Exercise 1

The Journal of Biogeography published a study of ants in Mongolia. Botanists placed seed baits of five sites in the Dry Steppe region and six sites in the Gobi Desert and observed the number of ant species attracted to each site. These data are listed in the next table. Is there evidence to conclude that a difference exists between the average number of ant species found at sites in the two regions of Mongolia? Draw an appropriate conclusion using a=.05. (file:ants)

Exercise 2

For a specific community the distribution of Y, the numbers of bedrooms in homes, is given by the following table:

1. Find the mean number of bedrooms.
2. Find the standard deviation.
3. What proportion of homes has 4 or more bedrooms?

Exercise 3

When female undergraduates switch from science, mathematics, and engineering majors into disciplines that are not science-based, are their reasons different from those of their male counterparts? A sample of 335 junior/senior undergraduates – 172 female and 163 males – at two large research universities were identified as "switchers", that is they left a declared SME major for a non-SME major. Each student listed one or more factors that contributed to the switching decision.

1. Of the 172 females in the sample 74 listed lack or loss of interest in SME as a major factor, compared to 72 of the 163 males. Conduct a test (at \[\alpha =.10\] ) to determine whether the proportion of female switchers who give "lack of interest in SME" as a major reason for switching differs from the corresponding proportion of males.
2. Thirty-three of the 172 females in the sample indicated that they were discouraged or lost confidence because of low grades in SME during their early years, compared to 44 of 163 males. Construct a 90% confidence interval for the difference between the proportion of female and male switchers who lost confidence due to low grades in SME.. Interpret the results.

Solution: Using Minitab we find that

This means that the 90% confidence interval for the difference between the proportion is

Exercise 4

A telephone company can repair 70% of service problems the same day as they are reported. Assume that each repair is independent of any others.

1. Find the probability that of the next 5 service problems, 2 or more can be repaired on the same day as reported.
2. Find the mean number of service problems that can be repaired on the same day as reported.
3. What is the probability that the third service problem that can be repaired the day it is reported is the 6 ^th service problem that is reported?

Exercise 5

Reinforced concrete T-beam cracking experiment. The experimental results were compared to the theoretical results obtained using the failure surface method of predicting ultimate load capacity. The actual and theoretical ultimate torsion moments for six T-beams with 40-cm slab widths are given in the table. Conduct a test to determine whether the experimental mean ultimate torsion moment differs from the theoretical mean ultimate torsion moment. Use \[\alpha =.05\] . (file:Torsion)

Exercise 6

In a shipment of 25 hard disks, eight are defective. If five of the disks are selected for inspection, find the following:

1. The probability that exactly 3 are defective.
2. The average number of defective hard disks that you would expect to obtain in the five.

Exercise 7

In the manufacture of machinery, it is essential to utilize parts that conform to specifications. In the past, diameters of the ball bearings produced by a certain manufacturer had a variance of .00156. To cut costs, the manufacturer instituted a less expensive production method. The variance of the diameters of 100 randomly sampled bearings produced by the new process was .00211. Do the data provide sufficient evidence to indicate that diameters of ball bearings produced by the new process are more variable than those produced by the old process? Test at \[\alpha =.05\] .

Exercise 8

Repair time for an oil change at a car dealership has a mean of 42 minutes with a standard deviation of 5.4 minutes.

1. If a random sample of 50 oil changes is selected, what is the probability that the sample mean will be longer than 39.5 minutes?
2. What is the probability that the sample mean of the 50 oil changes will be between 40 and 43 minutes?

Exercise 9

Refer to the Process Safety Process and U.S. Chemical Safety and Hazard Investigation Board study of industrial accidents caused by management system failures. The accompanying table gives a breakdown of the root causes of a sample of 83 incidents. Are there significant differences in the percentage of incidents in the four cause categories? Test using \[\alpha =.05\] (file MSFAIL)

Exercise 10

Let Y be a random variable with the following density function: f(y)=2(2-y) for 1<y<2 and 0 for all other values of y.

1. What is the probability that y is greater than 1.5?
2. Find the cumulative distribution function, F(y).

Exercise 11

Four pesticides used in dormant California orchards are chlorpyrifos, diazinon, methidathion, and parathion. Environmental Science and Technology reported the number of applications of these spray chemicals from January to June 1990 in California. The data for each of three types of fruit or nut orchard are shown in the accompanying table. (file: Pesticide)

1. Conduct a test to determine ( \[\alpha =.10\] ) whether pesticide use depends on type of orchard.
2. Because of the large number of pesticide applications reported, the total sample size for the test, part a, is extremely large ( n =417,697). Consequently, a "statistically significant" result may not be "practically significant." Perform an analysis to show the magnitude of differences in the rates of methidathion application for the three orchard types.

Exercise 12

Suppose that the time between accidents in a factory follows an exponential distribution with a mean of 10 days between accidents.

1. What is the probability that the time until the next accident is less than 6 days?
2. What is the variance of the time until the next accident?

Exercise 13

The distribution of the forecasted flood crest of a flood warning system follows a Weibull distribution with =2 and =6.

1. What is the probability that the flood crest is forecast as between 2 and 8 feet?
2. What is the average forecast flood crest?

Exercise 14

The following represent the voltage output of new 1.5volt batteries: 1.577, 1.584, 1.588, 1.593, 1.576, 1.570, 1.581, 1.586, 1.577, 1.582, 1.585, 1.583, and 1.585. Do these appear to come from a normal distribution? Explain your answer.

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