Statistics Chapter 6 Homework

6.1 Determine the required value of the missing probability to make the distribution a discrete probability distribution. x : P(x) 3 : 0.32 4 : ? 5 : 0.22 6 : 0.27 P(4) = ______ ​(Type an integer or a​ decimal.)

0.19

6.3 The random variable X follows a Poisson process with the given mean. Assuming μ=9, compute the following. ​(a) ​P(6​) ​(b) ​P(X<6​) ​(c) ​P(X≥6​) ​(d) ​P(3≤X≤7​) ​(a) ​P(6​)≈___ ​(Do not round until the final answer. Then round to four decimal places as​ needed.) ​(b) ​P(X<6​)≈___ ​(Do not round until the final answer. Then round to four decimal places as​ needed.) (c) ​P(X≥6​)≈____ ​(Do not round until the final answer. Then round to four decimal places as​ needed.) (d) ​P(3≤X≤7​)≈____ ​(Do not round until the final answer. Then round to four decimal places as​ needed.)

(a) 0.0911 ​(b)0.1157 ​(c)0.8843 ​(d)0.3177

6.3 The number of hits to a Web site follows a Poisson process. Hits occur at the rate of 1.7 per minute between​ 7:00 P.M. and 10​:00 P.M. Given below are three scenarios for the number of hits to the Web site. Compute the probability of each scenario between 8:36 P.M. and 8​:42 P.M. and interpret the results. ​(a) exactly eight hits ​(b) fewer than eight hits ​(c) at least eight hits ​(a)​ P(8​)=____ ​(Round to four decimal places as​ needed.) Fill in the blanks to complete the statement below. On about _______ of every 100​ days, it is expected that there will be ______ 8 hits to the Web site between 8:36 P.M. and 8​:42 P.M. ​(b) ​ P(X<8​)=___ ​(Round to four decimal places as​ needed.) On about ___ of every 100​ days, it is expected that there will be fewer than ___ hits to the Web site between 8:36 P.M. and 8​:42 P.M. ​(c) ​ P(X≥8​)=____ ​(Round to four decimal places as​ needed.) On about ___ of every 100​ days, it is expected that there will be at least ___ hits to the Web site between 8:36 P.M. and 8​:42 P.M.

(a) 0.1080 11; exactly (b) 0.2027 20, fewer than (c) 0.7973 80, at least

6.2 Suppose that a recent poll found that 58​% of adults believe that the overall state of moral values is poor. Complete parts​ (a) through​ (c). ​(a) For 200 randomly selected​ adults, compute the mean and standard deviation of the random variable​ X, the number of adults who believe that the overall state of moral values is poor. The mean of X is ____.​ (Round to the nearest whole number as​ needed.) The standard deviation of X is ___. ​(Round to the nearest tenth as​ needed.) ​(b) Interpret the mean. Choose the correct answer below. A. For every 200 ​adults, the mean is the minimum number of them that would be expected to believe that the overall state of moral values is poor. B. For every 200 ​adults, the mean is the range that would be expected to believe that the overall state of moral values is poor. C. For every 200 ​adults, the mean is the number of them that would be expected to believe that the overall state of moral values is poor. D. For every 116 ​adults, the mean is the maximum number of them that would be expected to believe that the overall state of moral values is poor. (c) Would it be unusual if 113 of the 200 adults surveyed believe that the overall state of moral values is​ poor? Yes No

(a) 116 7 (b) C. For every 200 ​adults, the mean is the number of them that would be expected to believe that the overall state of moral values is poor. Note: The mean of a discrete random variable can be thought of as the mean outcome of the probability experiment if the experiment is repeated many times. (c) no

6.2 Suppose that a recent poll found that 66​% of adults believe that the overall state of moral values is poor. Complete parts​ (a) through​ (c). ​(a) For 150 randomly selected​ adults, compute the mean and standard deviation of the random variable​ X, the number of adults who believe that the overall state of moral values is poor. The mean of X is ___.​ (Round to the nearest whole number as​ needed.) The standard deviation of X is____. ​(Round to the nearest tenth as​ needed.) ​(b) Interpret the mean. Choose the correct answer below. A. For every 150 ​adults, the mean is the number of them that would be expected to believe that the overall state of moral values is poor. B. For every 150 ​adults, the mean is the minimum number of them that would be expected to believe that the overall state of moral values is poor. C. For every 99 ​adults, the mean is the maximum number of them that would be expected to believe that the overall state of moral values is poor. D. For every 150 ​adults, the mean is the range that would be expected to believe that the overall state of moral values is poor. (c) Would it be unusual if 101 of the 150 adults surveyed believe that the overall state of moral values is​ poor? No Yes

(a) 99 5.8 (b) A. For every 150 ​adults, the mean is the number of them that would be expected to believe that the overall state of moral values is poor. (c) No​ because, provided that np(1−​p)≥​10, the interval μ−2σ to μ+2σ gives an interval of​ "usual" observations, and 101 is in that interval.

6.2 According to​ flightstats.com, American Airlines flights from Dallas to Chicago are on time 80​% of the time. Suppose 13 flights are randomly​ selected, and the number of​ on-time flights is recorded. ​(a) Explain why this is a binomial experiment. ​(b) Determine the values of n and p. ​(c) Find and interpret the probability that exactly 8 flights are on time. ​(d) Find and interpret the probability that fewer than 8 flights are on time. ​(e) Find and interpret the probability that at least 8 flights are on time. ​(f) Find and interpret the probability that between 6 and 8 ​flights, inclusive, are on time. ​(a) Identify the statements that explain why this is a binomial experiment. Select all that apply. A. There are two mutually exclusive​ outcomes, success or failure. B. The probability of success is the same for each trial of the experiment. C. The probability of success is different for each trial of the experiment. D. The trials are independent. E. The experiment is performed a fixed number of times. F. There are three mutually exclusive possible​ outcomes, arriving​ on-time, arriving​ early, and arriving late. G. The experiment is performed until a desired number of successes are reached. H. Each trial depends on the previous trial. ​(b) Using the binomial​ distribution, determine the values of n and p. n=____ p= ____ ​(Type an integer or a decimal. Do not​ round.) ​(c)1. Using the binomial​ distribution, the probability that exactly 8 flights are on time is ____ ​(Round to four decimal places as​ needed.) 2. Interpret the probability. In 100 trials of this​ experiment, it is expected that about ____ will result in exactly 8 flights being on time. ​(Round to the nearest whole number as​ needed.) ​(d) Using the binomial​ distribution, the probability that fewer than 8 flights are on time is _____. ​(Round to four decimal places as​ needed.) In 100 trials of this​ experiment, it is expected that about ____ will result in fewer than 8 flights being on time. (Round to the nearest whole number as​ needed.) ​(e) Using the binomial​ distribution, the probability that at least 8 flights are on time is _____. (Round to four decimal places as​ needed.) Interpret the probability. In 100 trials of this​ experiment, it is expected that about ____ will result in at least 8 flights being on time. ​(Round to the nearest whole number as​ needed.) ​(f) Using the binomial​ distribution, the probability that between 6 and 8 ​flights, inclusive, are on time is ____. ​(Round to four decimal places as​ needed.) Interpret the probability. In 100 trials of this​ experiment, it is expected that about ____ will result in between 6 and 8 ​flights, inclusive, being on time. ​(Round to the nearest whole number as​ needed.)

(a) A, B, D, E (b) n=13 p=.8 (c)1. 0.0691 2. 7 Note: The previously calculated probability value is for 1 trial of this experiment. Determine the factor to multiply this previous value by to compute the number of trials in 100 trials expected to have the corresponding result in the experiment. (d)0.03 3 (e)0.97 97 (f)0.0979 10

6.2 According to​ flightstats.com, American Airlines flights from Dallas to Chicago are on time 80​% of the time. Suppose 11 flights are randomly​ selected, and the number of​ on-time flights is recorded. ​(a) Explain why this is a binomial experiment. ​(b) Determine the values of n and p. ​(c) Find and interpret the probability that exactly 9 flights are on time. ​(d) Find and interpret the probability that fewer than 9 flights are on time. ​(e) Find and interpret the probability that at least 9 flights are on time. ​(f) Find and interpret the probability that between 7 and 9 ​flights, inclusive, are on time.

(a) An experiment is said to be a binomial experiment if each of the following statements are true. 1. The experiment is performed a fixed number of times. Each repetition of the experiment is called a trial. 2. The trials are independent. This means the outcome of one trial will not affect the outcome of the other trials. 3. For each​ trial, there are two mutually exclusive​ (disjoint) outcomes, success or failure. 4. The probability of success is the same for each trial of the experiment. Determine whether the experiment satisfies the given criteria. This experiment is performed a fixed number of times since there are 11 trials​ (the 11 randomly selected​ flights). The trials of this experiment are independent because each flight is randomly selected and no​ flight's arrival depends on any other flight. There are only two mutually exclusive​ outcomes, either the flight is on time or not. The probability of success​ (on-time flight) is the same for each trial because flights on this route are on time 80​% of the time. Since the four requirements are​ satisfied, the probability experiment represents a binomial experiment. ​(b) In a binomial probability​ experiment, n is the number of trials and p is the probability of success. Although there are multiple ways to calculate the​ probability, this example will use a binomial probability throughout. Identify the given values for n and p. n=11 and p=80​% Now convert the percent to a decimal. 80​%=0.8 ​So, the values are n=11 and p=0.8. ​(c) To find the probability that exactly 9 flights are on​ time, remember that the probability of obtaining x successes in n independent trials of a binomial​ experiment, where the probability of success is​ p, is given by the formula below. P(x)=nCxpx(1−p)n−x x=​0,​1, 2,​ ..., n While either technology or the formula can be used to find the​ probability, for this​ problem, use technology. Determine ​P(9​) with n=11 and p=0.8​, rounding to four decimal places. ​P(9​)=0.2953 The​ probability, 0.2953​, is for 1 trial of this experiment. Multiply by 100 to compute the number of trials of this experiment in 100 that are expected to result in the desired outcome. Use this information to reach the proper conclusion. ​(d) The phrase​ "fewer than" means​ "less than."​ Thus, the probability that fewer than 9 flights are on time is equal to P(X<9). Use technology to find P(X<9)​, rounding to four decimal places. ​P(X<9​)=0.3826 The​ probability, 0.3826​, is for 1 trial of this experiment. Multiply by 100 to compute the number of trials of this experiment in 100 that are expected to result in the desired outcome. Use this information to reach the proper conclusion. ​(e) Note that the phrase​ "at least" means​ "greater than or equal​ to." Therefore, the probability that at least 9 flights are on time is equal to P(X≥9). Use technology to find P(X≥9)​, rounding to four decimal places. P(X≥9)=0.6174 The​ probability, 0.6174​, is for 1 trial of this experiment. Multiply by 100 to compute the number of trials of this experiment in 100 that are expected to result in the desired outcome. Use this information to reach the proper conclusion. ​(f) The word​ "inclusive" means​ "including," so determine the probability that 7​, 8​, or 9 flights are on time. ​P(7 or 8 or 9​)=​P(7≤x≤9​) Use technology to find the​ probability, rounding to four decimal places. ​P(7≤x≤9​)=0.6275 The​ probability, 0.6275​, is for 1 trial of this experiment. Multiply by 100 to compute the number of trials of this experiment in 100 that are expected to result in the desired outcome. Use this information to reach the proper conclusion.

6.1 (a) Which variable is the explanatory​ variable? A. Asking price B. Square footage C. Determining the value of a home D. Number of homes ​(b) Draw a scatter diagram of the data. Choose the correct graph below. A. A scatter diagram has a horizontal axis labeled "Square Footage" from 1000 to 1600 in increments of 100 and a vertical axis labeled "Asking Price (Thousands)" from 140 to 200 in increments of 10. The following 9 approximate points are plotted, listed here from left to right: (1100, 160); (1140, 154); (1140, 169); (1290, 170); (1320, 170); (1340, 180); (1470, 180); (1490, 190); (1540, 189). The points follow the general pattern of a line that rises from left to right. B. A scatter diagram has a horizontal axis labeled "Square Footage" from 1000 to 1600 in increments of 100 and a vertical axis labeled "Asking Price (Thousands)" from 140 to 200 in increments of 10. The following 9 approximate points are plotted, listed here from left to right: (1100, 189); (1140, 180); (1140, 190); (1290, 180); (1320, 170); (1340, 169); (1470, 170); (1490, 154); (1540, 160). The points follow the general pattern of a line that falls from left to right. C. A scatter diagram has a horizontal axis labeled "Square Footage" from 1000 to 1600 in increments of 100 and a vertical axis labeled "Asking Price (Thousands)" from 140 to 200 in increments of 10. The following 9 approximate points are plotted, listed here from left to right: (1100, 170); (1140, 169); (1140, 174); (1290, 175); (1320, 170); (1340, 180); (1470, 180); (1490, 190); (1540, 189). The points follow the general pattern of a line that rises from left to right. ​(c) Determine the linear correlation coefficient between square footage and asking price. The linear correlation coefficient between square footage and asking price is r=_____. ​(Round to three decimal places as​ needed.) ​(d) Is there a linear relation between the square footage and asking​ price? a. No b. Yes ​(e) Find the​ least-squares regression line treating square footage as the explanatory variable. Choose the correct answer below. A. The​ least-squares regression line is y=83.53x−0.06843. B. The​ least-squares regression line is y=83.53x+0.06843. C. The​ least-squares regression line is y=0.06843x+83.53. D. The​ least-squares regression line is y=−0.06843x+83.53. ​(f) Interpret the slope. Choose the correct answer below. A. For each square foot added to the​ area, the expected asking price of the house will decrease by ​$68.43 ​(that is, 0.06843 thousand​ dollars). B. For each square foot added to the​ area, the expected asking price of the house will increase by ​$68.43 ​(that is, 0.06843 thousand​ dollars). C. For each square foot added to the​ area, the expected asking price of the house will decrease by ​$83.53 ​(that is, 0.08353 thousand​ dollars). D. It is not appropriate to interpret the slope. ​(g) Is it reasonable to interpret the​ y-intercept? a. Yes b. No ​(h)1. One home that is 1,092 square feet is listed at​ $189,900. Is this​ home's price above or below average for a home of this​ size? a. Above average b. Below average 2. May there be some reasons for this​ price? a. Yes b. No

(a) B. Square footage (b)A. A scatter diagram has a horizontal axis labeled "Square Footage" from 1000 to 1600 in increments of 100 and a vertical axis labeled "Asking Price (Thousands)" from 140 to 200 in increments of 10. The following 9 approximate points are plotted, listed here from left to right: (1100, 160); (1140, 154); (1140, 169); (1290, 170); (1320, 170); (1340, 180); (1470, 180); (1490, 190); (1540, 189). The points follow the general pattern of a line that rises from left to right. (c)0.921 (d)b. Yes (e)C. The​ least-squares regression line is y=0.06843x+83.53. (f)B. For each square foot added to the​ area, the expected asking price of the house will increase by ​$68.43 ​(that is, 0.06843 thousand​ dollars). (g)b. no (h) 1. a. Above average 2. a.Yes

6.1 Determine whether the random variable is discrete or continuous. In each​ case, state the possible values of the random variable. ​(a) The number of hits to a website in a day. ​(b) The amount of snowfall. ​(a) Is the number of hits to a website in a day discrete or​ continuous? A. The random variable is continuous. The possible values are x≥0. B. The random variable is discrete. The possible values are x=​0, ​1, ​2,.... C. The random variable is discrete. The possible values are x≥0. D. The random variable is continuous. The possible values are x=​0, ​1, ​2,... ​(b) Is the amount of snowfall discrete or​ continuous? A. The random variable is discrete. The possible values are s≥0. B. The random variable is continuous. The possible values are s=1, 2, 3, .... C. The random variable is discrete. The possible values are s=1, 2, 3, .... D. The random variable is continuous. The possible values are s≥0.

(a) B. The random variable is discrete. The possible values are x=​0, ​1, ​2,.... ​(b) D. The random variable is continuous. The possible values are s≥0.

6.1 Determine whether the random variable is discrete or continuous. In each​ case, state the possible values of the random variable. ​(a) The number of people with blood type A in a random sample of 11 people. ​(b) The time required to download a file from the Internet. ​(a) Is the number of people with blood type A in a random sample of 11 people discrete or​ continuous? A. The random variable is discrete. The possible values are 0≤x≤11. B. The random variable is continuous. The possible values are 0≤x≤11. C. The random variable is discrete. The possible values are x=​0, ​1, ​2,..., 11. D. The random variable is continuous. The possible values are x=​0, ​1, ​2,..., 11. ​(b) Is the time required to download a file from the Internet discrete or​ continuous? A. The random variable is continuous. The possible values are t=1, 2, 3, .... B. The random variable is continuous. The possible values are t>0. C. The random variable is discrete. The possible values are t=1, 2, 3, .... D. The random variable is discrete. The possible values are t>0.

(a) C. The random variable is discrete. The possible values are x=​0, ​1, ​2,..., 11. (b) B. The random variable is continuous. The possible values are t>0.

6.1 Determine whether the random variable is discrete or continuous. In each​ case, state the possible values of the random variable. ​(a) The number of points scored during a basketball game. ​(b) The time it takes to fly from City A to City B. ​(a) Is the number of points scored during a basketball game discrete or​ continuous? A. The random variable is discrete. The possible values are x≥0. B. The random variable is continuous. The possible values are x=​0, ​1, ​2,.... C. The random variable is discrete. The possible values are x=​0, ​1, ​2,.... D. The random variable is continuous. The possible values are x≥0. (b) Is the time it takes to fly from City A to City B discrete or​ continuous? A. The random variable is discrete. The possible values are t>0. B. The random variable is discrete. The possible values are t=1, 2, 3, .... C. The random variable is continuous. The possible values are t=1, 2, 3, .... D. The random variable is continuous. The possible values are t>0.

(a) C. The random variable is discrete. The possible values are x=​0, ​1, ​2,.... (b) D. The random variable is continuous. The possible values are t>0. Note: A discrete random variable has either a finite or countable number of values. The values of a discrete random variable can be plotted on a number line with space between each point. A continuous random variable has infinitely many values. The values of a continuous random variable can be plotted on a line in an uninterrupted fashion.

6.1 One of the biggest factors in determining the value of a home is the square footage. The accompanying data represent the square footage and asking price​ (in thousands of​ dollars) for a random sample of homes for sale. Complete parts​ (a) through​ (h). Click the icon to view the data table. Click the icon to view a table of critical values for the correlation coefficient. ​(a) Which variable is the explanatory​ variable? A. Asking price B. Number of homes C. Determining the value of a home D. Square footage (b) ​(b) Draw a scatter diagram of the data. Choose the correct graph below. A. A scatter diagram has a horizontal axis labeled "Square Footage" from 1000 to 1600 in increments of 100 and a vertical axis labeled "Asking Price (Thousands)" from 140 to 200 in increments of 10. The following 9 approximate points are plotted, listed here from left to right: (1100, 189); (1140, 180); (1140, 190); (1290, 180); (1320, 170); (1340, 169); (1470, 170); (1490, 154); (1540, 160). The points follow the general pattern of a line that falls from left to right. B. A scatter diagram has a horizontal axis labeled "Square Footage" from 1000 to 1600 in increments of 100 and a vertical axis labeled "Asking Price (Thousands)" from 140 to 200 in increments of 10. The following 9 approximate points are plotted, listed here from left to right: (1100, 160); (1140, 154); (1140, 169); (1290, 170); (1320, 170); (1340, 180); (1470, 180); (1490, 190); (1540, 189). The points follow the general pattern of a line that rises from left to right. C. A scatter diagram has a horizontal axis labeled "Square Footage" from 1000 to 1600 in increments of 100 and a vertical axis labeled "Asking Price (Thousands)" from 140 to 200 in increments of 10. The following 9 approximate points are plotted, listed here from left to right: (1100, 170); (1140, 169); (1140, 174); (1290, 175); (1320, 170); (1340, 180); (1470, 180); (1490, 190); (1540, 189). The points follow the general pattern of a line that rises from left to right. ​(c) Determine the linear correlation coefficient between square footage and asking price. The linear correlation coefficient between square footage and asking price is r=____. ​(Round to three decimal places as​ needed.) ​(d) Is there a linear relation between the square footage and asking​ price? a. No b. Yes ​(e) Find the​ least-squares regression line treating square footage as the explanatory variable. Choose the correct answer below. A. The​ least-squares regression line is y=83.53x−0.06843. B. The​ least-squares regression line is y=−0.06843x+83.53. C. The​ least-squares regression line is y=0.06843x+83.53. D. The​ least-squares regression line is y=83.53x+0.06843. (f) Interpret the slope. Choose the correct answer below. A. For each square foot added to the​ area, the expected asking price of the house will decrease by ​$68.43 ​(that is, 0.06843 thousand​ dollars). B. It is not appropriate to interpret the slope. C. For each square foot added to the​ area, the expected asking price of the house will increase by ​$68.43 ​(that is, 0.06843 thousand​ dollars). D. For each square foot added to the​ area, the expected asking price of the house will decrease by ​$83.53 ​(that is, 0.08353 thousand​ dollars). (g) Is it reasonable to interpret the​ y-intercept? a. Yes b. No (h)1. One home that is 1,094 square feet is listed at​ $189,900. Is this​ home's price above or below average for a home of this​ size? a. Below average b. Above average 2. May there be some reasons for this​ price? a. No b. Yes

(a) D. Square footage Note: The value of the response variable can be explained by the value of the​ explanatory, or​ predictor, variable. Determine the goal of the study and identify the variable that may affect the response variable. (b)B. A scatter diagram has a horizontal axis labeled "Square Footage" from 1000 to 1600 in increments of 100 and a vertical axis labeled "Asking Price (Thousands)" from 140 to 200 in increments of 10. The following 9 approximate points are plotted, listed here from left to right: (1100, 160); (1140, 154); (1140, 169); (1290, 170); (1320, 170); (1340, 180); (1470, 180); (1490, 190); (1540, 189). The points follow the general pattern of a line that rises from left to right. (c) 0.921 (d) b. Yes Note: Yes because the correlation coefficient is greater than the critical value for correlation when n=9. (e) C. The​ least-squares regression line is y=0.06843x+83.53. Note: The​ least-squares regression line is the line that minimizes the sum of the squared errors​ (or residuals). It is the line that minimizes the sum of the squared vertical distance between the observed values of y and those predicted by the​ line, y. The equation of the​ least-squares regression line is given by the formula y=b1x+b0 where b1=r•sy / sx is the slope of the​ least-squares regression​ line, r is the correlation​ coefficient, b0 is the​ y-intercept of the​ least-squares regression​ line, x is the sample mean and sx is the sample standard deviation of the explanatory variable​ x, and y is the sample mean and sy is the sample standard deviation of the response variable y. To find the equation of the​ least-squares regression​ line, use the formula or technology. Use the Tech Help button for further assistance. (f) C. For each square foot added to the​ area, the expected asking price of the house will increase by ​$68.43 ​(that is, 0.06843 thousand​ dollars). Note: To interpret the​ slope, let y=b1x+b0. The slope of the regression line is b1. If x increases by 1​ unit, the predicted​ value, y​, increases by b1 ​units, on average. Note that if the slope b1 is​ negative, y decreases as x increases. (g) b. No Note: No because a square footage of 0 is not reasonable. (h) 1. b. Above average 2. b. yes

6.2 Suppose that a recent poll found that 39​% of adults believe that the overall state of moral values is poor. Complete parts​ (a) through​ (c).

(a) For 600 randomly selected​ adults, compute the mean and standard deviation of the random variable​ X, the number of adults who believe that the overall state of moral values is poor. Calculate μx. Use the formula μx=np. μx=234 Calculate σx. Use the formula σx= sqrt [np(1−p)]. σx=11.9 ​(b) Interpret the mean. The mean of a discrete random variable can be thought of as the mean outcome of the probability experiment if the experiment is repeated many times. In this​ case, for every 600 ​adults, about 234 of them would be expected to believe that the overall state of moral values is poor. ​(c) Would it be unusual if 212 of the 600 adults surveyed believe that the overall state of moral values is​ poor? Provided that ​np(1−​p)≥​10, the interval μ−2σ to μ+2σ gives an interval of​ "usual" observations. Note that 600•0.39​(1−0.39)=142.74>10. ​Thus, this requirement is met. Determine μ−2σ​, rounding to the nearest integer. μ−2σ = 234−2(11.9) = 210 Determine μ+2σ​, rounding to the nearest integer. μ+2σ = 234+2(11.9) = 258 ​Thus, the interval of​ "usual" observations is between 210 and 258 adults surveyed who believe that the overall state of moral values is poor. An outcome resulting in 212 successes is not unusual because 210<212<258​, where 210 and 258 are the lower and upper bounds of the​ "usual" observations.

6.2 A binomial experiment is performed a fixed number of times. What is each repetition of the experiment​ called? For each repetition of a binomial​ experiment, what are the two mutually exclusive​ outcomes? (a) A binomial experiment is performed a fixed number of times. What is each repetition of the experiment​ called? a. Success b. Mean c. Trial d. Binomial random variable (b) For each repetition of a binomial​ experiment, what are the two mutually exclusive​ outcomes? a. ​Over/Under ​b. Yes/No ​c. Success/Failure ​d. Left/Right

(a) c.Trial Note: ​Trial because, to be a binomial​ experiment, the experiment is performed a fixed number of​ times, and each repetition of the experiment is called a trial. To be a binomial​ experiment, the experiment is performed a fixed number of​ times, and each repetition of the experiment is called a trial. (b) c. Success/Failure Note: For each​ trial, there are two mutually exclusive​ (disjoint) outcomes: success or failure.

6.1 Suppose the following data represent the ratings​ (on a scale from 1 to​ 5) for a certain smart phone​ game, with 1 representing a poor rating. Complete parts​ (a) through​ (d) below. Stars:Frequency 1:2494 2:2683 3: 4593 4: 4194 5:10,312 ​(a) Construct a discrete probability distribution for the random variable x. Stars​ (x):P(x) 1: ___ 2:___ 3:___ 4:___ 5:___ ​(Round to three decimal places as​ needed.) (b) Graph the discrete probability distribution. Choose the correct graph below. A. A histogram has a horizontal axis labeled from 0 to 4 in intervals of 1 and a vertical axis labeled from 0 to 0.5 in intervals of 0.1 has five vertical lines positioned on the horizontal axis tick marks. The approximate heights of the vertical lines are as follows, with the horizontal coordinate listed first and the line height listed second: 1, 0.43; 2, 0.17; 3, 0.19; 4, 0.11; 5, 0.1. B. A histogram has a horizontal axis labeled from 0 to 4 in intervals of 1 and a vertical axis labeled from 0 to 0.5 in intervals of 0.1 five vertical lines positioned on the horizontal axis tick marks. The approximate heights of the vertical lines are as follows, with the horizontal coordinate listed first and the line height listed second: 1, 0.19; 2, 0.17; 3, 0.43; 4, 0.1; 5, 0.11. C. A histogram has a horizontal axis labeled from 0 to 4 in intervals of 1 and a vertical axis labeled from 0 to 0.5 in intervals of 0.1 five vertical lines positioned on the horizontal axis tick marks. The approximate heights of the vertical lines are as follows, with the horizontal coordinate listed first and the line height listed second: 1, 0.1; 2, 0.11; 3, 0.19; 4, 0.17; 5, 0.43. D. A histogram has a horizontal axis labeled from 0 to 4 in intervals of 1 and a vertical axis labeled from 0 to 0.5 in intervals of 0.1 five vertical lines positioned on the horizontal axis tick marks. The approximate heights of the vertical lines are as follows, with the horizontal coordinate listed first and the line height listed second: 1, 0.11; 2, 0.17; 3, 0.1; 4, 0.19; 5, 0.43. ​(c) 1. Compute and interpret the mean of the random variable x. The mean is ____ stars. ​(Round to one decimal place as​ needed.) 2. Which of the following interpretations of the mean is​ correct? A. As the number of experiments​ decreases, the mean of the observations will approach the mean of the random variable. B. The observed value of an experiment will be less than the mean of the random variable in most experiments. C. The observed value of an experiment will be equal to the mean of the random variable in most experiments. D. As the number of experiments​ increases, the mean of the observations will approach the mean of the random variable. (d) Compute the standard deviation of the random variable x. The standard deviation is ____ stars. ​(Round to one decimal place as​ needed.)

(a) ​P(x) 1: 0.103 2: 0.111 3: 0.189 4: 0.173 5: 0.425 (b)C. A histogram has a horizontal axis labeled from 0 to 4 in intervals of 1 and a vertical axis labeled from 0 to 0.5 in intervals of 0.1 five vertical lines positioned on the horizontal axis tick marks. The approximate heights of the vertical lines are as follows, with the horizontal coordinate listed first and the line height listed second: 1, 0.1; 2, 0.11; 3, 0.19; 4, 0.17; 5, 0.43. (c) 1. 3.7 2. D. As the number of experiments​ increases, the mean of the observations will approach the mean of the random variable. (d)1.4

6.2 A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n=10​, p=0.3​, x=3 P(3)=____ ​(Do not round until the final answer. Then round to four decimal places as​ needed.)

0.2668

6.2 A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n=9​, p=0.3​, x≤3 The probability of x≤3 successes is ______. ​(Round to four decimal places as​ needed.)

0.7296

6.2 Which of the following are criteria for a binomial probability​ experiment? Select all that apply. A. The probability of success is the same for each trial of the experiment. B. Each trial depends on the previous trial. C. The probability of success is different for each trial of the experiment. D. There are three mutually exclusive​ outcomes, arriving​ on-time, arriving​ early, and arriving late. E. The experiment is performed a fixed number of times. F. The trials are independent. G. The experiment is performed until a desired number of successes is reached. H. There are two mutually exclusive​ outcomes, success or failure.

A, E, F, and H Note: At least one answer is incorrect​ and/or you have not selected all of the correct answers. An experiment is binomial​ provided: 1. The experiment consists of a fixed​ number, n, of trials. 2. The trials are independent. 3. Each trial has two possible mutually exclusive​ outcomes: success and failure. 4. The probability of​ success, p, remains constant for each trial of the experiment.

6.2 Describe how the value of n affects the shape of the binomial probability histogram. Choose the correct answer below. A. As n​ increases, the binomial distribution becomes more bell shaped. B. As n​ decreases, the binomial distribution becomes more bell shaped. C. As n​ increases, the binomial distribution becomes skewed right. D. As n​ decreases, the binomial distribution becomes skewed left. E. The value of n does not affect the shape of the binomial probability histogram

A. As n​ increases, the binomial distribution becomes more bell shaped. Note: To determine how the value of n affects the shape of the binomial probability​ histogram, compare binomial probability histograms with a fixed value of p and different values of n. For a fixed​ p, as the number of trials n in a binomial experiment​ increases, the probabilitiy distribution of the random variable X becomes more bell shaped.

6.2 Determine if the following probability experiment represents a binomial experiment. If​ not, explain why. If the probability experiment is a binomial​ experiment, state the number of​ trials, n. An experimental drug is administered to 190 randomly selected​ individuals, with the number of individuals responding favorably recorded. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your answer. A. ​Yes, because the experiment satisfies all the criteria for a binomial​ experiment, n=____. ​(Type a whole​ number.) B. ​No, because the experiment is not performed a fixed number of times. C. ​No, because the trials of the experiment are not independent because the probability of success differs from trial to trial. D. ​No, because there are more than two mutually exclusive outcomes for each trial.

A. ​Yes, because the experiment satisfies all the criteria for a binomial​ experiment, n=__190__. ​(Type a whole​ number.)

6.2 Determine if the following probability experiment represents a binomial experiment. If​ not, explain why. If the probability experiment is a binomial​ experiment, state the number of​ trials, n. A random sample of 40 college seniors is​ obtained, and the individuals selected are asked to state their ages.

An experiment is said to be a binomial experiment if each of the following statements are true. 1. The experiment is performed a fixed number of times. Each repetition of the experiment is called a trial. 2. The trials are independent. This means the outcome of one trial will not affect the outcome of the other trials. 3. For each​ trial, there are two mutually exclusive​ (disjoint) outcomes, success or failure. 4. The probability of success is the same for each trial of the experiment. The experiment is performed a fixed number of​ times, n=40. The trials are independent in this experiment. Since the​ variable, age, is​ continuous, there are not two mutually exclusive outcomes. The probability of success is not the same for each trial because there are several possible​ outcomes, not two mutually exclusive outcomes. ​Therefore, this probability experiment is not a binomial experiment because the third and fourth conditions are not satisfied.

6.1 What are the two requirements for a discrete probability​ distribution? Choose the correct answer below. Select all that apply. A. 0<P(x)<1 B. 0≤P(x)≤1 C. ∑P(x)=1 D. ∑P(x)=0 C

B and C B.0≤P(x)≤1 C.∑P(x)=1 Note: Each probability must be between 0 and​ 1, inclusive, and the sum of the probabilities must equal 1.

6.1 What is a random​ variable? Choose the correct answer below. A. A random variable is a numerical​ measure, having either a finite or countable number of​ values, of the outcome of a probability experiment. B. A random variable is a numerical measure of the outcome of a probability experiment. C. A random variable is a numerical​ measure, having values that can be plotted on a line in an uninterrupted​ fashion, of the outcome of a probability experiment.

B. A random variable is a numerical measure of the outcome of a probability experiment. Note :When experiments are conducted in a way such that the outcome is a numerical​ result, it is said that the outcome is a random variable. A random variable is a numerical measure of the outcome of a probability​ experiment, so its value is determined by chance.

6.2 Determine if the following probability experiment represents a binomial experiment. If​ not, explain why. If the probability experiment is a binomial​ experiment, state the number of​ trials, n. Five cards are selected from a standard​ 52-card deck without replacement. The number of hearts selected is recorded. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your answer. A. ​Yes, because the experiment satisfies all the criteria for a binomial​ experiment, n=____. B. ​No, because the trials of the experiment are not independent since the probability of success differs from trial to trial. C. ​No, because there are more than two mutually exclusive outcomes for each trial. D. ​No, because the experiment is not performed a fixed number of times.

B. ​No, because the trials of the experiment are not independent since the probability of success differs from trial to trial.

6.2 Determine if the following probability experiment represents a binomial experiment. If​ not, explain why. If the probability experiment is a binomial​ experiment, state the number of​ trials, n. A random sample of 25 high school seniors is​ obtained, and the individuals selected are asked to state their weights. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your answer. A. ​Yes, because the experiment satisfies all the criteria for a binomial​ experiment, n=___. B. ​No, because there are more than two mutually exclusive outcomes for each trial. C. ​No, because the trials of the experiment are not independent since the probability of success differs from trial to trial. D. ​No, because the experiment is not performed a fixed number of times.

B. ​No, because there are more than two mutually exclusive outcomes for each trial. Note: An experiment is said to be a binomial experiment if each of the following statements are true. 1. The experiment is performed a fixed number of times. Each repetition of the experiment is called a trial. 2. The trials are independent. This means the outcome of one trial will not affect the outcome of the other trials. 3. For each​ trial, there are two mutually exclusive​ (disjoint) outcomes, success or failure. 4. The probability of success is the same for each trial of the experiment.

6.3 State the conditions required for a random variable X to follow a Poisson process. Select all that apply. A. A sample of size n is obtained from the population of size N without replacement. B. The experiment is performed a fixed number of times. C. The number of successes in any interval is independent of the number of successes in any other interval provided the intervals are not overlapping. D. The probability of success is the same for any two intervals of equal length. E. The probability of two or more successes in any sufficiently small subinterval is 0.

C, D, E

6.1 ex Determine the required value of the missing probability to make the distribution a discrete probability distribution. x : P(x) 3 : 0.36 4 : ? 5 : 0.19 6 : 0.31

In a discrete probability​ distribution, the sum of the probabilities must equal​ 1, and all probabilities must be greater than or equal to 0 and less than or equal to 1. Notice that all the given probabilities are greater than or equal to 0 and less than or equal to 1. The probability​ P(4) is missing from the distribution. To find​ P(4), first add all of the given probabilities. ∑P(x)=0.36+0.19+0.31=0.86 Recall that the sum of all the probabilities must equal 1 in a discrete probability distribution. To find​ P(4), subtract the sum of the other probabilities from 1. 1.00−0.86=0.14 The value for​ P(4) is a valid probability because it is greater than or equal to 0 and less than or equal to 1. ​Thus, ​P(4)=0.14 makes the probability distribution valid.

6.2 A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n=10​, p=0.55​, x=6

Simplify each expression. P(6)=210​(0.02768​)(0.04101​) ​Finally, multiply to find the final answer. P(6)=0.2384 ​Therefore, the probability of 6 successes in the 10 independent trials of the binomial probability experiment is 0.2384.

6.2 Determine if the following statement is true or false. In the binomial probability distribution​ function, nCx represents the number of ways of obtaining x successes in n trials. Choose the correct answer on the below. True False

true

6.3 The hits to a Web site occur at the rate of 9 per minute between​ 7:00 P.M. and 10:00 P.M. The random variable X is the number of hits to the Web site between 7:23 P.M. and 7​:31 P.M. State the values of λ and t for this Poisson process.

λ represents the average number of occurrences of the event in some interval whose length is 1. λ is also sometimes thought of as the rate. The value of the rate in this problem is 9 hits per minute. ​Thus, λ=9 hits per minute. t represents the length of the interval being examined. In this​ case, the interval is being measured in minutes. To calculate the​ interval, subtract the start time of the interval from the ending time of the interval. The appropriate values from the problem statement are substituted. t = ending time−start time =7​:31−7​:23 Then subtract. t=7​:31−7​:23 = 8 minutes

6.3 The hits to a Web site occur at the rate of 10 per minute between​ 7:00 P.M. and 11:00 P.M. The random variable X is the number of hits to the Web site between 9:40 P.M. and 10​:27 P.M. State the values of λ and t for this Poisson process. λ=____ t=____

λ=10 t= 47

Save The hits to a Web site occur at the rate of 9 per minute between​ 7:00 P.M. and 9:00 P.M. The random variable X is the number of hits to the Web site between 8:21 P.M. and 8​:32 P.M. State the values of λ and t for this Poisson process. λ=____ t= _____

λ=9 t=11

6.2 According to an​ almanac, 80​% of adult smokers started smoking before turning 18 years old. ​(a) If 300 adult smokers are randomly​ selected, how many would we expect to have started smoking before turning 18 years​ old? ​(b) Would it be unusual to observe 255 smokers who started smoking before turning 18 years old in a random sample of 300 adult​ smokers? Why? ​(a) We would expect about _____ adult smokers to have started smoking before turning 18 years old. ​(Type a whole​ number.)​ (b) Would it be unusual to observe 255 smokers who started smoking before turning 18 years old in a random sample of 300 adult​ smokers? A. No​, because 255 is between μ−2σ and μ+2σ. B. No​, because 255 is greater than μ+2σ. C. Yes​, because 255 is greater than μ+2σ. D. ​Yes, because 255 is between μ−2σ and μ+2σ. E. ​No, because 255 is less than μ−2σ.

​(a) 240 (b)C. Yes​, because 255 is greater than μ+2σ. Note: Recall that the empirical rule states that in a​ bell-shaped distribution about​ 95% of all observations lie within two standard deviations of the mean. That​ is, about​ 95% of the observations lie between μ−2σ and μ+2σ.

6.1 ex The following data represent the number of games played in each series of an annual tournament from 1923 to 2000. Complete parts​ (a) through​ (d) below. x​ (games played): 4 5 6 7 Frequency: 14 23 23 17 ​(a) Construct a discrete probability distribution for the random variable X. ​(b) Graph the discrete probability distribution. ​(c) Compute and interpret the mean of the random variable X. (d) Compute the standard deviation of the random variable X.

​(a) Construct a discrete probability distribution for the random variable X. Use the formula P(xi)=fi / N to find the probabilities. Find the total​ frequency, N. N=14+23+23+17 =77 Substitute the corresponding frequency fi and the total frequency N in the formula and find Pxi​, rounding to four decimal places. P(4)= f4 / N = 14/77 =0.1818 Calculate the other probabilities likewise and complete the​ table, rounding to four decimal places. x​ (games played) : P(x) 4:0.1818 5:0.2987 6:0.2987 7:0.2208 ​(b) Graph the discrete probability distribution. In the graph of a discrete probability​ distribution, the horizontal axis is the values of the discrete random variable and the vertical axis is the corresponding probability of the discrete random variable. Draw the graph using vertical lines above each value of the random variable to a height that is the probability of the random​ variable, as shown to the right. A coordinate system has a horizontal x-axis labeled from 4 to 7 in increments of 1 and a vertical P(x)-axis labeled from 0 to 0.5 in intervals of 0.1. Four vertical lines are positioned on each horizontal axis tick mark. The approximate heights of the vertical lines are as follows, with the horizontal coordinate listed first and the line height listed second: 4, 0.18; 5, 0.3; 6, 0.3; 7, 0.22. ​(c) Compute and interpret the mean of the random variable X. While technology or the formula μX=∑[x•P(x)] can be used to calculate the mean of the random variable​ X, for this​ problem, use​ technology, rounding to one decimal place. μX=5.6 Interpret the mean of the random variable X. The mean of a discrete random variable is the average outcome if the experiment is repeated​ many, many times. The​ series, if played many​ times, would be expected to last about 5.6 ​games, on​ average, rounding to one decimal place. ​(d) Compute the standard deviation of the random variable X. Use technology or the formula σX=sqrt [∑((x−μX)^2•P(x))]= sqrt [∑x^2•P(x)−μ^2X] to find the standard deviation of the random variable X. While technology or the formula above can be used to calculate σX​, for this​ exercise, use​ technology, rounding to one decimal place. σX=1.0 ​Thus, the standard deviation of the random variable​ X, rounded to one decimal​ place, is 1.0.

6.2 According to an​ almanac, 75​% of adult smokers started smoking before turning 18 years old. ​(a) If 400 adult smokers are randomly​ selected, how many would we expect to have started smoking before turning 18 years​ old? ​(b) Would it be unusual to observe 311 smokers who started smoking before turning 18 years old in a random sample of 400 adult​ smokers? Why?

​(a) Recall that a binomial experiment with n independent trials and probability of success p has a mean and standard deviation given by the following formulas. μx=np and σx= sqrt [np(1−p)] First determine the values of n and p in this experiment. n=400​, p=0.75 Now calculate the​ mean, μx. μx = np =400(0.75) =300 ​Thus, we would expect about 300 adult smokers to have started smoking before turning 18 years old. ​(b) Note that as the number of trials n in a binomial experiment​ increases, the probability distribution of the random variable X becomes bell shaped. As a rule of​ thumb, if np(1−p)≥10​, the probability distribution will be approximately bell shaped. In this​ case, as was previously​ calculated, np(1−p)=75. ​Therefore, the probability distribution is approximately bell shaped. The result above means that the empirical rule can be used to identify unusual observations. Recall that the empirical rule states that in​ bell-shaped distribution about​ 95% of all observations lie within two standard deviations of the mean. That​ is, about​ 95% of the observations lie between μ−2σ and μ+2σ. Calculate the standard​ deviation, σx. First simplify the radicand. σx= sqrt [np(1−p)] σx= sqrt [400(0.75)(1−0.75)] =75 Simplify to find σx​, rounding to one decimal place. σx= sqrt [75] =8.7 To determine if observing 311 smokers who started smoking before turning 18 years old is​ unusual, calculate μ−2σ and μ+2σ. μ−2σ=300−​2(8.7​)=282.6​, μ+2σ=300+​2(8.7​)=317.4 ​Therefore, any value that is less than 282.6 or greater than 317.4 is unusual.

6.3 The number of hits to a Web site follows a Poisson process. Hits occur at the rate of 0.8 per minute between​ 7:00 P.M. and 9​:00 P.M. Given below are three scenarios for the number of hits to the Web site. Compute the probability of each scenario between 7:11 P.M. and 7​:21 P.M. and interpret the results. ​(a) exactly three hits ​(b) fewer than three hits ​(c) at least three hits

​(a) The Poisson probability distribution function or technology can be used to compute the probability. For this example use technology. ​First, identify the values for λ and t. Since λ represents the average number of occurrences of the event in some interval of length​ 1, it is often called a rate.​ Thus, the value for λ is 0.8 hits per minute. The length of the interval being​ examined, t, is calculated by subtracting the start time of the interval being examined from the ending time of the interval. t=7:21 P.M.−7:11 P.M.=10 minutes Next solve for the expected number of hits in 10 minutes which is represented by μx. μx=λt=0.8•10=8 Use technology to calculate ​P(3​). Use the Tech Help button for further assistance. ​P(3​)≈0.0286 ​Thus, the probability that the number of hits to the Web site between 7​:11 P.M. and 7:21 P.M. is exactly three is approximately 0.0286. The probability of 0.0286 translates to about 3 out of 100. Use this information in the context of the problem to interpret the results. ​(b) The probability of the number of​ occurrences, X, being less than a certain value (n) is the sum of the probabilities of all occurrences of values less than n. The Poisson probability distribution function or technology can be used to compute this probability. For this example use technology. Find the probability that X is less than 3. ​P(X<3​)=​P(X≤​2)≈0.0138 ​Thus, the probability that the number of hits to the Web site between 7​:11 P.M. and 7:21 P.M. is fewer than three is approximately 0.0138. The probability of 0.0138 translates to about 1 out of 100. Use this information in the context of the problem to interpret the results. ​(c) To obtain the value for an​ "at-least" probability for a Poisson​ process, the Complement Rule can be used. ​P(X≥​n) = 1−​P(X<​n) Substitute in the value of ​P(X<3​) which was obtained in part​ (b) above and subtract. ​P(X≥3​) = 1−​P(X<3​) ≈1−0.0138 ≈0.9862 ​Thus, the probability that the number of hits to the Web site between 7​:11 P.M. and 7:21 P.M. is at least three is approximately 0.9862. The probability of 0.9862 translates to about 99 out of 100. Use this information in the context of the problem to interpret the results.

6.1 One of the biggest factors in determining the value of a home is the square footage. The accompanying data represent the square footage and asking price​ (in thousands of​ dollars) for a random sample of homes for sale. Complete parts​ (a) through​ (h). Click the icon to view the data table. Click the icon to view a table of critical values for the correlation coefficient.

​(a) Which variable is the explanatory​ variable? The value of the response variable can be explained by the value of the​ explanatory, or​ predictor, variable. Determine the goal of the study and identify the variable that may affect the response variable. The explanatory variable is the square footage ​(b) Draw a scatter diagram of the data. A scatter diagram is a graph that shows the relationship between two quantitative variables measured on the same individual. Each individual in the data set is represented by a point in the scatter diagram. The explanatory variable is plotted on the horizontal​ axis, and the response variable is plotted on the vertical axis. Draw a scatter diagram of the data. The graph is shown below. ​(c) Determine the linear correlation coefficient between square footage and asking price. The linear correlation coefficient is a measure of the strength and direction of the linear relation between two quantitative variables. The linear correlation coefficient is given by the formula r=[∑(xi−x / sx) (yi−y / sy)] / n−1 where x is the sample mean of the explanatory​ variable, sx is the sample standard deviation of the explanatory​ variable, y is the sample mean of the response​ variable, sy is the sample standard deviation of the response​ variable, and n is the number of individuals in the sample. While either the formula or technology can be used to find the linear correlation​ coefficient, for the purpose of this​ problem, use technology. Use the Tech Help button for further assistance. Using​ technology, the linear correlation coefficient between square footage and asking price is r≈0.924. ​(d) Is there a linear relation between the square footage and asking​ price? To test whether the correlation between the explanatory and response variables is strong enough to conclude that there is a linear​ relation, determine the absolute value of the correlation coefficient. If the absolute value of the correlation coefficient is greater than the critical value for correlation for the given sample​ size, then a linear relation exists between the two variables.​ Otherwise, no linear relation exists. Find the critical value for correlation for the sample size n=5. Use the table of critical values for correlation coefficient. For n=​9, the critical value is 0.666. Since the correlation coefficient is greater than the critical​ value, there is a linear relation between the square footage and asking price. ​(e) Find the​ least-squares regression line treating square footage as the explanatory variable. The​ least-squares regression line is the line that minimizes the sum of the squared errors​ (or residuals). It is the line that minimizes the sum of the squared vertical distance between the observed values of y and those predicted by the​ line, y. The equation of the​ least-squares regression line is given by the formula y=b1x+b0 where b1=r•sy / sx is the slope of the​ least-squares regression​ line, r is the correlation​ coefficient, b0=y−b1x is the​ y-intercept of the​ least-squares regression​ line, x is the sample mean and sx is the sample standard deviation of the explanatory variable​ x, and y is the sample mean and sy is the sample standard deviation of the response variable y. While either the formula or technology can be used to find the​ least-squares regression​ line, for the purpose of this​ problem, use technology. Use the Tech Help button for further assistance. Using​ technology, the​ least-squares regression line is y=0.06778x+84.25. ​(f) Interpret the slope. To interpret the​ slope, let y=b1x+b0. The slope of the regression line is b1. If x increases by 1​ unit, the predicted​ value, y​, increases by b1 ​units, on average. Note that if the slope b1 is​ negative, y decreases as x increases. ​Thus, for each square foot added to the​ area, the expected asking price of the house will increase by ​$67.78 ​(that is, 0.06778 thousand​ dollars). ​(g) Is it reasonable to interpret the​ y-intercept? In​ general, to interpret the​ y-intercept, the​ y-intercept is the value of the response variable when the explanatory variable is 0. To interpret a​ y-intercept, 0 should be a reasonable value for the explanatory variable. The regression model should not be used to make predictions outside the scope of the model. Outside the scope of the model refers to using the regression model to make predictions for values of the explanatory variable that are much larger or much smaller than those observed. This is a dangerous practice because one cannot be certain of the behavior of data. Since a square footage of 0 is not​ reasonable, it is not appropriate to interpret the​ y-intercept. ​(h) One home that is 1,094 square feet is listed at​ $189,900. Is this​ home's price above or below average for a home of this​ size? Substitute 1,094 into the equation of the​ least-squares regression line and find the predicted asking price for a home that is 1,094 square feet. The predicted asking price for a home that is 1,094 square feet is y=0.06778(1,094)+84.25≈158.401​, or​ approximately, ​$158,401. Compare​ $189,900 and the value of the predicted asking price ​$158,401. Since the value of the predicted asking price for a home that is 1,094 square feet is less than the observed​ home's price, this​ home's price is above average for a home of this size. May there be some reasons for this​ price? There are some factors that could affect the​ price, including location and updates such as thermal windows or new siding.

6.1 The following data represent the number of games played in each series of an annual tournament from 1923 to 2018. Complete parts​ (a) through​ (d) below. x​ (games played) : Frequency 4:16 5:17 6:17 7:45 ​(a) Construct a discrete probability distribution for the random variable X. x​ (games played) : P(x) Values 4:___ 5:___ 6:___ 7:___ ​(Round to four decimal places as​ needed.) (b) Graph the discrete probability distribution. Choose the correct graph below. A. A coordinate system has a horizontal x-axis labeled from 4 to 7 in increments of 1 and a vertical P(x)-axis labeled from 0 to 0.5 in intervals of 0.1. Four vertical lines are positioned on each horizontal axis tick mark. The approximate heights of the vertical lines are as follows, with the horizontal coordinate listed first and the line height listed second: 4, 0.27; 5, 0.18; 6, 0.28; 7, 0.47. B. A coordinate system has a horizontal x-axis labeled from 4 to 7 in increments of 1 and a vertical P(x)-axis labeled from 0 to 0.5 in intervals of 0.1. Four vertical lines are positioned on each horizontal axis tick mark. The approximate heights of the vertical lines are as follows, with the horizontal coordinate listed first and the line height listed second: 4, 0.17; 5, 0.18; 6, 0.18; 7, 0.47. C. A coordinate system has a horizontal x-axis labeled from 4 to 7 in increments of 1 and a vertical P(x)-axis labeled from 0 to 0.5 in intervals of 0.1. Four vertical lines are positioned on each horizontal axis tick mark. The approximate heights of the vertical line are as follows, with the horizontal coordinate listed first and the line height listed second: 4, 0.27; 5, 0.28; 6, 0.28; 7, 0.5. D. A coordinate system has a horizontal x-axis labeled from 4 to 7 in increments of 1 and a vertical P(x)-axis labeled from 0 to 0.5 in intervals of 0.1. Four vertical lines are positioned on each horizontal axis tick mark. The approximate heights of the vertical lines are as follows, with the horizontal coordinate listed first and the line height listed second: 4, 0.07; 5, 0.08; 6, 0.08; 7, 0.37. ​(c) 1. Compute and interpret the mean of the random variable X. μX=____ game(s) ​(Round to one decimal place as​ needed.) 2. Interpret the mean of the random variable X. Select the correct choice below and fill in the answer box within your choice. ​(Round to one decimal place as​ needed.) A. The​ series, if played one​ time, would be expected to last about ______ ​game(s). B. The​ series, if played many​ times, would be expected to last at least ____ ​game(s), on average. ​(d) Compute the standard deviation of the random variable X. σX=____ game(s) ​(Round to one decimal place as​ needed.)

​(a) x​ (games played): P(x) 4: 0.1684 5: 0.1789 6: 0.1789 7:0.4737 (b) B. A coordinate system has a horizontal x-axis labeled from 4 to 7 in increments of 1 and a vertical P(x)-axis labeled from 0 to 0.5 in intervals of 0.1. Four vertical lines are positioned on each horizontal axis tick mark. The approximate heights of the vertical lines are as follows, with the horizontal coordinate listed first and the line height listed second: 4, 0.17; 5, 0.18; 6, 0.18; 7, 0.47. (c) 1. 6.0 2. B. The​ series, if played many​ times, would be expected to last at least _6_ ​game(s), on average. (d) 1.2