You are the manager of the Blue Mountain Rigger store. You want to know how you are doing this year and whether you should order more hiking boots. The number of pairs of boots that you have sold each season during the last four years is presented as follows:
YEAR SEASON 2017 2018 2019 2020
SPRING 65 69 76 83
SUMMER 45 49 58 72
FALL 78 64 84 98
WINTER 32 46 47 54
Use Excel and estimate a simple linear trend model with t=0 in Spring of year 2017.
a. Write the estimated equation for the linear trend model.
b. How much of the variation in time can be explained by the trend?
c. You believe that there is an upward trend in sales of boots over the four years. What are the hypotheses you would use to test that belief?
HO:
HA:
d. Draw a picture of the appropriate sampling distribution and label on your drawing the CRITICAL value. Use alpha = 0.05.
e. What is the CALCULATED test statistic you would use to compare to the part (d) answer?
f. What should you conclude about an upward trend?
g. Develop a 95% confidence interval that captures the change in the sales of boots with the passage of a quarter of time.
h. What is the p-value associated with a MODEL test designed to answer whether the simple trend has explanatory power?
Someone suggests to you that the sale of boots may vary from season to season. You decide that is a distinct possibility and add to your model variables to isolate seasonal differences.
i. What is the name we give to variables designed to incorporate qualitative information?
Use Excel and estimate a linear model that includes both a trend variable and seasonal dummy variables. Use FALL as the dummy variable base group.
j. Test the explanatory power of the four independent variable model. Use alpha = 0.05.
k. How much additional variation in boot sales can be explained by adding the seasonal variables to the original model?
l. What type of test would you use to determine whether or not the difference in explanatory power between these two models is statistically significant?
m. What are the hypotheses for the test you identified in part (k)?
HO:
HA:
n. At the 1% level of significance, what is the CRITICAL value for your test as described in part (l)?
o. What is the CALCULATED value of the test statistic for this test?
p. What decision and conclusion apply to your test?
q. What season serves as the base season in this model?
r. Interpret the estimated coefficient of the SPRING variable?
Suppose that your computer output included a Durbin Watson statistic with dw=0.96.
s. What violation of the linear regression assumptions would we use the Durbin Watson statistic to examine?
t. To examine if there is either positive or negative first order serial correlation in the error terms, what are our test hypotheses?
HO:
HA:
u. What are the CRITICAL values if alpha = 0.05? Please draw an illustration.
v. Based upon the dw=0.96 supplied above, what decision and conclusion are appropriate?
w. Given that Durbin Watson statistic, what is an approximate value for the correlation coefficient between consecutive error terms?