HW 6, HW 7, HW 5
A football association polls fans to develop a rating for each football game. Each game is rated on a scale from 0 (forgettable) to 100 (memorable). The fan ratings for a random sample of 12 games follow. 516380766675145974817776 (a)Develop a point estimate of mean fan rating for the population of football games. (b)Develop a point estimate of the standard deviation for the population of football games.
Question 1
Consider the experiment of tossing a coin twice. (Let H represent the head of the coin and T represent it's tail.) (a)List the experimental outcomes. (Select all that apply.) (b)Define a random variable that represents the number of heads occurring on the two tosses. The random variable x, where x = the number of heads that occur for two coin tosses, describes the scenario. (c)Show what value the random variable would assume for each of the experimental outcomes. (If an experimental outcome does not occur, enter NONE.)
Question 1
The random variable x is known to be uniformly distributed between 40 and 50. (a)Show the graph of the probability density function. (b)Compute P(x < 45). (c)Compute P(43 ≤ x ≤ 47). (d)Compute E(x). (e)Compute Var(x).
Question 1
10.)Axline Computers manufactures personal computers at two plants, one in Texas and the other in Hawaii. The Texas plant has 55 employees; the Hawaii plant has 35. A random sample of 15 employees is to be asked to fill out a benefits questionnaire. (a)What is the probability that none of the employees in the sample work at the plant in Hawaii? (b)What is the probability that 1 of the employees in the sample works at the plant in Hawaii? (c)What is the probability that 2 or more of the employees in the sample work at the plant in Hawaii? (d)What is the probability that 14 of the employees in the sample work at the plant in Texas?
Question 10
In a benchmark study, a fast food restaurant had an average service time of 2.7 minutes. Assume that the service time for the fast food restaurant has an exponential distribution. (Round your answers to four decimal places.) (a)What is the probability that a service time is less than or equal to one minute? (b)What is the probability that a service time is between 45 seconds and one minute? (c)Suppose the manager of the restaurant is considering instituting a policy such that if the time it takes to serve you exceeds five minutes, your food is free. What is the probability that you will get your food for free? If the manager is okay with 1% chance of customers receiving free food, does the policy seem like a good idea?
Question 10
A company introduced a much smaller variant of its tablet, known as the tablet junior. Weighing less than 11 ounces, it was about 50% lighter than the standard tablet. Battery tests for the tablet junior showed a mean life of 10.25 hours. Assume that battery life of the tablet junior is uniformly distributed between 8.5 and 12 hours. (a)Give a mathematical expression for the probability density function of battery life. (b)What is the probability that the battery life for a tablet junior will be 9.5 hours or less? (c)What is the probability that the battery life for a tablet junior will be at least 9 hours? (d)What is the probability that the battery life for a tablet junior will be between 9.5 and 11.5 hours? (e)In a shipment of 100 tablet juniors, how many should have a battery life of at least 10.5 hours?
Question 2
Listed is a series of experiments and associated random variables. In each case, identify the values that the random variable can assume and state whether the random variable is discrete or continuous. Take a 30-question examination. random variable x = number of questions answered correctly Identify the values that the random variable can assume. 0, 1, 2, ..., 30
Question 2
Draw a graph for the standard normal distribution. Label the horizontal axis at values of −3, −2, −1, 0, 1, 2, and 3. Use the table of probabilities for the standard normal distribution to compute the following probabilities. (Round your answers to four decimal places.)
Question 3
The probability distribution for the random variable x follows. (a) Is this probability distribution valid? Explain. Since f(x) ≥ 0 for all values of x and E f(x) = 1, this is a proper probability distribution. (b) What is the probability that x = 30? .30 (c) What is the probability that x is less than or equal to 25? .35 (d) What is the probability that x is greater than 30? .35
Question 3
You may need to use the appropriate appendix table or technology to answer this question. The Economic Policy Institute periodically issues reports on wages of entry level workers. The institute reported that entry level wages for male college graduates were $21.68 per hour and for female college graduates were $18.80 per hour in 2011.† Assume the standard deviation for male graduates is $2.30, and for female graduates it is $2.05. (a)What is the probability that a sample of 70 male graduates will provide a sample mean within $0.60 of the population mean, $21.68? (b)What is the probability that a sample of 70 female graduates will provide a sample mean within $0.60 of the population mean, $18.80? (c)In which of the preceding two cases, part (a) or part (b), do we have a higher probability of obtaining a sample estimate within $0.60 of the population mean? Why? (d)What is the probability that a sample of 140 female graduates will provide a sample mean less than the population mean by more than $0.40?
Question 3
For unemployed persons in the United States, the average number of months of unemployment at the end of December 2009 was approximately seven months.† Suppose the following data are for a particular region in a state. The values in the first column show the number of months unemployed and the values in the second column show the corresponding number of unemployed persons.
Question 4
Given that z is a standard normal random variable, compute the following probabilities. (a) P(−1.95 ≤ z ≤ 0.48) (b) P(0.55 ≤ z ≤ 1.21) (c) P(−1.65 ≤ z ≤ −1.03)
Question 4
You may need to use the appropriate appendix table or technology to answer this question. The mean preparation fee H&R Block charged retail customers last year was $183.† Use this price as the population mean and assume the population standard deviation of preparation fees is $50. (a)What is the probability that the mean price for a sample of 35 H&R Block retail customers is within $8 of the population mean? (b)What is the probability that the mean price for a sample of 65 H&R Block retail customers is within $8 of the population mean? (c)What is the probability that the mean price for a sample of 64 H&R Block retail customers is within $8 of the population mean? (d)Which, if any, of the sample sizes in parts (a), (b), and (c) would you recommend to have at least a 0.95 probability that the sample mean is within $8 of the population mean?
Question 4
5.) The following table provides a probability distribution for the random variable x. (a) Compute E(x), the expected value of x. The expected value of a random variable is a measure for the central location. Thus, the expected value of a random variable is another way to refer to the mean, μ. Recall the formula for the expected value, E(x), of a discrete random variable x. Multiply each value of the random variable by its respective probability and then sum these values. E(x) = Exff(x) It can be helpful to add a column to the given table of information to organize the products of xf(x). Complete the table.
Question 5
A sample of size 110 is selected from a population with p = 0.50. (a)What is the expected value of p? (b)What is the standard error of p? (Round your answer to four decimal places.) (c)Show the sampling distribution of p. (d)What does the sampling distribution of p show?
Question 5
You may need to use the appropriate appendix table to answer this question. Given that z is a standard normal random variable, find z for each situation. (Round your answers to two decimal places.) The area to the left of z is 0.9750. The area between 0 and z is 0.4750. (c) The area to the left of z is 0.7517. The area to the right of z is 0.1251. The area to the left of z is 0.6700. The area to the right of z is 0.3300.
Question 5
6.) West Virginia has one of the highest divorce rates in the nation, with an annual rate of approximately 5 divorces per 1,000 people.† The Marital Counseling Center, Inc. (MCC) thinks that the high divorce rate in the state may require them to hire additional staff. Working with a consultant, the management of MCC has developed the following probability distribution for x = the number of new clients for marriage counseling for the next year.
Question 6
Assume that the population proportion is 0.64. Compute the standard error of the proportion, σp, for sample sizes of 100, 200, 500, and 1,000. What can you say about the size of the standard error of the proportion as the sample size is increased?
Question 6
You may need to use the appropriate appendix table to answer this question. The average return for large-cap domestic stock funds over the three years 2009-2011 was 14.4%.† Assume the three-year returns were normally distributed across funds with a standard deviation of 4.4%. (a)What is the probability an individual large-cap domestic stock fund had a three-year return of at least 22%? (b)What is the probability an individual large-cap domestic stock fund had a three-year return of 15% or less? (c)How big does the return have to be to put a domestic stock fund in the top 20% for the three-year period?
Question 6
7.) A university found that 15% of its students withdraw without completing the introductory statistics course. Assume that 18 students registered for the course. Compute the probability that 2 or fewer will withdraw. This is a binomial experiment with n = 18 trials. Let a success be that a student withdraws from the course. It is given that 15% of students withdraw. Recall the method used to find the probability of an event given a percentage. Probability = given percentage/100%. P = 15%/100% = .15 For 2 or fewer students to withdraw from the course means that exactly 0, 1, or 2 studies withdraw. Therefore, x will be 0, 1, and 2. The probability of each x value is f(x), so the probability that 2 or fewer students withdraw from the course will be f(0) + f(1) + f(2). We have a binomial experiment with n = 18 trials, each with probability p = 0.15 of a success. A success occurs if a student withdraws from a class, so the number of successes, x, will take on the values 0, 1, and 2. The probability of each x value, denoted f(x), can be found using a table like the one below. Note that these values are rounded to four decimal places.
Question 7
You may need to use the appropriate appendix table to answer this question. Automobile repair costs continue to rise with the average cost now at $367 per repair.† Assume that the cost for an automobile repair is normally distributed with a standard deviation of $88. Answer the following questions about the cost of automobile repairs.
Question 7
ou may need to use the appropriate appendix table or technology to answer this question. The Food Marketing Institute shows that 17% of households spend more than $100 per week on groceries. Assume the population proportion is p = 0.17 and a sample of 500 households will be selected from the population. (a)Show the sampling distribution of p, the sample proportion of households spending more than $100 per week on groceries. (b)What is the probability that the sample proportion will be within ±0.02 of the population proportion? (c)Answer part (b) for a sample of 1,000 households.
Question 7
8.) You may need to use the appropriate appendix table or technology to answer this question.Consider a binomial experiment with n = 20 and p = 0.80. (Round your answers to four decimal places.) (a)Compute f(11).= .0074 (b)Compute f(16).= .2182 (c)Compute P(x ≥ 16).= .6296 (d)Compute P(x ≤ 15).= .3704 (e)Compute E(x).= 16 (f)
Question 8
Consider the following exponential probability density function. f(x) = 14e−x/4 for x ≥ 0 (a)Write the formula for P(x ≤ x0). (b)Find P(x ≤ 2). (Round your answer to four decimal places.) (c)Find P(x ≥ 4). (Round your answer to four decimal places.).3679 (d)Find P(x ≤ 6). (Round your answer to four decimal places.) (e)Find P(2 ≤ x ≤ 6). (Round your answer to four decimal places.)
Question 8
9.) Phone calls arrive at the rate of 24 per hour at the reservation desk for Regional Airways. (a)Compute the probability of receiving four calls in a 5-minute interval of time. The Poisson distribution is a discrete probability distribution used to calculate probabilities related to a random variable counting the number of times an event occurs over a specified interval. The parameter μ represents the mean number of times we expect the event to occur over the specified interval. For a Poisson random variable where μ is the mean number of occurrences over a specified interval, the formula for calculating the probability of x occurrences in the interval, f(x), is given below.
Question 9
The time between arrivals of vehicles at a particular intersection follows an exponential probability distribution with a mean of 11 seconds. (a)Sketch this exponential probability distribution. (b)What is the probability that the arrival time between vehicles is 11 seconds or less? (c)What is the probability that the arrival time between vehicles is 7 seconds or less? (d)What is the probability of 33 or more seconds between vehicle arrivals?
Question 9
One of the questions on a survey asked adults if they used the internet at least occasionally. The results showed that 453 out of 478 adults aged 18-29 answered Yes; 741 out of 833 adults aged 30-49 answered Yes; 1,056 out of 1,644 adults aged 50 and over answered Yes. (Round your answers to four decimal places.) (a)Develop a point estimate of the proportion of adults aged 18-29 who use the internet. (b)Develop a point estimate of the proportion of adults aged 30-49 who use the internet. (c)Develop a point estimate of the proportion of adults aged 50 and over who use the internet. (d)Comment on any relationship between age and Internet use that seems apparent. (e)
Questiono 2
