Chapter 13, Problem 32CE

(a)

To determine

Identify the p-value that indicates predictor significance at $\alpha =0.05$.

Answer to Problem 32CE

The p-value for ‘% with Bachelor's Degree’ indicates predictor significance at $\alpha =0.05$.

Explanation of Solution

Calculation:

The given information is that, the dataset of ‘Noodles & Company Sales, Seating, and Demographic data’ contains $n=74$ observations. The response variable is ‘annual sales per square foot’, there are $k=5$ predictor variables ‘Interior Seat Count, Patio Seat Count, Median HH Income, Median Age of Population, % with Bachelor's Degree’.

Software procedure:

Step by step procedure to obtain p-value for predictor using MegaStat software is given as,

• • Choose MegaStat >Correlation/Regression>Regression Analysis.
• • SelectInput ranges, enter the variable range for ‘Seats-Inside, Seats-Patio, MedIncome, MedAge, BachDeg%’ as the column of X, Independent variable(s)
• • Enter the variable range for ‘Sales/SqFt’ as the column of Y, Dependent variable.
• • Click OK.

Output using MegaStatsoftware is given below:

The p-value for predictor seats-inside is 0.0733.

The p-value for predictor seats-patio is 0.2350.

The p-value for predictor MedIncome is 0.0589.

The p-value for predictor MedAge is 0.9972.

The p-value for predictor BachDeg% is 0.0015.

For seats-inside:

Let ${\beta }_1$ is the parameter for the predictor seats-inside.

Null hypothesis:

$H_0:{\beta }_1=0$.

The predictor variable seats-inside is not related to annual sales.

Alternative hypothesis:

$H_1:{\beta }_1\ne 0$.

The predictor variable seats-inside is related to annual sales.

Rejection rules:

• • If p-value is less than the level of significance then the null hypothesis is rejected. The predictor is significant.
• • If p-value is greater than the level of significance then the null hypothesis is not rejected. The predictor is not significant.

Conclusion:

The p-value for predictor seats-inside is 0.0733.

The level of significance is 0.05.

The p-value is greater than the level of significance.

That is, $p\text{-value}\left(=0.0733\right)>\alpha \left(=0.05\right)$.

The null hypothesis is not rejected.

The predictor variable seats-inside is not related to annual sales.

The predictor seats-inside is not significant.

For seats-Patio:

Let ${\beta }_2$ is the parameter for the predictor seats-Patio.

Null hypothesis:

$H_0:{\beta }_2=0$

The predictor variable seats-Patio is not related to annual sales.

Alternative hypothesis:

$H_1:{\beta }_2\ne 0$

The predictor variable seats-Patio is related to annual sales.

Conclusion:

The p-value for predictor seats-patio is 0.2350.

The level of significance is 0.05.

The p-value is greater than the level of significance.

That is, $p\text{-value}\left(=0.2350\right)>\alpha \left(=0.05\right)$.

The null hypothesis is not rejected.

The predictor variable seats-patio is not related to annual sales.

The predictor seats-patio is not significant.

For Median Income:

Let ${\beta }_3$ is the parameter for the predictor median income.

Null hypothesis:

$H_0:{\beta }_3=0$

The predictor variable median income is not related to annual sales.

Alternative hypothesis:

$H_1:{\beta }_3\ne 0$

The predictor variable median income is related to annual sales.

Conclusion:

The p-value for predictor median income is 0.0589.

The level of significance is 0.05.

The p-value is greater than the level of significance.

That is, $p\text{-value}\left(=0.0589\right)>\alpha \left(=0.05\right)$.

The null hypothesis is not rejected.

The predictor variable median income is not related to annual sales.

The predictor median income is not significant.

For Median Age of population:

Let ${\beta }_4$ is the parameter for the predictor median age of population.

Null hypothesis:

$H_0:{\beta }_4=0$

The predictor variable median age of population is not related to annual sales.

Alternative hypothesis:

$H_1:{\beta }_4\ne 0$

The predictor variable median age of population is related to annual sales.

Conclusion:

The p-value for predictor median age of population is 0.9972.

The level of significance is 0.05.

The p-value is greater than the level of significance.

That is, $p\text{-value}\left(=0.9972\right)>\alpha \left(=0.05\right)$.

The null hypothesis is not rejected.

The predictor variable median age of population is not related to annual sales.

The predictor median age of population is not significant.

For % with Bachelor's Degree:

Let ${\beta }_5$ is the parameter for the predictor % with Bachelor's Degree.

Null hypothesis:

$H_0:{\beta }_5=0$

The predictor variable % with Bachelor's Degree is not related to annual sales.

Alternative hypothesis:

$H_1:{\beta }_5\ne 0$

The predictor variable % with Bachelor's Degree is related to annual sales.

Conclusion:

The p-value for predictor ‘% with Bachelor's Degree’ is 0.0015.

The level of significance is 0.05.

The p-value is less than the level of significance.

That is, $p\text{-value}\left(=0.0015\right)<\alpha \left(=0.05\right)$.

The null hypothesis is rejected.

The predictor variable ‘% with Bachelor's Degree’ is related to annual sales.

The predictor ‘% with Bachelor's Degree’of population is significant.

(b)

To determine

Explain whether p-values support the conclusions reached from the t tests.

Answer to Problem 32CE

Yes, p-values support the conclusions reached from the t tests.

Explanation of Solution

Justification: The conclusion reached from the t test is that the predictor variable ‘% with Bachelor's Degree’ is significant predictor and the predictor variables ‘seats-inside, seats-patio, median income, median age of population, ‘% with Bachelor's Degree’’ are not significant predictors. The conclusions for the predictor variables with the p-values at $\alpha =0.05$, level of significance are same as the t tests.

Hence, p-values support the conclusions reached from the t tests.

(c)

To determine

Explain whether the t test or the p-value approach is preferred.

Answer to Problem 32CE

The p-value approach is preferred because it determines the strength of the significance for the predictor.

Explanation of Solution

Justification: The conclusion using the t test and p-value approach are same, but in most of the tests p-value is preferred because the strength of the significance for the predictor is better determined using the p-value when compared with the t test statistic. Hence, the p-value approach is preferred in tests.