More Practice with Hypothesis Testing
For each of the problems, complete the following:
a) state the null and alternative hypotheses
b) state the p-value
c) state the conclusion…. ie Is the claim true or false or state what the results indicate. This requires a complete sentence or two.
1. The Magic Dragon Cigarette Company claims that their cigarettes contain an average of only 10 mg of tar. A random sample of
100 Magic Dragon cigarettes shows the average tar content to be 11.5 mg with standard deviation 4.5 mg. Construct a hypothesis
test to determine whether the average tar content of Magic Dragon cigarettes exceeds 10 mg. Use a 5% level of significance.
2. Statistical Abstracts (117th edition) reports that the average amount spent annually for food by householders under 25 years of age
is $2,690. A random sample of 16 people under 25 years of age who live in a university neighborhood were surveyed. The survey
showed that they spent a sample mean $3,220 with sample standard deviation $750. Test the claim that the mean for this
neighborhood is greater than the national average. Use a 5% significance level.
3. A random sample of 68 divorced men with incomes of $75,000 or more per year were asked how long their marriage lasted before
the divorce. The sample mean was 8.1 years with sample standard deviation 2.2 years. An independent random sample of 51
divorced men with incomes in the range $30,000 to 74,999 were asked the same question. For this group the sample mean was 7.8
years with sample standard deviation 2.6 years. Test to see if there is a difference either way in the mean length of marriage for
these income groups. Use a 5% significance level.
4. Does it make a difference which word-processing software you use? A random sample of 25 experienced secretaries were given
the task of typing, proofreading and formatting a fifty-page document. Thirteen used Word Master software while the rest used
Super Word software. The mean time required to produce the document using Word Master was 12.6 hr with standard deviation
1.7 hr. The mean time for the group using Super Word was 13.3 hr with standard deviation 1.9 hr. Test the claim that there is no
difference in the mean time using the two packages. Use a 1% level of significance.
5. A researcher conducted an experiment to see if specific eye exercises can improve peripheral vision. A random sample of 5 people
were rated for peripheral vision on a scale from 1 to 20 where 9 is considered average and higher scores indicate better peripheral
vision. Then they followed the prescribed eye exercise program and were rated again. The results follow:
Test the hypothesis that the exercises improved peripheral vision. Use a 5% significance level.
6. Have eating habits changed? In 1990 a random sample of 100 people surveyed showed that 39 ate red meat twice a week or less.
In 1996 an independent random sample of 80 people showed 36 ate red meat twice a week or less. In this time period has the
proportion of people limiting their consumption of red meat changed? Use a 5% significance level.
7. Dental associations recommend that the time lapse between routine dental checkups should average six months. A random sample
of 36 patient records at one dental clinic showed the average time between routine check-ups to be 7.2 months with standard
deviation 2 months. Do the sample data indicate that patients wait longer than recommended between dental checkups? In your
test, calculate the P value for the sample test statistic and determine if the data is statistically significant. Use a 1% significance
level.
8. Last year a random sample of 200 new car deals showed that 56 were leases. The rest were sales. An independent random sample
of 200 new car deals this year showed that 68 were leases. Test to see if the proportion of cars leased this year is greater than the
proportion that were leased last year. Use a 5% significance level.
9. The owner of a private fishing pond says that the average length of fish in the pond is 15 inches. A sample of 23 fish are caught
from various locations in the pond. The average length of these fish is 10.7 in. with standard deviation 4.8 in. Does the data
support the claim that the average fish length is less than 15 in.? Use a 1% significance level.
10. A large shipping company has a fleet of 90 almost identical cargo ships. A fluid dynamics engineer claims that a certain
modification of the front hull will allow the ships to glide better and faster at the same engine speed than the present design does.
The fleet is randomly divided into two groups of 30 and 60 ships. The group of 30 ships have their front hulls modified. When the
engines are at 3/4 power, the mean speed in a calm sea is measured for each ship. The group with the modified hulls have mean
speed 29.0 knots with standard deviation 5.0 knots. The mean speed for the unmodified ships is 25.6 knots with standard deviation
4.3 knots. Does this data support the engineer’s claim at the 1% significance level?