Math 254 Assignment 4 (2011)

Math 254
Assignment 4 (2011)
1. High airline occupancy rates on scheduled flights are essential to corporate profitability. Suppose
a scheduled flight must average at least 60% occupancy in order to be profitable, and an
examination of the occupancy rate for 120 10:00 A.M flights from Toronto to Calgary showed a
mean occupancy per flight of 58% and a standard deviation of 11%.
(a) If µ is the mean occupancy per flight and if the company wishes to determine whether or
not this scheduled flight is unprofitable, give the alternative and the null hypotheses for
the test.
(b) Does the alternative hypothesis in part (a) imply a one or two-tailed test? Explain.
(c) Do the occupancy data for the 120 flights suggest that this scheduled flight is unprofitable?
Test using α = .05.
2. “Welcome to the new movie pre-show!” Before you can see the newly released movie you
have just paid to see, you must sit through a variety of trivia slides, snack bar ads, paid
product advertising, and movie trailers. Although the total barrage of advertising may last
up to 20 minutes or more, a particular theatre chain claim that the average length of any
one advertisement is no more than 3 minutes. To test this claim, 50 theatre advertisements
were randomly selected and found to have an average of 3 minutes 15 seconds with a standard
deviation of 30 seconds. Do the data provide sufficient evidence to indicate that the average
duration of theatre ads is more than claimed by the theatre? Test at the 1% level of significance.
3. Analyses of drinking water samples for 100 homes in each of two different sections of a city
gave the following means and standard deviations of lead levels (in part per million):
Section 1 Section 2
Sample size 100 100
Mean 34.1 36.0
Standard deviation 5.9 6.0
(a) Calculate the test statistic and its p-value (observed significance level) to test for a difference
in the two population means. Use the p-value to evaluate the statistical significance
of the results at the 5% level.
(b) Use a 95% confidence interval to estimate the difference in the mean lead levels for the
two sections of the city.
(c) Suppose the city environment engineers will be concerned only if they detect a difference
of more than 5 parts per million in the two sections of the city. Based on your confidence
interval in part (b), is the statistical significance in part (a) of practical significance to the
city engineers?
4. A random sample of n = 1400 observations from a binomial population produced x = 529.
(a) If your research hypothesis is that p differs from 0.4, what hypotheses should you test?
(b) Calculate the test statistic and its p-value.
(c) Do the data provide sufficient evidence to indicate that p is different from 0.4?
5. Independent random samples of n 1 = 140 and n 2 = 130 observations were randomly selected
from binomial populations 1 and 2, respectively. Sample 1 had 74 successes and sample 2 had
72 successes.

Math 254 Assignment 4 (2011) Page 2 of 2
(a) Suppose you have no preconceived idea as to which parameter, p 1 or p 2 , is the larger,
but you want to detect only a difference between the two parameters if one exists. What
should you choose as the null and alternative hypotheses for a statistical test?
(b) Calculate the standard error for the difference in the two sample proportions, (ˆp 1 − ˆp 2 ).
Make sure to use the pooled estimate for the common value of p.
(c) p-value approach: Find the p-value for the test. Test for a significant difference in the
population proportions at the 1% significance level.
(d) Critical value approach: Find the rejection region when α = 0.01. Do the data provide
sufficient evidence to indicate a difference in the population proportions?
6. The braking ability was compared for two 2005 automobile models. Random samples of 64
automobiles were tested for each type. The recorded measurement was the distance (in meters)
required to stop when the brakes were applied at 65 km/h. These are the computed means and
variances:
Model 1 Model 2
x 1 = 36 x 2 = 33
s 2 1 = 31 s2 2 = 27
Do the data provide sufficient evidence to indicate a difference between the mean stopping
distances for the two models?
7. A soft-drink manufacturer claims that its 355 mL cans do not contain, on average, more than
30 calories. A random sample of 16 cans of this soft drink, which were checked for calories,
contained a mean of 31.8 calories with a standard deviation of 3 calories. Assume that the
number of calories in 355 mL cans is normally distributed. Does the sample information
support the alternative hypothesis that the manufacturer’s claim is false? Use a significance
level of 5%.
8. A random sample of 14 cans of Brand I soda produced a sample mean of 23 calories per can
with a standard deviation of 3 calories. Another random sample of 16 cans of Brand II soda
produced a sample mean of 25 calories per can with a standard deviation of 4 calories. At the
1% significance level, can you conclude that there is a difference between the mean number of
calories per can for these two brands? Assume that the calories per can of soda are normally
distributed for the two brands and that the standard deviations for the two populations are
equal.
9. A random sample of 15 observations from a normal population produced the following data.
119 107 104 115 93 104 116 97 101 105 91 104 103 98 109
(a) Find a 95% confidence interval for the population mean µ.
(b) Find a 95% confidence interval for the population standard deviation σ.
10. The following table lists the observed frequency for a sample of 120 rolls of a die.
Number 1 2 3 4 5 6
Frequency 14 23 17 31 19 16
Use a chi-square test at the 5% significance level to test if the die is fair?