Question

Consumer Reports provided extensive testing and ratings for more than 100 HDTVs. An overall score, based primarily on picture quality, was developed for each model. In general, a higher overall score indicates better performance. The following (hypothetical) data show the price and overall score for the ten 42-inch plasma televisions (Consumer Report data slightly changed here): Brand Price (X) Score (Y) Dell 2900 51 Hisense 2800 49 Hitachi 2500 45 JVC 3800 60 LG 3300 58 Maxent 2000 30 Panasonic 4500 70 Phillips 3100 54 Proview 2300 35 Samsung 2800 48 Use the above data to develop and estimated regression equation and interpret the coefficients. Compute Coefficient of Determination and correlation coefficient and show their relation. Interpret the explanatory power of the model. Estimate the overall score for a 42-inch plasma television with a price of $3400. Finally, test the significance of the slope coefficient. (Note that you need to provide necessary interpretations to get full points).

Answer 1

Estimated regression equation

Answer:

Intercept: The expected value of overall score = 3.7552 when the price of the 42-inch plasma television is zero.

Slope: For one unit increase in the price of the 42-inch plasma television, the overall score increases by 0.0154.

Explanation:

The regression equation is defined as,

Now, the regression analysis is done in excel by following steps

Step 1: Write the data values in excel. The screenshot is shown below,

Step 2: DATA > Data Analysis > Regression > OK. The screenshot is shown below,

Step 3: Select Input Y Range: 'Y' column, Input X Range: 'Xe' column then OK. The screenshot is shown below,

The result is obtained. The screenshot is shown below,

The regression equation is,

Coefficient of Determination and correlation coefficient And explanatory power of the model

From the regression output summary,

Multiple R	0.962623
R Square	0.926642

Correlation coefficient:

The correlation coefficent tells the stength and the magnitute of the relationship between to variable. The positive correlation coefficient indicates the positive relationship and maginitude is close to 1 which indicates the strong relationship between two variable. Hence,

There is a strong positive linear association between the overall score and the price of 42-inch plasma television which means as the price of a 42-inch plasma television increases, the overall score also increases.

Coefficient of Determination (R-square)

The R-square value tell, how well the regression model fit the data values. The R-square value of the model is 0.926642 which means, the model explains approximately 92.6642% of the variance of the data value.

Relation between Coefficient of Determination and correlation coefficient

Explanatory power of the model

The model explains approximately 92.6642% of the variance of the data value. Based on this evidence we can conclude the model is a good fit.

Predictetion

Answer:

Overall score = 56.166

Explanation:

The regression equation is,

For X = $3400,

Significance of the slope coefficient

Answer:

There is sufficient evidence to conclude that the slope coefficient is statistically significant in the model.

Explanation:

Hypothesis

This is a two-tailed test.

Let the significance level = 0.05

Test statistic

From the regression output summary,

…	t Stat
X	10.05259

P-value

…	P-value
X	8.16E-06

Conclusion:

Since the p-value is less than 0.05, the null hypothesis is rejected at a 5% significance level. Hence there is sufficient evidence to conclude that the slope coefficient is statistically significant in the model.