3.*       Public transportation and the automobile are two methods an employee can use to get to work each day. Samples of times recorded for each method are shown. Times are in minutes.

            Public Transportation:        18   15   13   20   22   17   11   14    3   16

            Automobile:                        12   24   19   21   19   25   23    9    17 10

            a)         Compute the sample mean time to get to work for each method.

            b)         Compute the sample standard deviation for each method.

              c)         On the basis of your results from a) and b), which method of transportation should be preferred? Please explain.

For a new study conducted by a fitness magazine,

240

females were randomly selected. For each, the mean daily calorie consumption was calculated for a September-February period. A second sample of

210

females was chosen independently of the first. For each of them, the mean daily calorie consumption was calculated for a March-August period. During the September-February period, participants consumed a mean of

2385.5

calories daily with a standard deviation of

222

. During the March-August period, participants consumed a mean of

2414.5

calories daily with a standard deviation of

252.5

. The population standard deviations of daily calories consumed for females in the two periods can be estimated using the sample standard deviations, as the samples that were used to compute them were quite large. Construct a

90%

confidence interval for

−μ1μ2

, the difference between the mean daily calorie consumption

μ1

of females in September-February and the mean daily calorie consumption

μ2

of females in March-August. Then complete the table below

. Assume that the lifetimes (measured from the beginning of use) of lightbulbs are i.i.d. random variables with distribution P(T ≥ k) = (k + 1)−β , k = 0, 1, 2, . . . , for some β > 0. (Note that time is measured in discrete units.) In a lightbulb socket in a factory, a bulb is used until it fails, and then it is replaced at the next time unit. Let (Xn)n≥0 be the irreducible Markov chain which records the age of the bulb currently in use in the socket (Xn = 0 at times when a bulb is replaced, corresponding to a new bulb). (a) Derive the transition probabilities of the chain. (b) For each value of β, determine if the chain is positive recurrent, null recurrent, or transient.

Market research has indicated that customers are likely to bypass Roma tomatoes that weigh less than 70 grams. A produce company produces Roma tomatoes that average 78.0 grams with a standard deviation of 5.2 grams.

Suppose there were 3 undersized tomatoes in the random sample of 20. What is the probability of getting at least 3 undersized tomatoes in a random sample of 20 if the company's claim is true? Do you believe the company's claim? Why or why not?

The Chartered Financial Analyst (CFA®) designation is fast becoming a requirement for serious investment professionals. Although it requires a successful completion of three levels of grueling exams,it also entails promising careers with lucrative salaries. A student of finance is curious about the average salary of a CFA® charterholder. He takes a random sample of 36 recent charterholders and computes a mean salary of $162,000 with a standard deviation of $36,000. Use this sample information to determine the upper bound of the 90% confidence interval for the average salary of a CFA® charterholder. (Round the "t" value to 3 decimal places.)

You work for a large retailer and have been asked to estimate the proportion of your customers that are less than 30 years old. You have sampled a large number of stores and have found that of the 106 customers you have surveyed, 58 are less than 30 years old.

Assuming your sample is valid, what is the upper bound of a 99% confidence interval for the population proportion of customers who are less than 30 years old?
(Report your answer as a decimal and not as a percentage. For example, report 0.05 rather than 5%.)

1) For the following data on Year-end Audit times (in days), 17, 20, 25, 27, 19, 19, 20, 32, 26, 23, 24, 23, 27, 38, 21, 23, 22, 28, 33, 18, 27, 20, 23, 27, 31 Prepare a table showing in columns Audit Time Intervals (days), Frequencies, Cumulative Frequencies, Relative Frequencies, Cumulative Relative Frequencies, Percent Frequencies, and Cumulative Percent Frequencies.

Researchers selected a simple random sample of 4048 medical records of adults diagnosed with gum disease. In all, 2226 were current smokers, 891 were former smokers, and 931 never smoked regularly. Their research question is:

Do these data indicate that gum disease is equally likely regardless of smoking status?

Using a significance level of 0.05, what is the appropriate conclusion for this test?

The data are consistent with an equal representation of current, former, and never smokers among adults diagnosed with gum disease.

Current smokers make up a significantly greater proportion of adults diagnosed with gum disease than former or never smokers.

Current smokers are most likely to have gum disease.

There is significant evidence that current, former, and never smokers are not equally represented among adults diagnosed with gum disease.

In automobile mileage and gasoline-consumption testing, 6 automobiles were road tested for 300 miles in both city and highway driving conditions. The following data were recorded for miles-per-gallon performance. City 16.2 16.7 15.9 14.4 16 16.2 Highway 19.4 20.6 18.3 18.6 18.6 18.7 Use mean, median, and mode to make a statement about the difference in performance for city and highway driving. Which area of Statistics helps you to either validate or disprove such a statement and why?

A restaurant manager is interested in taking a more statistical approach to predicting customer load. She begins the process by gathering data. One of the restaurant hosts or hostesses is assigned to count customers every five minutes from 7 P.M. until 8 P.M. every Saturday night for three weeks. The data are shown here. After the data are gathered, the manager computes lambda using the data from all three weeks as one data set as a basis for probability analysis.What value of lambda did she find? Assume that these customers randomly arrive and that the arrivals are Poisson distributed. Use the value of lambda computed by the manager and help the manager calculate the probabilities in parts (a) through (e) for any given five-minute interval between 7 P.M. and 8 P.M. on Saturday night. Number of Arrivals Week 1 Week 2 Week 3 3 1 5 6 2 3 4 4 5 6 0 3 2 2 5 3 6 4 1 5 7 5 4 3 1 2 4 0 5 8 3 3 1 3 4 3 a. What is the probability that no customers arrive during any given five-minute interval? b. What is the probability that five or more customers arrive during any given five-minute interval? c. What is the probability that during a 10-minute interval fewer than four customers arrive? d. What is the probability that between four and six (inclusive) customers arrive in any 10-minute interval? e. What is the probability that exactly six customers arrive in any 15-minute interval? *Round your answers to 4 decimal places when calculating using Table A.3. **Round your answer to 4 decimal places, the tolerance is +/-0.0005. a. P(x = 0) = * b. P(x ≥ 5) = * c. P(x < 4 | 10 minutes) = * d. P(4 ≤ x ≤ 6 | 10 minutes) = * e. P(x = 6 | 15 minutes) = **

3)

A sample of midterm grades for five students showed the results: 72, 65, 82, 90, and 76. Based on the data, which of the following statements are correct, and which should be challenged as being too generalized? Justify your answer. a. The average midterm grade for the sample of five students is 77. b. The average midterm grade for all students who took the exam is 77. c. An estimate of the average midterm grade for all students who took the exam is 77. d. More than half of the students who take this exam will score between 70 and 85. e. If five other students are included in the sample, their grades will be between 65 and 90.

Hello,

I need assistance in explaining this output,

Let X be a exponential random variable with pdf f(x) = λe−λx for x > 0, and cumulative distribution function F(x).

(a) Show that F(x) = 1−e −λx for x > 0, and show that this function satisfies the requirements of a cdf (state what these are, and show that they are met). [4 marks]

(b) Draw f(x) and F(x) in separate graphs. Define, and identify F(x) in the graph of f(x), and vice versa. [Hint: write the mathematical relationships, and show graphically what the functions represent.] [4 marks]

(c) X has mgf M(t) = λ(λ−t) −1 . Derive the mean of the random variable from first principles (i.e. using the pdf and the definition of expectation). Also show how this mean can be obtained from the moment generating function. [10 marks]

(d)

(i) Show that F −1 (x) = − 1 λ ln(1 − x) for 0 < x < 1, where ln(x) is the natural logarithm. [4 marks]

(ii) If 0 < p < 1, solve F(xp) = p for xp, and explain what xp represents. [4 marks] (iii) If U ∼ U(0, 1) is a uniform random variable with cdf FU (x) = x (for 0 < x < 1), prove that X = − 1 λ ln(1 − U) is exponential with parameter λ. Hence, describe how observations of X can be simulated. [4 marks]

If I have two dice A and B, and I roll it twice, we have the outcomes A1, A2, B1, B2. Let X = A1+ A2, Y= B1+B2. What is the probability of X+Y <= 22.

Copier maintenance. The Tri-City Office Equipment Corporation sells an imported copier on a franchise basis and performs preventive maintenance and repair service on this copier. The data below have been collected from 45 recent calls on users to perform routine preventive maintenance service; for each call, X is the number of copiers serviced and Y is the total number of minutes spent by the service person. Assume that first-order regression model (1.1) is appropriate. (a) Obtain the estimated regression function. (b) Plot the estimated regression function and the data. How well does the estimated regression function fit the data? (c) Interpret b o in your estimated regression function. Does b o provide any relevant information here? Explain. (d) Obtain a point estimate of the mean service time when X = 5 copiers are serviced. Use R programming . The data set is 20 2 60 4 46 3 41 2 12 1 137 10 68 5 89 5 4 1 32 2 144 9 156 10 93 6 36 3 72 4 100 8 105 7 131 8 127 10 57 4 66 5 101 7 109 7 74 5 134 9 112 7 18 2 73 5 111 7 96 6 123 8 90 5 20 2 28 2 3 1 57 4 86 5 132 9 112 7 27 1 131 9 34 2 27 2 61 4 77 5