STA 103: Lab 1
Solutions
1) tax ti paid by individual i is a function of income xi
ti = 0.1(xi 2000) i.e. at tax on all income over $2,000.
500 individuals, total income of $5,000,000
What is total tax revenue?
Total tax revenue is:
500
500
0.1(xi 2000)
t
Statistics 103, Lab 1:
Practice with n=1 notation
k
1
Linear properties of n n=1
k
Instructions: Hi Everyone. Here are the lab/discussion problems. The TAs will go over the
solutions in detail during lab/discussion (and hopefully have enough extra time to
Exercises
251
Chapter 14 Exercises
14.1 If you deal one card from a standard deck, the probability that the card is a heart
is 0.25. Find the odds of drawing a heart.
14. 3
A study was designed to compare two energy drink commercials. Each
participant was
Exercises
239
Summarize your results. Include a plot of the data with the least-squares regression
line. Plot the residuals versus each of the other two chemicals. Are any patterns
evident? (The concentrations of the other chemicals are lurking variables
236
Exercises
6.81
0.97
26.41
0.36
9.57
1.9
0
1.04
5.53
0.41
(a) Examine the distribution of CRP and serum retinol. Use graphical and
numerical methods.
(b) Forty percent of the CRP values are zero. Does this violate any assumptions
that we need to do a r
240
Exercises
Chapter 12 Exercises
12.3 An experiment was run to compare three groups. The sample sizes were 25, 22,
and 19, and the corresponding estimated standard deviations were 22, 20, and 18.
(a) Is it reasonable to use the assumption of equal stand
Exercises
255
Is food, expressed by the Diet/Cuisine score, more important than activities, expressed
as the Program/Facilities score, for a top ranking? Formulate this question in terms of
null and alternative hypotheses. Then compute the differences and
Exercises
243
amount of energy intake for infants who have other foods introduced into the diet at
different ages. Part of one study compared the energy intakes, measured in kilocalories
per day (kcal/d), for infants who were breast-fed exclusively for 4,
262
Exercises
17.15 A pharmaceutical manufacturer forms tablets by compressing a granular material
that contains the active ingredient and various fillers. The hardness of a sample from
each lot of tables is measured in order to control the compression pr
234
Exercises
Table 10.1 In-state tuition and fees (in dollars) for 32 public universities
University
Year 2000 Year 2005 University
Year 2000 Year 2005
Penn State
7018
11508 Purdue
3872
6458
Pittsburgh
7002
11436 Cal-San Diego
3848
6685
Michigan
6926
979
264
Exercises
17.43 Make a Normal quantile plot of the 85 distances in data file ex17_19 that remain
after removing sample 5. How does the plot reflect the limited precision of the
measurements (all of which end in 4)? Is there any departure from Normalit
246
Exercises
13.13 Analysis of data for a 3 2 ANOVA with 5 observations per cell gave the F
statistics in the following table:
Effect F
A
1.5
3
B
3.8
7
AB
2.9
4
What can you conclude from the information given?
13.17 Refer to the Exercise 13.16. Here are
244
Exercises
(b) Run the analysis of variance. Report the F statistic with its degrees of
freedom and p-value. What do you conclude?
12.53 Refer to Exercise 12.25. There are two comparisons of interest to the experimenter:
They are (1) Placebo versus the
238
Exercises
Chapter 11 Exercises
11.3 Recall Exercise 11.1. Due to missing values for some students, only 86 students
were used in the multiple regression analysis. The following table contains the
estimated coefficients and standard errors:
Variable
In
260
Exercises
16.77 Exercise 2.17 (page 97) describes a study that suggests that the pain caused by
social rejection really is pain, in the sense that it causes activity in brain areas known to
be activated by physical pain. Here are data for 13 subjects
258
Exercises
Chapter 16 Exercises
16.5 The distribution of carbon dioxide (CO2) emissions in Table 1.6 is strongly skewed
to the right. The United States and several other countries appear to he high outliers.
Generate a bootstrap distribution for the me
Exercises
245
Chapter 13 Exercises
13.7 A recent study investigated the influence that proximity and visibility of food have
on food intake. A total of 40 secretaries from the University of Illinois participated in the
study. A candy dish full of individu
Exercises
241
(a) To compare the mean scores across classes, what are the degrees of freedom
for the ANOVA F statistic?
(b) The MSG =11.806. If sp = 2.16, what is the F statistic?
(c) Give an approximate (from a table) or exact (from software) p-value. Wh
Exercises
237
(b) Find the least-squares line. Summarize the significance test for the slope.
What do you conclude?
(c) State carefully what the slope tells you about the relationship between wages
and length of service.
(d) Give a 95% confidence interval
242
Exercises
(a) Give a table with sample size, mean, standard deviation, and standard error
for each condition.
(b) Perform a one-way ANOVA for these data. Be sure to state your hypotheses,
the test statistic with degrees of freedom, and the p-value.
(c
Exercises
259
Suppose you are interested in the wealth of typical billionaires. Bootstrap an
appropriate statistic, inspect the bootstrap distribution, and draw conclusions based on
this sample.
16.31 Consider the small random subset of the Verizon data i
Exercises
261
Chapter 17 Exercises
17.5 A sandwich shop owner takes a daily sample of 6 consecutive sandwich orders at
random times during the lunch rush and records the time it takes to complete each
order. Past experience indicates that the process mean
Exercises
Female
Male
191
152
0
253
27
515
Use logistic regression to study the question of whether or not gender is related to
alcohol use in people who are fatally injured in bicycle accidents.
248
Exercises
experiment as one part of this examination. Some of the data are given in Table 13.2
reproduced below. The diameters in millimeters (mm) of five tools were measured by
the same operator at three times (8:00 a.m., 11:00 a.m., and 3:00 p.m.).
Exercises
263
it is clear that in effect the measurements were made to the nearest 0.001 inch, not to the
nearest 0.0001 inch.
Calculate x and s for the first two samples. The data file ex17_19 contains x
and s for all 18 samples. Based on long experience
254
Exercises
Chapter 15 Exercises
15.3 Refer to Exercise 15.1. State appropriate null and alternative hypotheses for this
setting and calculate the value of W, the test statistic.
Group A 55
2
Group B 32
9
44
8
78
0
68
56
0
24
3
54
0
30
24
0
15.5 Refer t
252
Exercises
14.13 Refer to Exercise 14.11. Use x = 1 for women and x = 0 for men.
(a) Find the estimates b0 and b1.
(b) Give the fitted logistic regression model.
(c) What is the odds ratio for men versus women?
14.21 Different kinds of companies compen
Exercises
247
Because the sample sizes in this experiment are very small, we expect a large
amount of variability in the sample standard deviations. Although they vary
more than we would prefer, we will proceed with the ANOVA.
(b) Make a plot of the means
256
Exercises
15.21 Refer to exercise 15.19. Find + , + , and the Normal approximation for the
W
W
p-value for the Wilcoxon signed rank test.
15.25 Can the full moon influence behavior? A study observed at nursing home
patients with dementia. The number o
250
Exercises
(a) For each of the F statistics given above find the degrees of freedom and an
approximate P-value. Summarize the results of these tests.
Exercises
235
other weather phenomena. Table 10.4 summarizes the annual number of
tornadoes in the United States between 1953 and 2005.
(a) Make a plot of the total number of tornadoes by year. Does a linear trend
over the years appear reasonable?
(b) Are