# Social Studies 201

Winter 2004

Problem Set 4

Due: Monday, March 15, 2004

If you hand in the answers to this problem set by Friday, March 12, Mark Nelson will attempt to grade it by March 14.  There will be a review session in the Tuesday labs, March 14, prior to the second midterm examination the next day.

1.  Standardized normal distribution.  For the standardized normal distribution,

a.       What is the area between Z of 0 and Z of +1.25?

b.       What is the area between Z of +0.5 and Z of +2.5?

c.       What is the proportion of cases between Z = -1.8 and Z = +2.5?

d.       What percentage of the area under the normal curve lies to the left of Z = -1.33?

e.       What is the area under the normal curve above Z = -1.83?

f.        In a normally distributed population, what is the percentage of the population is within one and a half standard deviations of the mean?

g.       What is the Z-value so that 0.25 of the area lies to the right of this Z?  (That is, what is the Z-value of the 75th percentile or 3rd quartile?)

h.       What are the Z-values so that there is 0.035 of the area in each tail of the distribution beyond these Z-values, for a total of 0.070 in the two tails of the distribution?

i.         In a standardized normal distribution, where is the fifteenth percentile?

j.         The Explore procedure in SPSS displays the “trimmed mean,” defined as the mean when the largest 5% and the smallest 5% of the cases have been eliminated.  In the standardized normal distribution, what are the Z-values for the trim points?

2.  Distribution of income.  The distribution of household income for Saskatchewan respondents has a mean of fifty thousand dollars and a standard deviation of thirty-five thousand dollars.  Using these values, and assuming that household income is normally distributed, obtain the following.

1. Proportion of households with incomes below \$10,000.
2. Percentage of households with incomes above \$100,000.
3. Proportion of households with income between \$30,000 and \$80,000.
4. Proportion of households with income between \$10,000 and \$30,000.
5. Compare the results of a. – d. with the actual distribution of household income in Table 1, commenting on whether household income in Saskatchewan appears normally distributed.

3.  Television and internet use.  The data in Table 2 come from Saskatchewan respondents aged 15-24 surveyed in Statistics Canada, 2000 General Social Survey, Cycle 14: Access to and Use of Information Communication Technology.  Use these data to answer the following for Saskatchewan residents aged 15-24:

1. Obtain 92% confidence intervals for the estimate of mean weekly hours (i) watching television, (ii) using internet at home, and (iii) using internet at school.
2. Obtain a 90% confidence interval for mean weekly hours using internet at work.
3. How large a sample size would be required to obtain the true mean weekly hours of internet use for all Saskatchewan residents aged 15-24, correct to within one hour, with 98% confidence?
4. A researcher claims the mean hours of television use and of internet use at work and at home are each 11 hours per week.  From the results of a. and b., what do you conclude about this claim?

4.  Problems using data from t:\students\public\201\ssae98.sav

a.  Use Analyze-Descriptive Statistics-Frequences, with options Charts-Histograms-With Normal Curve, to obtain frequency distributions of the three variables: study hours, V3 (affirmative action), and V4 (gays and lesbians married).  The frequency distribution table and the histogram, with the normal curve superimposed, should be available on the printout.

i.  For study hours, use the frequency distribution table to determine the percentage of cases that are within one standard deviation of the mean; within two standard deviations of the mean.  Compare with the percentages of cases within one and two standard deviations of the mean in a normal distribution.  Use the figure and diagram on the printout to write a note comparing the actual distribution of study hours with that of a normal distribution.

ii.  For V3 and V4, use the statistics generated and the table of the normal distribution to determine the percentage of cases that take on the neutral response of 3 (between 2.5 and 3.5) if these variables were exactly normally distributed.  Compare with the percentage of neutral responses in the table of the frequency distribution.  Write a short note comparing these two frequency distributions with the normal distribution.

b.  Use Analyze-Descriptive Statistics-Explore with Statistics selected and Plots deselected to obtain the following confidence intervals.  Assume the data set is a random sample of all undergraduates at the University of Regina.

i.  90%, 95%, and 99% confidence interval estimates for true mean weekly study hours of all undergraduates.   Use the formula from class to verify the value of one of the interval estimates and the standard error.  Very briefly explain why the intervals differ in width.

ii.  Obtain 80% and 99% confidence interval estimates for the true mean debt level (debt1) for students in each year of their program (first through fifth year).  From these tables, what might you say about the mean debt level for all University of Regina students at each of the four undergraduate years?  Why is the 80% interval for first year students so much narrower than the 99% interval for fourth year students?

Table 1. Frequency and percentage distribution of Saskatchewan household income

 Income in thousands of dollars Number of respondents Percentage of respondents 0 15 1.7 Less than 5 10 1.2 5-10 36 4.2 10-15 67 7.7 15-20 56 6.5 20-30 113 13.1 30-40 112 12.9 40-50 101 11.7 50-60 103 11.9 60-80 104 12.0 80-100 69 8.0 100 plus 79 9.1 Total 865 100.0

Table 2.  Statistics of weekly hours of television and internet use, Saskatchewan respondents aged 15-24 who used each service

 Variable Mean hours per week Standard deviation of hours per week Sample size Watch television 13.46 10.18 180 Use internet at home 9.03 9.61 63 Use internet at work 7.60 10.80 15 Use internet at school 3.98 3.89 45

Source for tables: Statistics Canada, 2000 General Social Survey, Cycle 14: Access to and Use of Information Communication Technology