Probability and Statistics

Question: The number of faculty listed for a variety of private colleges that o

The number of faculty listed for a variety of private colleges that offer only bachelor’s degrees is listed below. Use the data to construct a frequency distribution with 7 classes, a histogram, a frequency polygon and an ogive plot. Discuss the shape of the distribution. What proportion of schools have over 180 faculty members? EXPERT …

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Question: Sound Warehouse in Georgetown sells CD players (with speakers

Sound Warehouse in Georgetown sells CD players (with speakers), which it orders from Fuji Electronics in Japan. Because of shipping and handling costs, each order must be for five CD players. Because of the time it takes to receive an order, the warehouse outlet places an order every time the present stock drops to five …

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An investment firm has classified its clients according to their gender and the composition of their investment portfolio (primarily bonds, primarily stocks, or a balanced mix of bonds and stocks). The proportions of clients falling into the various categories are shown in the following table Portfolio Composition Bonds Gender Balanced Stocks 0.18 0.12 0.20 0.10 Male Female 0.25 0.15 One client is selected at random, and two events A and B are defined as follows: A: The client selected is male. B: The client selected has a balanced portfolio. Are the events A and B independent? Events A and B are dependent Events A and B are independent Events A and B are mutually exclusive cannot be determined with given information O O

EXPERT ANSWER right choice is first P(A)=0.63, P(B)=0.4, P(A and B)=0.25 P(A)*P(B)=0.63*0.4  P(A)*P(B)  P(A and B), so A and B are not independent since P(A and B)0, so A and B are not mutually exclusive so Events A and B are dependent right choice is first

3: Uniform distribution Let X!, , Xn be iid. from a uniform(0,0) distribution where θ is unknown, that is 0, otherwise a) Find the maximum likelihood estimator for 0, 0MLE b) Explain why will be a sufficient and complete statistic for 0 (see example 6.2.23) c) Find the method of moments estimator for θ.0MME- Is it a function of T(X)? d) Find the expectation and variance of 0MME e) Start with θΜΜΕ, and use the Rao-Blackwell theorem to construct a new estimator for θ, θRBE-What do you know about this estimator? f) Show that the probability density function of T(X) is given by 0 otherwise Use this distribution to find the expectation and variance for θMLE and θ11E g) Compare the MSE for MME, 0MLE and ORB h) Simulate n = 7 independent uniformly distributed values on (0,3) and cornpute 0MME and θRBE Repeat this experiment B 1000 times, and compare the performance of the two estima- tors by means, variances, histograms etc. (Hint: In R you can use function runif(7,0,3) to simulate an experiment.)

EXPERT ANSWER

Ashley’s Department Store in Kansas City maintains a successful catalog sales department in which a clerk takes orders by telephone. If the clerk is occupied on one line, incoming phone calls to the catalog department are answered automatically by a recording machine and asked to wait. As soon as the clerk is free, the party that has waited the longest is transferred and answered first. Calls come in at a rate of about 12 per hour. The clerk is capable of taking an order in an average of 4 minutes. Calls tend to follow a Poisson distribution, and service times tend to be exponential. The clerk is paid $10 per hour, but because of lost goodwill and sales, Ashley’s loses about $50 per hour of customer time spent waiting for the clerk to take an order.

(a) What is the average time that catalog customers must wait before their calls are transferred to the order clerk? (b) What is the average number of callers waiting to place an order? (c) Ashley’s is considering adding a second clerk to take calls. The store would pay that person the same $10 per hour. …

Ashley’s Department Store in Kansas City maintains a successful catalog sales department in which a clerk takes orders by telephone. If the clerk is occupied on one line, incoming phone calls to the catalog department are answered automatically by a recording machine and asked to wait. As soon as the clerk is free, the party that has waited the longest is transferred and answered first. Calls come in at a rate of about 12 per hour. The clerk is capable of taking an order in an average of 4 minutes. Calls tend to follow a Poisson distribution, and service times tend to be exponential. The clerk is paid $10 per hour, but because of lost goodwill and sales, Ashley’s loses about $50 per hour of customer time spent waiting for the clerk to take an order. Read More »

Question: Determine if each data column is discrete or continuous and ID th

Determine if each data column is discrete or continuous and ID the appropriate level of measurement for each column. Age (years)                         discrete/continuous……    nominal/ordinal/interval/ratioPulse rate                           discrete/continuous……    nominal/ordinal/interval/ratiosystolic B/P                        discrete/continuous……    nominal/ordinal/interval/ratiodiastolic B/P                  discrete/continuous……    nominal/ordinal/interval/ratioHDL (mg/dL)                          discrete/continuous……    nominal/ordinal/interval/ratioLDL (mg/dL)                           discrete/continuous……    nominal/ordinal/interval/ratiowhite blood cell count   discrete/continuous……    nominal/ordinal/interval/ratiored blood cell count           discrete/continuous……    nominal/ordinal/interval/ratioPlatelet count                      discrete/continuous……    nominal/ordinal/interval/ratioweight                                   discrete/continuous……    nominal/ordinal/interval/ratioheight                                    discrete/continuous……    …

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1 A study of the relationship between athletic involvement and academic achievement for col- lege students sampled randomly 852 students. The selected students were categorized accord- ing to amount of athletic involvement and grade- point averages at graduation. The results are listed in table below. Athletic Involvement 1-3 4 or More None Semesters Semesters Below Mean 42 GPA Above Mean 238 528 219 105 a Do final grade-point averages appear to be independent of athletic involvement? (Use a = 0.05.) b For students with four or more semesters of athletic involvement, is the proportion with GPAs above the mean significantly different, at the 0.05 level, from the proportion with GPAs below the mean?

EXPERT ANSWER Using R

Twelve goats were fed on diet A and fifteen on diet B. Gain in weight of each goat after a month is given below:

Test the assumption that there is no significance difference on weight on goats by using diet A and diet B. Explain the results in detail and also state on the basis result that which diet is better to increase the weight of goat.    EXPERT ANSWER Solution:- State the hypotheses. The first step is to state the …

Twelve goats were fed on diet A and fifteen on diet B. Gain in weight of each goat after a month is given below: Read More »