Question

Please conduct a Hypothesis test in the following scenario to determine if the population slope is statistically significant. State the appropriate test statistic, the critical values, and whether or not you can reject a null hypothesis that the population slope is zero.

Use a 1% significance level. Suppose that you collected data to determine the relationship between the amount of time a person spends online as an independent variable and the amount of money a person spends online as the dependent variable.

The regression equation is = 24 + 8.5x, where x represents the number of hours a person spends online and represents the predicted amount of money that person spends online.

Sample size is 6.

SSE is 200

SSR is 800

SST is 1000

Standard deviation of the sampling distribution of the sample intercept is 9.54

Standard deviation of the sampling distribution of the sample slope is 2.26

Answer 1

The solution to this problem takes four steps: :

• State the hypotheses -

  H_o: The slope of the regression line is equal to zero.

  H_a: The slope of the regression line is not equal to zero.

  If the relationship between amount of time a person spends online and amount of money is significant, the slope will not equal zero.
• Formulate an analysis plan. For this analysis, the significance level is 0.01. Using sample data, we will conduct a linear regression t-test to determine whether the slope of the regression line differs significantly from zero.
• Analyze sample data. To apply the linear regression t-test to sample data, we require the standard error of the slope, the slope of the regression line, the degrees of freedom, the t test statistic, and the P-value of the test statistic.

  We get the slope (b₁) from the regression output and we calculate standard error (SE) from standard deviation

  b₁ = 8.5       SE = 2.26 / sqrt(6) = 0.92

• We compute the degrees of freedom and the t statistic test statistic, using the following equations.

  DF = n - 2 = 6 - 2 = 4

  t = b₁/SE = 8.5/0.92 = 9.23

  where DF is the degrees of freedom, n is the number of observations in the sample, b₁ is the slope of the regression line, and SE is the standard error of the slope.

  Based on the t test statistic and the degrees of freedom, we determine the P-value. The P-value is the probability that a t statistic having 4 degrees of freedom is more extreme than 9.23. Since this is a two-tailed test, "more extreme" means greater than 9.23 or less than -9.23. We use the t Distribution Calculator to find P(t > 9.23) = 0.0004 and P(t < -9.23) = 0.0004. Therefore, the P-value is 0.0004 + 0.0004 or 0.0008.
• Interpret results. Since the P-value (0.0008) is less than the significance level (0.01), we reject the null hypothesis.

The relationship between amount of time a person spends online and amount of money is significant