Question

Question: Comparing mosquito repellents. Which insect repellents protect best against mosquitoes? Consumer Reports (June 2000) tested 14 products that all claim to be an effective mosquito repellent. Each product was classified as either lotion/cream or aerosol/spray. The cost of the product (in dollars) was divided by the amount of the repellent needed to cover exposed areas of the skin (about 1>3 ounce) to obtain a cost-per-use value. Effectiveness was measured as the maximum number of hours of protection (in half-hour increments) provided when human testers exposed their arms to 200 mosquitoes. The data from the report are listed in the table.

1. Suppose you want to use repellent type to model the cost per use (y). Create the appropriate number of dummy variables for repellent type and write the model.
2. Fit the model, part a, to the data.
3. Give the null hypothesis for testing whether repellent type is a useful predictor of cost per use (y).
4. Conduct the test, part c, and give the appropriate conclusion. Use α = .10.
5. Repeat parts a–d if the dependent variable is the maximum number of hours of protection (y).

Short answer

Answer

1. For qualitative regression model, variables are introduced in the model where, . Since in this question there are only 2 levels, 1 qualitative variable is introduced in the model. Where when repellent is lotion based, 0 otherwise.
2. From the excel output, the regression equation becomes$E\left(y\right)=0.7775+0.109167x_1$
3. The null and alternate hypothesis for whether repellent type is a useful predictor of cost per use (y) can be written as $H_0\text{:}{\beta }_1=0$against.$H_a\text{:}{\beta }_1\ne 0$
4. At 95%significance level,${\beta }_3=0$ meaning that is statistically significant for the model
5. The regression equation for maximum hours of protection as a function of repellent type is.$E\left(y\right)=7.5625-1.64583x_1$ At 95% significance level${\beta }_3=0$, meaning that ${\beta }_3$is statistically significant for the model.

Step-by-step solution

Given Information and definition

It is given that y is the dependent variable (here, cost per use). The test to be conducted at $\alpha =.10$.

Dummy variables indicate the presence or absence of the categorical effect which may shift the outcome. Dummy variable can take only the value 0 or 1.

Dummy variables 

a.

For qualitative regression model,(k-1) variables are introduced in the model where k = no. of levels. Since in this question there are only 2 levels, 1 qualitative variable x₁ is introduced in the model where x₁ = 1 when repellent lotion is based, 0 otherwise.

Regression model

b. From the excel output, the regression equation becomes $E\left(y\right)=0.7775+0.109167x_1$.

For the anova table one need to calculate the mean of the independent variable and then calculate the SSR, SSE AND SST, after that one need to calculate the degrees of freedom and the mean squares and then F.

The SSR is calculated by using$n\Sigma {\left(X_j--{\overline{x}}_j..\right)}^2$, and the SSE is calculated by squaring the each term and adding them all. The SST is the sum of SSR and SSE. The MS regression is calculated by dividing SST by degrees of regression and similarly the MS residual is calculated by dividing SSE by degrees of residual and F is calculated by dividing MS regression by MS residual.

The coefficients of x is calculated by using this formula:$\frac{n\left(\sum xy\right)-\left(\sum x\right)\left(\sum y\right)}{n\left(\sum x_2\right)-\left(\sum x_2\right)}$whereas the coefficient of intercept is calculated by $\frac{\left(\sum y\right)\left(\sum x_2\right)-\left(\sum x\right)\left(\sum xy\right)}{n\left(\sum x_2\right)-\left(\sum x_2\right)}$.

The standard error is calculated by dividing the standard deviation by the sample size's square root.

Hypothesis statement

c.

The null and alternate hypothesis for whether repellent type is a useful predictor of cost per use (y) can be written as$H_0\text{:}{\beta }_1=0$ against $H_a\text{:}{\beta }_1\ne 0$.

Significance of β1 

d.

$$\begin{gathered} H_0\text{:}{\beta }_1=0 \\ {H}_a\text{:}{\beta }_1\ne 0 \end{gathered}$$

Here, t-test statistic

.$$\begin{gathered} t=\frac{\overbrace{{\beta }_1}}{S\overbrace{{\beta }_1}} \\ =\frac{0.109167}{0.454457} \\ =0.24021 \end{gathered}$$

Value of$t_{0.05,13}$is 1.771.

H₀ is rejected if t statistic$>t_{0.05,13}$.

For$\alpha =0.05$,since$<t_{0.05,13}$.

Not sufficient evidence to reject H₀ at 95% confidence interval.

Therefore,${\mathbf{\beta }}_{\mathbf{3}}\mathbf{=}\mathbf{0}$meaning thatrole="math" localid="1660794217559" ${\mathbf{\beta }}_{\mathbf{3}}$is statistically significant for the model.

Regression model when dependent variable is maximum number of hours of protection

e.

For the anova table one need to calculate the mean if the independent variable and then calculate the SSR, SSE AND SST, after that onee need to calculate the degrees of freedom and the mean squares and the F.

The SSR is calculated by using$n\Sigma (X_i-\stackrel{⏜}{X_j})$, and the SSE is calculated by squaring the each term and adding them all. The SST is the sum of SSR and SSE. The MS regression is calculated by dividing SST by degrees of regression and similarly the MS residual is calculated by dividing SSE by degrees of residual and F is calculated by dividing MS regression by MS residual.

The coefficients of x is calculated by using this formula: $\frac{n\left(\sum xy\right)-\left(\sum x\right)\left(\sum y\right)}{n\left(\sum x_2\right)-\left(\sum x_2\right)}$ whereas the coefficient of intercept is calculated by $\frac{\left(\sum y\right)\left(\sum x2\right)-\left(\sum x\right)\left(\sum xy\right)}{n\left(\sum x_2\right)-\left(\sum x_2\right)}$.

The standard error is calculated bydividing the standard deviation by the sample size's square root.

The R-squared is calculated as$R_2=SSyy--\frac{SSE}{SSyy}$.

The excel output is

The regression equation for maximum hours of protection as a function of repellent type is$E\left(y\right)=7.5625-1.64583x_1$

To test the significance of${\beta }_1$

$$\begin{gathered} H_0\text{:}{\beta }_1=0 \\ {H}_a\text{:}{\beta }_1\ne 0 \end{gathered}$$

Here, t-test statistic

$$\begin{gathered} t=\frac{\overbrace{{\beta }_1}}{S\overbrace{{\beta }_1}} \\ =\frac{-1.64583}{3.573625} \\ =-0.46054 \end{gathered}$$

Value of $t_{0.05,13}$is 1.771

H₀ is rejected if t statistic $>t_{0.05,13}$

For,since$>t_{0.05,13}$.

Not sufficient evidence to reject H₀ at95%confidence interval.

Therefore,${\mathbf{\beta }}_{\mathbf{3}}\mathbf{=}\mathbf{0}$meaning that${\mathbf{\beta }}_{\mathbf{3}}$ is statistically significant for the model.