In this Question, use the ‘full’ regression model you obtained in Qu.#1, part ‘b
ID: 3263426 • Letter: I
Question
In this Question, use the ‘full’ regression model you obtained in Qu.#1, part ‘b’.b. Obtain the linear regression equation for all the above data; call it the ‘full’ regression equation.
a. Manually conduct the Hypothesis Test on ‘1’ at the 5% ‘Level of Significance’. Use ‘Critical Value’ of ‘t’ to reach your conclusion. Also calculate the p-Value and based on it, justify your conclusion.
b. Manually conduct the Hypothesis Test on ‘Significance of Regression’ at the 5% ‘Level of Significance’. Use ‘Critical Value’ of ‘F’ to reach your conclusion. Also calculate the p-Value and based on it, justify your conclusion.
c. Manually obtain the 99% Confidence Interval for the ‘Percentage Market Share’ if the ‘Monthly Ad Expense’ is $170,000. What does this Interval mean? Explain in brief.
d. Manually obtain the 99% Prediction Interval for the ‘Percentage Market Share’ if the ‘Monthly Ad Expense’ is $170,000. What does this Interval mean? Explain in brief.
e. Why is the ‘Prediction Interval’ broader than the ‘Confidence Interval’? Explain in brief.
f. With the whole of the data or the thirteen ordered pairs, obtain a linear regression equation, by dropping the constant term. What will be its R2 and R2Adj ? Is dropping the constant term particularly essential? Comment briefly.
Leisure Activity Products Manufacturing Company (LAPMC) tracked its monthly 'Market Share in %, or , and its monthly Advertising Expenses, in $K (thousands of dollars) or X. The following data was obtained over a period of time. Use this data for both Qu and Qu.#2. Data Display X 64 88 156142 76 96 104 120130 146 150 160 174 Y 68 2 22 20 10 14 121816 20 22 28 26Explanation / Answer
Let dependent variable Y = Market share in %
Independent variable X = advertising expenses in $K
Here we have to fit regression onf Y on X.
We can fit regression of Y on X in EXCEL.
steps :
ENTER data into EXCEL sheet --> Stat --> Regression --> Regression --> Response : Y --> Predictors : X --> Options --> Predicted intervals of new observations : 170 --> Confidence level : 99.0 --> Storage : click on confidence limits and prediction limits --> ok --> Results : select second option --> ok --> ok
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Welcome to Minitab, press F1 for help.
Regression Analysis: Y versus X
The regression equation is
Y = - 5.98 + 0.187 X
Predictor Coef SE Coef T P
Constant -5.979 2.100 -2.85 0.016
X 0.18663 0.01640 11.38 0.000
S = 1.991 R-Sq = 92.2% R-Sq(adj) = 91.5%
Analysis of Variance
Source DF SS MS F P
Regression 1 513.33 513.33 129.53 0.000
Residual Error 11 43.59 3.96
Total 12 556.92
Predicted Values for New Observations
New Obs Fit SE Fit 99.0% CI 99.0% PI
1 25.748 0.941 ( 22.823, 28.673) ( 18.903, 32.593)
Values of Predictors for New Observations
New Obs X
1 170
b. Obtain the linear regression equation for all the above data; call it the ‘full’ regression equation.
The regression equation is
Y = - 5.98 + 0.187 X
a. Manually conduct the Hypothesis Test on ‘1’ at the 5% ‘Level of Significance’. Use ‘Critical Value’ of ‘t’ to reach your conclusion. Also calculate the p-Value and based on it, justify your conclusion.
Here we have to test the hypothesis that,
H0 : B = 0 vs H1 : B not= 0
where B is population slope for independent variable.
Assume alpha =level of significance = 5% = 0.05
Here test statistic follows t-distribution.
Test statistic = 11.38
P-value= 0.000
P-value < alpha
Reject H0 at 5% level of significance.
Conclusion : The population slope for X is differ than 0.
We get significant result about X.
b. Manually conduct the Hypothesis Test on ‘Significance of Regression’ at the 5% ‘Level of Significance’. Use ‘Critical Value’ of ‘F’ to reach your conclusion. Also calculate the p-Value and based on it, justify your conclusion.
Here we can test the hypothesis that,
H0 : Bj= 0 Vs H1 : Bj not= 0
where Bj is population slope for jth independent variable.
Test statistic follows F distribution.
Test statistic = 129.53
P-value = 0.000
P-value < alpha
Reject H0 at 0.05 level of significance.
COnclusion : The population slope for jth independent variable is differ than 0.
Here also we get significant result about jth independent variable.
c. Manually obtain the 99% Confidence Interval for the ‘Percentage Market Share’ if the ‘Monthly Ad Expense’ is $170,000. What does this Interval mean? Explain in brief.
The 99% Confidence Interval for the ‘Percentage Market Share’ if the ‘Monthly Ad Expense’ is $170,000 is (22.823, 28.673).
We are 99% confident that the population slope is lies between 22.823 and 28.673
d. Manually obtain the 99% Prediction Interval for the ‘Percentage Market Share’ if the ‘Monthly Ad Expense’ is $170,000. What does this Interval mean? Explain in brief.
The 99% Prediction Interval for the ‘Percentage Market Share’ if the ‘Monthly Ad Expense’ is $170,000 is (18.903, 32.593).
We are 99% confident that the population slope is lies between 18.903 ad 32.593.
e. Why is the ‘Prediction Interval’ broader than the ‘Confidence Interval’? Explain in brief
A prediction interval includes process variability and is therefore wider than a confidence interval. A confidence interval is an interval for a parameter, which is a constant (though unknown), while a prediction interval is for a random variable. The liability on a line of business is a random variable, not a parameter.
f. With the whole of the data or the thirteen ordered pairs, obtain a linear regression equation, by dropping the constant term. What will be its R2 and R2Adj ? Is dropping the constant term particularly essential? Comment briefly.
R-sq = 92.2%
It expresses the proportion of variation in Y which is explained by variation in X.
No need to remove the variable from the model since we get significant result.