ISyE 6414 Regression Analysis Homework 3 solution

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Enterprise Inc. produces Clean, a brand of liquid laundry detergent. In order to study the
relationship between demand and price, the company gathered the data for 30 sales periods
(each sales period is a month) where:

• y: demand for the large bottle of Clean (in hundreds of thousands of bottles) in the
sales periods.
• x: The price difference per bottle (in dollars) between the Clean price and the average
price of competitors’ similar detergents in the sales period.

By using the data set (6414 HW3 Clean.csv), answer Questions 1–9. Submit the relevant
outputs and R codes.

1. Construct and submit a scatter plot of y versus x. Does a simple linear regression
model seem appropriate here?
2. Construct and run a simple linear regression model using x as the predictor variable
and y as the response variable. By using the output, identify and report βˆ
0, βˆ
1, ˆσ,
se(βˆ
0), and se(βˆ
1).

3. Calculate a 95% confidence interval for the slope β1 and give a statement in plain
English interpreting this interval.
4. Conduct a hypothesis test using α = 0.05 to see whether or not the predictor x is
statistically significant. State the null and alternative hypotheses, the test statistic,
critical t-value, and your conclusion. Give a statement in plain English interpreting
this result.

5. Identify the p-value for testing the intercept β0. By using the p-value, state whether
or not the intercept is statistically significant.
6. Identify the p-value for testing β1. Using the p-value, determine whether we can reject
H0 by setting α to 0.10, 0.05, and 0.005. What do you conclude about the strength of
the relationship between y and x?
1

7. By using R, find a point estimate of and a 95% confidence interval for the mean demand
when the price difference is 0.10.
8. By using R, find a point prediction of and a 95% prediction interval for the actual
demand for Clean in an individual sales period when the price difference is 0.10. Is
this interval wider than the confidence interval for the mean demand found in Question
7? Answer by comparing the half-lengths numerically.

9. Find a point estimate of and a 95% confidence interval for the mean demand when the
price difference is 0.25. Is the half-length of this confidence interval identical to the one
found in Question 7? If not, why? Answer by comparing the half-lengths numerically.

Derivation Question

10. Suppose you know that the true β0 (intercept) is zero in a simple linear regression
model, and you do not want to include the intercept in the model, i.e., the no-intercept
model is y = β1x + . Show that the least squares estimate of β1 for such a model that
minimizes SSE =
Pn
i=1(yi − yˆi)
2
is
βˆ
1 =
Pn
i=1
P
xiyi
n
i=1 x
2
i
2