AMS 315 HW 3 Chapter 7. (Due Thursday. 2/25)
7.5 A packaging line fills nominal 32-ounce tomato just jars with a quantity of juice having a
normal distribution with a mean of 32.30 ounces. The process should have a standard deviation
smaller than .15 ounces per jar. (A larger standard deviation leads to too many underfilled and
overfilled jars.) A random sample of 50 jars is taken every hour to evaluate the process. The data
from one such sample are summarized here.
Descriptive Statistics for Juice Data
Variable
Juice Jars
Variable
Speed (mph)
a.
N
Mean
Median
TrMean
StDev
50
32.267
32.248
32.270
0.135
Minimum
Maximum
31.874
32.515
Q1
Q3
32.177
32.376
SE Mean
0.019
If the process yields jars having a normal distribution with a mean of 32.30 ounces and a
standard deviation of .15 ounces, what proportion of the jars filled on the packaging line
will be underfilled?
b. Does the plot suggest any violation of the conditions necessary to use the chi-square
procedures for generating a confidence interval and a test of hypotheses about σ?\
c.
Construct a 95% confidence interval on the process standard deviation/
d. Do the data indicate that the process standard deviation is greater than .15? Use α = .15.
e.
Place bounds on the p-value of the test.
7.9 Baseballs vary somewhat in their rebounding coefficient. A baseball that has a large rebound
coefficient will travel further when the same force is applied to it than a ball with a smaller
coefficient. To achieve a game in which each batter has an equal opportunity to hit a home run,
the balls should have nearly the same rebound coefficient. A standard test has been developed to
measure the rebound coefficient of baseballs. A purchaser of large quantities of baseballs requires
that the mean coefficient value be 85 units and the standard deviation be less than 2 units. A
random sample of 81 baseballs is selected from a large batch of balls and tested. The data are
summarized here.
a.
Does the plot indicate any violation of the conditions underlying the use of the chisquare procedures for constructing confidence intervals or testing hypotheses about σ?
b. Is there sufficient evidence that the standard deviation in rebound coefficient for the
batch of balls is less than 2?
c.
Estimate the standard deviation of the rebound coefficients using a 95% confidence
interval.
7. 15 A soft-drink firm is evaluating an investment in a new type of canning machine. The
company has already determined that it will be able to fill more cans per day for the same cost if
the new machines are installed. However, it must determine the variability of fills using the new
machines, and wants the variability from the new machines to be equal to or smaller than that
currently obtained using the old machines. A study is designed in which random samples of 61
cans are selected from the output of both types of machines and the amount of fill (in ounces) is
determined. The data are summarized in the following table and boxplots.
Boxplots of odd machine and new machine (means are indicated by solid circles)
7. 18 In Example 7.9 we stated that the Hartley test was not appropriate because there was
evidence that two of the population distributions were nonnormal. The BFL test was then applied
to the data and it was determined that the data did not support a difference in the population
variances at an α = .05 level. The data yielded the following summary statistics:
a.
Using the plots in Example 7.9, justify that the population distributions are not normal.
b. Use the Hartley test to test for differences in the population variances.
c.
Are the results of the Hartley test consistent with those of the BFL test?
d. Which test is more appropriate for this data set? Justify your answer.
e.
Which of the additives appears to be a better product? Justify your answer.
7. 20 A wildlife biologist was interested in determining the effect of raising deer in captivity on
the size of the deer. She decided to consider three populations; deer raised in the wild, deer
raised on large hunting ranches, and deer raised in zoos. She randomly selected eight deer in
each of the three environments and weighted the deer at age 1 year. The weights (in pounds) are
given in the following table.
a.
The biologist hypothesized that the weights of deer from captive environments would
have a larger level of variability than the weights from deer raised in the wild. Do the data
support her contention?
b. Are the requisite conditions for the test you used in (a) satisfied in this situation? Provide
plots to support your answer.