A candy bar manufacturer is interested in trying to estimate how sales are influenced by the price of their product. To do this, the company randomly chooses six small cities and offers the candy bar at different prices. Using candy bar sales as the dependent variable, the company will conduct a simple linear regression on the data below:

Question

City Price (R) Sales
River Falls 1.30 100
Hudson 1.60 90
Ellsworth 1.80 90
Prescott 2.00 40
Rock Elm 2.40 38
Stillwater 2.90 32

To test that the regression coefficient, β, is not equal to 0, what would be the critical values? Use α = 0.05.

Answer 1

To test whether the regression coefficient \( \beta \) is significantly different from 0, you will typically conduct a hypothesis test using the t-statistic. The null hypothesis \( H_0 \) states that \( \beta = 0 \), indicating that price has no effect on sales, while the alternative hypothesis \( H_a \) states that \( \beta \neq 0 \).

Since this is a two-tailed test at a significance level of \( \alpha = 0.05 \), you will be looking for critical t-values that correspond to this significance level. Here are the steps to find the critical t-values:

1. Determine the degrees of freedom (df). For simple linear regression with one predictor, the degrees of freedom can be calculated as:

   \[ df = n - 2 \]

   where \( n \) is the number of observations. In your case, \( n = 6 \), so:

   \[ df = 6 - 2 = 4 \]

2. Find the critical t-values for a two-tailed test at \( \alpha = 0.05 \) with 4 degrees of freedom. You can look these values up in a t-distribution table or use statistical software.

For a two-tailed test with \( \alpha = 0.05 \) and \( df = 4 \):

• The critical t-values can be found by looking up \( \alpha/2 = 0.025 \) in a t-table.

Using a t-table or calculator:

• The critical t-value for \( df = 4 \) and \( \alpha = 0.025 \) (each tail) is approximately \( \pm 2.776 \).

Thus, the critical values to test the regression coefficient \( \beta \) at the \( \alpha = 0.05 \) level are:

Critical values: \( -2.776 \) and \( +2.776 \)