課程名稱︰統計學一下
課程性質︰必修
課程教師︰蔣明晃
開課學院：管理學院
開課系所︰工管系
考試日期（年月日）︰105/6/16
考試時限（分鐘）：三小時
試題 :
考試方式為自己帶筆電、事前從CEIBA下載data檔案。
1. (34%) Psychologists are interested in how much influences does the media,
especially reality TV programs, have one’s decision to undergo cosmetic
surgery. In a study, 170 college students answered question about their
impressions of reality TV shows featuring cosmetic surgery, level of
self-esteem, satisfaction with one’s own body, and desire to have cosmetic
surgery to alter one’s body. The variations analyzed in the study were
measured as follows:
DESIRE: scale ranging from 5 to 25, where the higher the value, the greater
the interest in having cosmetic surgery
Gender: 1 if male; 0 if female
SELFESTM: scale ranging from 4 to 40, where the higher the value, the greater
level of self-esteem
BODYSAT: scale ranging from 1 to 9, where the higher the value, the greater
satisfaction with one’s own body
IMPREAL: scale ranging from 1 to 7, where the higher the value, the more one
believes reality TV shows featuring cosmetic surgery are realistic
The data of study are saved in the file of BDYIMG. The psychologists used
multiple regression to model desire to have cosmetic surgery (y) as a
function of gender (x_1), self-esteem (x_2), body satisfaction (x_3), and
impression of reality TV (x_4).
Part I:
first-order model (E(y)=β_0+(β_1)(x_1)+(β_2)(x_2)+(β_3)(x_3)+(β_4)(x_4)+
ε)
a. Fit the first-order regression model to the data using the method of least
square. (3%)
b. Interpret the β estimates in the words of the problem. (4%)
c. Is the overall model statistically useful for predicting desire to have
cosmetic surgery? Using α=0.01. (3%)
d. Which statistic, R^2 or R^2-adj, is the preferred measure of model fit?
Practically interpret this statistic. (2%)
e. Conduct a test to determine whether desire to have cosmetic surgery
decreases linearly as level of body satisfaction increases. Use α=0.05. (2%)
Part II:
Interaction model (E(y)=β_0+(β_1)(x_1)+(β_2)(x_2)+(β_3)(x_1)(x_4)+ε)
f. Fit the interaction model to the data using the method of least square.
(3%)
g. Find the predicted level of desire for a male college student with an
impression-of-reality-TV-scale score of 5. (2%)
h. Conduct a test of overall model adequacy. Use α=0.10. (3%)
i. Conduct a test at α=0.10 to determine if gender and impression of reality
TV show interact in the prediction of level of desire for cosmetic surgery.
(2%)
j. Find an estimate of the change in desire for every 1-point increase in
impression of reality TV shows for female students and male students,
respectively. (2%)
Part II: The psychologists theorized that one’s impression of reality TV
will “moderate” the impact that the first three independent variables has
on one’s desire to have cosmetic surgery. If so, then x_4 will interact
with each of the other independent variables.
k. Write down the equation of the model for E(y) that matches the above
theory. (2%)
l. Fit the model, part k, to the data in the file. Evaluate the overall
model. Use α=0.01 (3%)
m. Setup the null hypothesis for testing the psychologists’ theory. (2%)
n. Conduct a partial-F test to test theory. Use α=0.05. (3%)
2. (9%) In general, before an academic publisher agrees to publish a book,
each manuscript is thoroughly reviewed by university professors. Suppose that
the Duxbury Publishing Company has recently received two manuscripts for
statistics books. To help them decide which one to publish both are sent to
30 professors of statistics who rate the manuscripts to judge which one is
better. Suppose that 10 Professors rate manuscript 1 better and 20 rate
manuscript 2 better.
a. Which kind of nonparametric tests is appropriate for the station? (2%)
b. Can Duxbury conclude at the 5% significance level that manuscript 2 is not
highly rated than manuscript 1? (4%)
c. What is the p-value of this test? (3%)
3. (22%) To determine whether extra personnel are needed for the day, the
owners of a water adventure park would like to find a model that would allow
them to predict the day’s attendance each morning before opening based on
the day of the week and weather conditions. The model is of the form
E(y)=β_0+(β_1)(x_1)+(β_2)(x_2)+(β_3)(x_3) where y= daily admission,
x_1={1, if weekend x_2={1, if sunny
{0 otherwise {0 otherwise
x_3= predicted daily high temperature (℉)
Part I:
These data were recorded for a random sample of 30 days, and a linear
regression model was fitted to the data. The least squares analysis produced
the following results:
y-hat=-105+25x_1+100x_2+10x_3 with s_b1=10,s_b2=30,s_b3=4,R^2=0.65
a. Interpret the estimated model coefficients. (3%)
b. Is there sufficient evidence to conclude that this model is useful for the
prediction of daily attendance? Use α=0.05 [Hint: think about how to
calculate F value if R^2 is known] (3%)
c. Is there sufficient evidence to conclude that the mean attendance
increases on weekend? Use α=0.05 (3%)
d. Use the model to predict the attendance on a sunny weekday with a
predicted high temperature of 95℉. (2%)
e. Suppose that 90% prediction interval for part d is (645, 1245). Interpret
this interval. (2%)
Part II: The owners of the water adventure park are advised that the
prediction model could probably be improved if interaction terms were added.
In particular, it is thought that the rate at which mean attendance increases
as predicted high temperature increases will be greater on weekend than on
weekdays. The following model is therefore proposed:
E(y)=β_0+(β_1)(x_1)+(β_2)+(x_2)+(β_3)(x_3)+(β_4)(x_1)(x_3)
The same 30 days of data in part a are used to obtain the least squares model
y-hat= 250-700x_1+100x_2+5x_3+15(x_1)(x_3)
g. Use the model to predict the attendance for a sunny weekday with a
predicted high temperature of 95℉. (2%)
h. Suppose that 90% prediction interval for part g is (800, 850). Compare
this result with the prediction model in part e. Do the relative widths of
the confidence interval support or refute your conclusion about the
effectiveness of interaction term (part f)? (2%)
i. The owners, noting that the coefficient b_1=-700, conclude the model is
ridiculous because it seems to imply that the mean attendance will be 700
less on weekends than weekdays. Does their argument make sense? If yes, how
could you solve the problem? If no, state your reason. (3%)
4. (9%) A movie critic wanted to determine whether or not moviegoers of
different age groups evaluate a movie differently. With this objective, he
commissioned a survey that asked people their ratings of the most recently
watched movies. The rating categories where: 1 = terrible, 2 = fair, 3 = good,
4
Each respondent was also asked to categorize his or her age as either: 1 =
teenager, 2 = young adult (20-34), 3 = middle age (35-50), 4 = over 50. The
result are shown below.
Movie Ratings
Teenager Young Adult Middle Age Senior
3 2 3 3
4 3 2 4
3 3 1 4
3 2 2 3
3 2 2 3
4 1 3 4
2 3 1 4
4 2 4 3
a. Which test can the movie critic use in this situation? (3 points)
b. Does this data provide sufficient evidence to infer at the 5% significance
level that there are differences in ratings among the different age
categories? (6 points)
5. (10%) Consider the 2011 monthly closing prices (i.e., closing prices on
the last day of each month) given in file HITECH for IBM company stocks
listed on the New York Stock Exchange,
a. Use the exponentially smoothed series (with w = .5) from January to
September 2011 to forecast the monthly values of the IBM stock price from
October to December 2011. Calculate the forecast errors. (4%)
b. Use a simple linear regression model fit to the IBM stock prices from
January to September 2011. Let time t range from 1 to 9, representing the 9
months in the sample. Interpret the least squares estimates. (3%)
c. Compare the exponential smoothing forecasts, part a, to the regression
forecasts, part b, using MAD and SSE. (3%)
6. (16%) The revenue (in $thousands) of a chain of fast food stores are
listed for each quarter during the previous 5 years in file Fastfood.
a. Use the regression analysis to determine the trend line. (3%)
b. Determine the seasonal indexes. (4%)
c. Using the seasonal indexes and trend line to forecast the next 4 quarters.
(4%)
d. Discuss whether exponential smoothing is an appropriate forecasting tool
in this problem. State your reasons. (3%)