List Practice Problems:

Week of January 4 In text : 1.15, 1.25, 1.55, 1.61 and

1.
The nine players chosen to start for the National League in the major league All Star game had the following batting averages:

.304 .321 .289 .307 .352 .284 .329 .306 .190

a)
What is the mean batting average for these starters?
b)
What is the standard deviation for the batting averages of the starters?
c)
An astute observer questions the inclusion of the pitcher (who has the . 190 average) in the calculation because pitchers are not paid to hit. If the pitcher is omitted, what is the batting average of the eight other starters and what is the standard deviation for them?
d)
Do you think that the batting average for the eight starters would be a good estimate for the population mean of all major league ballplayers? Would you expect the population mean to be larger or smaller? Why?
e)
Do you think that the standard deviation of the eight batting averages for the starters (excluding the pitcher) is a good estimate for the population standard deviation? Would you expect the population standard deviation to be larger or smaller? Why?
2.
Students are surveyed as to their number of close friends (individuals to whom they would not be afraid to confide a deep secret). The frequency distribution of replies is given below.

 Number of friends (x) f 0 2 1 3 2 5 3 5 4 4 5 1
a)
How many students were surveyed?
b)
Compute the relative frequencies and construct a histogram.
c)
What percentage of the students had 2 or fewer close friends?
d)
Find the median, mean and standard deviation.
3.
Columnist Ann Landers was asked whether having children was worth the problems involved. She asked her readers "If you had it to do over again, would you have children?" A few weeks later her column was headlined "Seventy Percent of Parents Say Kids Not Worth It", because 70% of the parents who wrote said they would not have children if they could make the choice again. Is this a valid conclusion? Comment.

Week of January 11 In text: 1.69, 1.71, 1.77, 1.79, 1.81, 1.87, 1.91, 1.95 and

1.
a)
In 1975 Governor Brown of California proposed that all state employ given a flat raise of \$70 per month. What would this do to the mean monthly salary of state employees? to the standard deviation?
b)
What would a 5% increase in the salaries, across the board, do to the average monthly salary? to the standard deviation?
2.
100 dumpsites were tested for the presence of six toxic pollutants. The results are given below:
Number of pollutants found:  0 1 2 3 4 5 6

Number of dumpsites  2 28 25 22 11 9 3
a)
What was the mean number of pollutants found per dumpsite.
b)
If a dumpsite is chosen at random from the 100 sites, what is the probability that it has more than 3 pollutants? fewer than 2 pollutants? no pollutants?
c)
Construct a boxplot for these data. Are the data symmetric?
d)
Compare the mean and median for the dumpsite data. Are the results of this comparison supported by your conclusions in (c)?
e)
Compute the standard deviation. What proportion of the observations fall in the I interval ± 2s ?

Week of January 18 In text: 2.3, 2.5, 2.15, 2.21, 2.23, 2.33, 2.39, 2.43, 2.47, 2.51, 2.59

Week of January 25 In text: 2.87, 2.89, 2.91, 2.93, 2.95

Week of February 1 In text: 3.11, 3.21, 3.47, 3.53

Week of February 8 In text: 4.13, 4.15, 4.39, 4.41, 4.43, 4.51, 4.55, 4.79, 4.85, 4.87, 4.89, 4.91

Week of February 15 In text: 4.61, 4.65, 4.69, 5.3 and review problems from old midterms

Week of March 1 In text: 5.5, 5.9, 5.13, 5.19, 5.21, 5.29, 5.33, 5.35, 5.37

Week of March 8 In text: 5.67, 6.1, 6.3, 6.7, 6.9, 6.10, 6.11

Week of March 15 In text: 6.23, 6.27, 6.29, 6.33, 6.35, 6.37, 6.45, 6.55

Week of March 22 In text: 7.1, 7.4, 7.19, 7.20 7.21, 7.31 and

1.
An optometrist claimed that the mean waiting time for her patients was at most 10 minutes. A sample of 16 patients showed an average waiting time of 12 minutes with a standard deviation of 4.4 minutes.
a)
Test the optometrist's claim. State null and alternative hypotheses, value of the test statistic, p-value and conclusion.
b)
For what values of a would the null hypothesis be rejected?

For what values of a would we fail to reject the claim?

c)
Calculate a 90% confidence interval for the mean waiting time.

Week of March 29 In text: 7.61, 7.63, 8.1, 8.5, 8.11, 8.12, 8.23 and

1.
In an investigation of the possible influence of dietary chromium on diabetic symptoms, 14 randomly selected rats were fed a low-chromium diet and 10 were fed a normal diet. One response was the activity of the liver enzyme GITH, which was measured using a radioactively labelled molecule. The accompanying table shows the results, expressed as thousands of counts per minute per gram of liver.

Low-Chromium Diet

 42.3 51.5 53.7 48 56 55.7 54.8 52.8 51.3 58.5 55.4 38.3 54.1 52.1

Normal Diet

 53.1 50.7 55.8 55.1 47.5 53.6 47.8 61.8 52.6 53.7
a)
Are these observational data or do the data come from a designed experiment?
b)
Plot the data and comment on whether there appears to be any difference in GITH activity between the diets. Comment on the symmetry of the data, whether there are any outliers, and whether the variances appear to be similar.
c)
Is there evidence that mean GITH activity is different for the two diets? (State null and alternative hypotheses and base your conclusion on the p-value).
d)
Calculate the 95% confidence interval for the mean difference in GITH activity.
e)
Suppose that the investigators believe that the effect of the low chromium diet is "important" if the population mean difference is more than 8 thousand counts per minute per gram of liver. Do the data support the conclusion that the difference is "important"?

Week of April 5 In text: 8.29, 9.21, 9.9 parts a) to e) and 9.9 part f)

Calculate the 90% confidence interval for the difference in the proportions of CHD survivors who are pet owners and those who are non pet owners.

Review problems from old exams - available in the Math/Stats Learning Centre

Jonathan Payne
January 14, 1999