Question

A local trucking company fitted a regression to relate the travel time (days) of its shipments as a function of the distance traveled (miles). The fitted regression is Time = -7.126 + .0214 Distance, based on a sample of 20 shipments. The estimated standard error of the slope is 0.0053. Find the critical values for a two-tailed test to see if the slope is different from 0, using α = 0.10.

Answer 1

The critical values for α = 0.10 are t=±1.7341.

Explanation:

The critical values for a two tailed test are the values of t that are placed in the limit of the significance level. If the test statistic t is greater than this critical values, the null hypothesis is rejected.

To determine this values, we first divide the significance level by 2. With a t-table or app, we can determine the t values for P(x>|t|)=α/2=0.05.

In this case, for a degrees of freedom DF=20-2=18, we have the test statistic t=±1.7341 that satisfies the condition.

Solution of the hypothesis test

In this problem we want to prove if the slope of the regression model is different from 0, by means of a hypothesis test.

First, we state the null and alternative hypothesis:

`H_0: B_1=0\\\\H_1: B_1 \\eq 0`

The significance level is
`\alpha=0.10`.

The degrees of freedom for a simple regression model is:

`DF=n-2=20-2=18`

The standard error estimated of the slope is 0.0053.

The test statistic t is calculated as:

`t=(B_1)/(SE)=(0.0214)/(0.0053) = 4.0377`

The P-value for a two-tailed test for a t=4.0377 and DF=18 is:

`P=0.00078`

The P-value is smaller than the significance level, so the effect is statistically significant. The null hypothesis is rejected.