Textbook #1: Lane et al. Introduction to Statistics, David M. Lane et al., 2013. ( http://onlinestatbook.com/Online_Statistics_Education.pdf ) Textbook #2: Illowsky et al. Introductory Statistics, Barbara Illowsky et al., 2013. ( http://openstaxcollege.org/files/textbook_version/hi_res_pdf/15/col11562-op.pdf ) Z-Tables (Normal Distribution) are at the end of this document Lane - Chapter 7: 8,11,12 8. Assume the speed of vehicles along a stretch of I-10 has an approximately normal distribution with a mean of 71 mph and a standard deviation of 8 mph. a. The current speed limit is 65 mph. What is the proportion of vehicles less than or equal to the speed limit? b. What proportion of the vehicles would be going less than 50 mph? c. A new speed limit will be initiated such that approximately 10% of vehicles will be over the speed limit. What is the new speed limit based on this criterion? d. In what way do you think the actual distribution of speeds differs from a normal distribution? 11. A group of students at a school takes a history test. The distribution is normal with a mean of 25, and a standard deviation of 4. (a) Everyone who scores in the top 30% of the distribution gets a certificate. What is the lowest score someone can get and still earn a certificate? (b) The top 5% of the scores get to compete in a statewide history contest. What is the lowest score someone can get and still go onto compete with the rest of the state? 12. Use the normal distribution to approximate the binomial distribution and find the probability of getting 15 to 18 heads out of 25 flips. Compare this to what you get when you calculate the probability using the binomial distribution. Write your answers out to four decimal places. Illowsky - Chapter 6 (60,66,76,88) 60. The patient recovery time from a particular surgical procedure is normally distributed with a mean of 5.3 days and a standard deviation of 2.1 days. What is the median recovery time? a. 2.7 b. 5.3 c. 7.4 d. 2.1 66. Height and weight are two measurements used to track a child's development. TheWorld Health Organization measures child development by comparing the weights of children who are the same height and the same gender. In 2009, weights for all 80 cm girls in the reference population had a mean = 10.2 kg and standard deviation = 0.8 kg. Weights are normally distributed. X ~ N(10.2, 0.8). Calculate the zscores that correspond to the following weights and interpret them. a. 11 kg b. 7.9 kg c. 12.2 kg 76. Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 250 feet and a standard deviation of 50 feet. a. If X = distance in feet for a fly ball, then X ~ _____(_____,_____) b. If one fly ball is randomly chosen from this distribution, what is the probability that this ball traveled fewer than 220 feet? c. Find the 80th percentile of the distribution of fly balls. Sketch the graph, and write the probability statement. 88. Facebook provides a variety of statistics on its Web site that detail the growth and popularity of the site. On average, 28 percent of 18 to 34 year olds check their Facebook profiles before getting out of bed in the morning. Suppose this percentage follows a normal distribution with a standard deviation of five percent. a. Find the probability that the percent of 18 to 34-year-olds who check Facebook before getting out of bed in the morning is at least 30. b. Find the 95th percentile, and express it in a sentence. Illowsky - Chapter 7 (62,70,96) 62. Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 250 feet and a standard deviation of 50 feet. We randomly sample 49 fly balls. Assume X-bar is the \"Random Variable\" X and is defined by the Central Limit Theorem a. If X-bar = average distance in feet for 49 fly balls, then X-bar ~ _______(_______,_______) b. What is the probability that the 49 balls traveled an average of less than 240 feet? c. Find the 80th percentile of the distribution of the average of 49 fly balls. 70. Which of the following is NOT TRUE about the distribution for averages? a. The mean, median, and mode are equal. b. The area under the curve is one. c. The curve never touches the x-axis. d. The curve is skewed to the right. 96. A typical adult has an average IQ score of 105 with a standard deviation of 20. If 20 randomly selected adults are given an IQ test, what is the probability that the sample mean scores will be between 85 and 125 points? Quiz 1 Scoring: Each problem is worth 10 points. Some problems contains multiple parts, but the total value of the problem will be 10 points. The total test score will be a maximum of 100 points. ========================================================================== (Problem 1) A random sample of 12 customers was chosen in a supermarket. The (incomplete) results for their checkout times are shown in the table below. Checkout Time (minutes) Frequency 4.0 - 5.9 6.0 - 7.9 8.0 - 9.9 10.0 - 11.9 12.0 - 13.9 TOTALS Relative Frequency Cumulative Relative Frequency 2 0.25 1 2 12 (a - 4 points) Complete the frequency table (b - 2 points) What percent of the checkout times are at least 10 minutes? (c - 2 points) What percent of the checkout times are between 8 and 10 minutes? (d - 2 points) What percent of the checkout times are less than 12 minutes? (Problem 2) Using the data from Problem #1 ... (a - 4 points) Construct a histogram ... you may draw it by hand, if desired. (b - 2 points) In what interval must the median lie? Assume the largest recorded checkout time was 13.2 minutes. Suppose that data point was incorrect and the actual checkout time was 13.8 minutes. Answer the following ... (c - 2 points) Will the mean of the dataset increase, decrease or remain the same and why? (d - 2 points) Will the median of the dataset increase, decrease or remain the same and why? (Problem 3) A fitness center is interested in the mean amount of time a group of clients exercise each week. A survey will be conducted of a group of clients. Answer the following questions (2 points each). (a) What is the population? (b) What is the sample? (c ) What is the parameter? (d) What is the statistic ? (e) What is the variable? (Problem 4) A random sample of starting salaries for an engineer are: $40,000, $40,000, $48,000, $55,000 and $67,000. Find the following and show all work (2 points each). Include equations, a table or EXCEL work, to show how you found your solution. (a) Mean (b) Median (c) Mode (d) Standard Deviation (e) If a recent graduate is considering a career in engineering, which statistic (mean or median) should they consider when determining the starting salary they are likely to make? Explain your answer. (Problem 5) The checkout times (in minutes) for 12 randomly selected customers at a large supermarket during the store's busiest time are as follows: 4.6, 8.5, 6.1, 7.8, 10.7, 9.3, 12.4, 5.8, 9.7, 8.8, 6.7, 13.2 HINT: DO NOT use EXCEL to calculate quartiles, as the method used by EXCEL is not the same as the standard method we use in our course. It is best to find the quartile values by hand. (a - 2 points) What is the mean checkout time? (b - 2 points) What is the value for the 25% percentile (first quartile) Q1? (c - 2 points) What is the value for the 50% percentile (median)? (d - 2 points) What is the value for the 75% percentile (third quartile) Q3? (e - 2 points) Construct a boxplot of the dataset. (Problem 6) Consider two standard dice where each die has six faces (numbered 1 to 6). (a - 2 points) List the number of outcomes in the sample space when you roll both dice. (b - 2 points) What is the probability of rolling a 2, or 3 or 4 with one die? (c - 2 points) You roll both dice, one at a time. What is the probability of rolling a 3 with the first die and an EVEN number with the second die? (d - 2 points) You roll both dice at the same time. What is the probability the sum of the two dice is less than 5? (e - 2 points) You roll both dice, one at a time. What is the probability that the second die is greater than 3, given that the first die is an odd number? Think about this one ... it is tricky. (Problem 7) In a box of 100 cookies, 36 contain chocolate and 12 contain nuts. Of those, 8 cookies contain both chocolate and nuts. (a - 3 points) Draw a Venn diagram representing the sample space and label all regions. You may draw the diagram by hand, if desired. (b - 1 points) What is the probability that a randomly selected cookie contains chocolate? (c - 3 points ) What is probability that a randomly selected cookie contains chocolate OR nuts? Note, it cannot contain both chocolate and nuts, but must have either chocolate OR nuts. (d - 3 points) What is the probability that a randomly selected cookie contains nuts, given that it contains chocolate? (Problem 8) Assume a baseball team has a lineup of 9 batters. (a - 4 points) How many different batting orders are possible with these 9 players? (b - 4 points) How many different ways can I select the first 3 batters? HINT: This is a permutation. (c - 2 points) Is a \"Combination Lock\" really a permutation or combination of numbers? Explain your answer. (Problem 9) You are playing a game with 3 prizes hidden behind 4 doors. One prize is worth $100, another is worth $40 and another $20. You have to pay $100 if you choose the door with no prize. (a - 4 points) Construct a probability table. See your homework for Illowsky, Chapter 4, #80. (b - 3 points) What is your expected winning? (c - 3 points) What is the standard deviation of your winning? (HINT: Use the expanded table, similar to your homework, Illowsky, Chapter 4, #80) (Problem 10) Suppose that 85% of graduating students attend their graduation. A group of 22 graduating students is randomly chosen. Let X be the number of students that attend graduation. As we know, the distribution of X is a binomial probability distribution. Answer the following: (a - 1 point) What are the number of trials (n)? (b - 1 point) What is the probability of successes (p)? (c - 1 point) What is the probability of failures (q)? (d - 2 points) How many students are expected to attend graduation? (e - 5 points) What is the probability that exactly 18 students attend graduation? (HINT: This is tougher than it looks. You will need use a Binomial Probability Function, so review Illowsky, Chapter 4, #88)