Exam 20: Probability and Combinatorics

SCENARIO 11-13 An important factor in selecting database software is the time required for a user to learn how to use the system. To evaluate three potential brands (A, B and C) of database software, a company designed a test involving five different employees. To reduce variability due to differences among employees, each of the five employees is trained on each of the three different brands. The amount of time (in hours) needed to learn each of the three different brands is given below: Software Operator A B C 1 17 17 23 2 18 17 23 3 14 13 19 4 12 11 18 5 19 17 22 Mean 16 15 21 $\text { Below is the Excel output for the randomized block design: }$ Source of Variation SS df MS F P-value Fcrit Rows 84.66667 4 21.16667 50.8 9.98-06 3.837853 Columns 103.3333 2 51.66667 124 9.54-07 4.45897 Error 3.333333 8 0.416667 Total 191.3333 14 -Referring to Scenario 11-13, the decision made at a 0.05 level of significance on the randomized block F test for the difference in means implies that all 3 means are significantly different from each other.

SCENARIO 11-11 An agronomist wants to compare the crop yield of 3 varieties of chickpea seeds. She plants all 3 varieties of the seeds on each of 5 different patches of fields. She then measures the crop yield in bushels per acre. Treating this as a randomized block design, the results are presented in the table that follows. 1 11.1 19.0 14.6 2 13.5 18.0 15.7 3 15.3 19.8 16.8 4 14.6 19.6 16.7 5 9.8 16.6 15.2 -Referring to Scenario 11-11, using an overall level of significance of 0.01, what is the critical range for the Tukey multiple comparison procedure?

SCENARIO 11-13 An important factor in selecting database software is the time required for a user to learn how to use the system. To evaluate three potential brands (A, B and C) of database software, a company designed a test involving five different employees. To reduce variability due to differences among employees, each of the five employees is trained on each of the three different brands. The amount of time (in hours) needed to learn each of the three different brands is given below: Software Operator A B C 1 17 17 23 2 18 17 23 3 14 13 19 4 12 11 18 5 19 17 22 Mean 16 15 21 $\text { Below is the Excel output for the randomized block design: }$ Source of Variation SS df MS F P-value Fcrit Rows 84.66667 4 21.16667 50.8 9.98-06 3.837853 Columns 103.3333 2 51.66667 124 9.54-07 4.45897 Error 3.333333 8 0.416667 Total 191.3333 14 -Referring to Scenario 11-13, what is the null hypothesis for testing the block effects? a) $H _ { 0 } : \mu _ { A } = \mu _ { B } = \mu _ { C }$ b) $H _ { 0 } : \mu _ { 1 } = \mu _ { 2 } = \mu _ { 3 } = \mu _ { 4 } = \mu _ { 5 }$ c) $H _ { 0 } : M _ { A } = M _ { B } = M _ { C }$ d) $H _ { 0 } : M _ { 1 } = M _ { 2 } = M _ { 3 } = M _ { 4 } = M _ { 5 }$

SCENARIO 11-11 An agronomist wants to compare the crop yield of 3 varieties of chickpea seeds. She plants all 3 varieties of the seeds on each of 5 different patches of fields. She then measures the crop yield in bushels per acre. Treating this as a randomized block design, the results are presented in the table that follows. 1 11.1 19.0 14.6 2 13.5 18.0 15.7 3 15.3 19.8 16.8 4 14.6 19.6 16.7 5 9.8 16.6 15.2 -Referring to Scenario 11-11, what is the null hypothesis for testing the block effects? a) $H _ { 0 } : \mu _ { \text {Field 1. } } = \mu _ { \text {Field 2. } } = \mu _ { \text {Field 3. } } = \mu _ { \text {Field 4. } } = \mu _ { \text {Field 5. } }$ b) $H _ { 0 } : \mu _ { \text {smith. } } = \mu _ { \text {Walsh. } } = \mu _ { \text {Trevor } }$ . c) $H _ { 0 } : M _ { \text {Field 1. } } = M _ { \text {Field 2. } } = M _ { \text {Field 3. } } = M _ { \text {Field 4. } } = M _ { \text {Field 5. } }$ . d) $\quad H _ { 0 } : M _ { \text {Smith } } = M _ { \text {Walsh. } } = M _ { \text {Trevor } }$ .

To construct a bootstrap confidence interval estimate for the population mean, you will first select a random sample of size n without replacement from a population of size N as the initial sample.

SCENARIO 16-16 Given below are the average prices for three types of energy products for five consecutive years. Year Electricity Natural Gas Fuel Oil 1 43.205 25.893 0.892 2 16.959 28.749 0.969 3 47.202 28.933 1.034 4 48.874 29.872 0.913 5 48.693 28.384 0.983 -Referring to Scenario 16-16, what are the simple price indices for electricity, natural gas and fuel oil, respectively, in year 5 using year 1 as the base year?

SCENARIO 5-11 There are two houses with almost identical characteristics available for investment in two different neighborhoods with drastically different demographic composition. The anticipated gain in value when the houses are sold in 10 years has the following probability distribution: -Referring to Scenario 5-11, if your investment preference is to maximize your expected return and not worry at all about the risk that you have to take, will you choose a portfolio that will consist of 10%, 30%, 50%, 70%, or 90% of your money on the house in neighborhood A and the remaining on the house in neighborhood B?

SCENARIO 5-11 There are two houses with almost identical characteristics available for investment in two different neighborhoods with drastically different demographic composition. The anticipated gain in value when the houses are sold in 10 years has the following probability distribution: -Referring to Scenario 5-11, if you can invest 70% of your money on the house in neighborhood A and the remaining on the house in neighborhood B, what is the portfolio expected return of your investment?

SCENARIO 9-12 A drug company is considering marketing a new local anesthetic. The effective time of the anesthetic the drug company is currently producing has a normal distribution with a mean of 7.4 minutes with a standard deviation of 1.2 minutes. The chemistry of the new anesthetic is such that the effective time should be normally distributed with the same standard deviation. The company will market the new local anesthetic as being better if there is evidence that the population mean effective time is greater than the 7.4 minutes of the current local anesthetic. -Referring to Scenario 9-12, if you select a sample of 25 new local anesthetics and are willing to have a level of significance of 0.1, the probability of a Type II error is _____ if the population mean effective time is 8 minutes.

SCENARIO 11-12 A student team in a business statistics course designed an experiment to investigate whether the brand of bubblegum used affected the size of bubbles they could blow. To reduce the person-to-person variability, the students decided to use a randomized block design using themselves as blocks. Four brands of bubblegum were tested. A student chewed two pieces of a brand of gum and then blew a bubble, attempting to make it as big as possible. Another student measured the diameter of the bubble at its biggest point. The following table gives the diameters of the bubbles (in inches) for the 16 observations. -Referring to Scenario 11-12, the randomized block F test is valid only if there is no interaction between the diameters of the 4 brands of bubble gums and the 4 ability of the students.

SCENARIO 5-11 There are two houses with almost identical characteristics available for investment in two different neighborhoods with drastically different demographic composition. The anticipated gain in value when the houses are sold in 10 years has the following probability distribution: -Referring to Scenario 5-11, what is the covariance of the two houses?

A simple price index tracks the price of a group of commodities at a given period of time to the price paid for that group of commodities at a particular point of time in the past.

SCENARIO 6-7 The interval between consecutive hits at a web site is assumed to follow an exponential distribution with a mean of 40 hits per minute. -Referring to Scenario 6-7, what is the probability that the next hit at the web site will occur within no sooner than 5 seconds after just being hit by a visitor?

The interval between patients arriving at an outpatient clinic follows an exponential distribution with mean 15 minutes. What is the probability that a randomly chosen arrival to be less than 15 minutes?

SCENARIO 11-12 A student team in a business statistics course designed an experiment to investigate whether the brand of bubblegum used affected the size of bubbles they could blow. To reduce the person-to-person variability, the students decided to use a randomized block design using themselves as blocks. Four brands of bubblegum were tested. A student chewed two pieces of a brand of gum and then blew a bubble, attempting to make it as big as possible. Another student measured the diameter of the bubble at its biggest point. The following table gives the diameters of the bubbles (in inches) for the 16 observations. -Referring to Scenario 11-12, what is the estimated relative efficiency?

The Paasche price index is a form of unweighted aggregate price index.

SCENARIO 8-17 A wealthy real estate investor wants to decide whether it is a good investment to build a high-end shopping complex in a suburban county in Chicago. His main concern is the total market value of the 3,605 houses in the suburban county. He commissioned a statistical consulting group to take a sample of 200 houses and obtained a sample mean market price of $225,000 and a sample standard deviation of $38,700. The consulting group also found out that the mean differences between market prices and appraised prices was $125,000 with a standard deviation of $3,400. Also the proportion of houses in the sample that are appraised for higher than the market prices is 0.24. -Referring to Scenario 8-17, if he wants a 95% confidence on estimating the true population mean market price of the houses in the suburban county to be within $10,000, how large a sample will he need?

The base period should be recent so that a price index is not severely affected by change in technology, and consumer attitudes and habits.

SCENARIO 8-16 A random sample of 100 stores from a large chain of 500 garden supply stores was selected to determine the mean number of lawnmowers sold at an end-of-season clearance sale. The sample results indicated a mean of 6 and a standard deviation of 2 lawnmowers sold. A 95% confidence interval (5.623 to 6.377) was established based on these results. -Referring to Scenario 8-16, if the population had consisted of 400 stores, the confidence interval estimate of the mean with finite population correction would have been wider in range.

SCENARIO 5-14 An accounting firm in a college town usually recruits employees from two of the universities in town. This year, there are fifteen graduates from University A and five from University B and the firm decides to hire six new employees from the two universities. -Referring to Scenario 5-14, what is the probability that at least one of the new employees will be from University A?