th of the same month. Use a 0.05 significance level to test the claim that when the 13th day of a month falls on a Friday, the number of hospital admissions from motor vehicle crashes is increased. Friday the 6th: Friday the 13th: 9 13 6 12 11 14 11 10 3 4 5 12 2. (5 points) Testing Effects of Alcohol Researchers conducted an experiment to test the effects of alcohol. The errors were recorded in a test of visual and motor skills for a treatment group of 22 people who drank ethanol and another group of 22 people given a placebo. The errors for the treatment group have a standard deviation of 2.20, and the errors from the placebo group have a standard deviation of 0.72. Use a 0.05 significance level to test the claim that the treatment group has errors that vary more than the errors of the placebo group. Some formulas you may need: t d d sd E t / 2 sd df n 1 n n s12 F 2 s2 df1 n1 1 df 2 n2 1 n x 2 x 2 s nn 1 Name:___________________ Date: 7/28/2011 STAT 50 Quiz 17 Some formulas you may need: r b1 n xy x y n x 2 x n xy x y yˆ b1 x b0 n y 2 y 2 n x 2 x 2 2 y x x xy n x x 2 b0 2 2 1. (4, 4, 2) The data below are blood pressure measurements of 5 patients taken once from their left arms and once from their right arms (measured in millimeters of Mercury (mm Hg)). Use x to denote a patient’s right arm blood pressure and let y denote a patient’s left arm blood pressure. Right Arm x Left Arm y a) Find r 102 175 101 169 94 182 79 146 79 144 b) Find the least squares regression line for the data c) Use the least squares regression line to predict a patient’s blood pressure in their left arm if the blood pressure in their right arm is 100 mm Hg. Name:___________________ Date: 8/2/2011 STAT 50 Quiz 18 Some formulas you may need: 2 O E 2 E df k 1 E n k E pn 1. (5 points) The table below lists the frequency of wins for different post positions in the Kentucky Derby horse race. A post position of 1 is closest to the inside rail, so that horse has the shortest distance to run. Use a 0.05 significance level to test the claim that the likelihood of winning is the same for the different post positions. Based on the results, should bettors consider the post position of a horse racing in the Kentucky Derby? Post Position Wins 1 19 2 14 3 11 4 14 5 14 6 7 7 8 8 11 9 5 10 11 Leading Digit Benford's Law distribution of leading digits 1 30.1% 2 17.6% 3 12.5% 4 9.7% 5 7.9% 6 6.7% 7 5.8% 8 5.1% 9 4.6% 2. (5 points) Amounts of political contributions are randomly selected, and the leading digits are found to have frequencies of 52, 40, 23, 20, 21, 9, 8, 9, and 30. (Those observed frequencies correspond to the leading digits of 1, 2, 3, 4, 5, 6, 7, 8, and 9 respectively). Using a 0.01 significance level, test the observed frequencies for goodness-of-fit with Bedford’s law. Does it appear that the political contributions are legitimate? Name:___________________ Date: 8/3/2011 STAT 50 Quiz 19 Some formulas you may need: 2 O E 2 E row total column total E grand total df r 1c 1 1. (10 points) Which Treatment Is Better? A randomized controlled trial was designed to compare the effectiveness of splinting versus surgery in the treatment of carpal tunnel syndrome. Results are given in the table below. The results are based on evaluations made one year after the treatment. Using a 0.01 significance level, test the claim that success is independent of the type of treatment. What do the results suggest about treating carpal tunnel syndrome? Splint Treatment Surgery Treatment Successful Treatment 60 67 Unsuccessful Treatment 23 6 Name:___________________ Date: 8/4/2011 STAT 50 Quiz 20 Some formulas you may need: V .B.S. ns X F 2 V .B.S . V .W .S . n x 2 x 2 df (numerator) k 1 s df (deno min ator) k (n 1) 2 nn 1 1. (10 points) In order to investigate the effectiveness of different diets, 40 people were randomly selected who were on various diets for a year. The amount of weight the people lost and what diets they were on is summarized in the table below. Use a 0.05 significance level to test the claim that a person’s average weight loss is the same for the various diet plans listed in the table. x s2 Weight Watchers Atkins Jenny Craig Nutrisystem 16 41 22 28 62 59 26 31 35 55 32 24 23 17 17 11 18 48 32 18 33 41 25 23 37 29 28 23 25 44 19 39 21 37 11 31 42 49 17 28 31.2 42 22.9 25.6 192.4 152 48.1 59.6