Question

Do blondes raise more funds? During fundraising, does the physical appearance of the solicitor impact the level of capital raised? An economist at the University of Nevada- Reno designed an experiment to answer this question and published the results in Economic Letters (Vol. 100, 2008). Each in a sample of 955 households was contacted by a female solicitor and asked to contribute to the Center for Natural Hazards Mitigation Research. The level of contribution (in dollars) was recorded as well as the hair color of the solicitor (blond Caucasian, brunette Caucasian, or minority female).

a) Consider a model for the mean level of contribution, E(y), that allows for different means depending on the hair color of the solicitor. Create the appropriate number of dummy variables for hair color. (Use minority female as the base level.)

b) Write the equation of the model, part a, incorporating the dummy variables.

c) In terms of the b’s in the model, what is the mean level of contribution for households contacted by a blond Caucasian solicitor?

d) In terms of the b’s in the model, what is the difference between the mean level of contribution for households contacted by a blond solicitor and those contacted by a minority female?

e) One theory posits that blond solicitors will achieve the highest mean contribution level, but that there will be no difference between the mean contribution levels attained by brunette Caucasian and minority females. If this theory is true, give the expected signs of the’s in the model.

f) The researcher found the b estimate for the dummy variable for blond Caucasian to be positive and significantly different from 0 (p-value < 0.01). The$\mathit{\beta }$estimate for the dummy variable for brunette Caucasian was also positive, but not significantly different from 0 (p-value < 0.10). Do these results support the theory, part e?

Short answer

a) For k no of level, the dummy variable created in the model is (k-1). Since here, there are 3 levels, 2 dummy variable x₁ and x₂ will be created where x₁ represents a female with blond Caucasian hair color and x₂ represents a female with brunette Caucasian hair color.

b) A model with 2 dummy variables can be written as $y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2$where $x_1$represents female with blond Caucasian hair color and $x_2$represents female with brunette Caucasian hair color.

c) The mean contribution level for households contacted by a blond Caucasian solicitor is represented when thus the value of contribution can be summarized.

d) The mean level of difference in contribution for households contacted by a blond solicitor and those contacted by a minority female is represented by${\beta }_1$_.

e) ${\beta }_0$will be positive since it represents the level of contribution of a household at the base level (which here is when the solicitor is minority female). ${\beta }_1$will also be positive since the theory posits that the blond solicitors will collect the highest mean contribution level.${\beta }_2$ will not be recorded in the model since the mean contribution levels of brunette Caucasian and minority female is the same.

f) To check whether${\beta }_1$ is different from 0, the$H_0:{\beta }_1$ while $H_0:{\beta }_1\ne 0$. Since the value of the p-value is given to be less than 0.01 which indicates that the value of ${\beta }_1$is not zero. However, for${\beta }_2$ the value is coming to be positive but not significantly different from 0 since the p-value is greater than 0.10 thus the value of ${\beta }_2$will be 0.

Step-by-step solution

Creating dummy variables

For k no of level, the dummy variable created in the model is (k-1). Since here, there are 3 levels, 2 dummy variable x₁ and x₂ will be created where x₁ represents female with blond Caucasian hair color and x₂ represents female with brunette Caucasian hair color.

Dummy variable model

A model with 2 dummy variables can be written as representing a female with blond Caucasian hair color and a female with brunette Caucasian hair color.

Interpretation of β

The mean contribution level for households contacted by a blond Caucasian solicitor is represented when thus the value of contribution can be summarized.

Analysis of β

The mean level of difference in contribution for households contacted by a blond solicitor and those contacted by a minority female is represented by_{.${\beta }_1$}

Signs of β’s

_{${\beta }_0$}will be positive since it represents the level of contribution of a household at the base level (which here is when the solicitor is a minority female)

${\beta }_1$will also be positive since the theory posits that the blond solicitors will collect the highest mean contribution level.

_{${\beta }_2$}will not be recorded in the model since the mean contribution levels of brunette Caucasian and minority female is the same.

Hypothesis testing for β’s

To check whether${\beta }_1$ is different from 0, the$H_0:{\beta }_1$while$H_0:{\beta }_1\ne 0$. Since the value of the p-value is given to be less than 0.01 which indicates that the value of${\beta }_1$is not zero. However, for${\beta }_2$the value is coming to be positive but not significantly different from 0 since the p-value is greater than 0.10 thus the value of${\beta }_2$will be 0.