consumer's spending is widely believed to be a function of their income. To estimate this relationship, a university professor randomly selected 19 of his students and collected information on their spending (Y, in dollars) and income (X, in dollars) patterns in week 6 of the semester. Assuming a linear relationship between Y and X, the professor used the least-squares method and found that the Y intercept = 20.90 and the slope = 0.66. The professor also found that the standard error of the slope was 0.08. Based on this information, what conclusion should you reached at the 5% level of significance when testing the null hypothesis that there is no linear relationship between the two variables, X and Y? a. There is insufficient evidence at the 5% level of significance to conclude that there is a significant linear relationship between X and Y. b. There is sufficient evidence at the 5% level of significance to conclude that there is a significant linear relationship between X and Y. c. There is sufficient evidence at the 5% level of significance to conclude that there is a significant linear relationship between the Y intercept and the slope. d. There is sufficient evidence at the 5% level of significance to conclude that there is no significant linear relationship between X and Y.
Question
consumer's spending is widely believed to be a function of their income. To estimate this relationship, a university professor randomly selected 19 of his students and collected information on their spending (Y, in dollars) and income (X, in dollars) patterns in week 6 of the semester. Assuming a linear relationship between Y and X, the professor used the least-squares method and found that the Y intercept = 20.90 and the slope = 0.66. The professor also found that the standard error of the slope was 0.08. Based on this information, what conclusion should you reached at the 5% level of significance when testing the null hypothesis that there is no linear relationship between the two variables, X and Y?
a.
There is insufficient evidence at the 5% level of significance to conclude that there is a significant linear relationship between X and Y.
b.
There is sufficient evidence at the 5% level of significance to conclude that there is a significant linear relationship between X and Y.
c.
There is sufficient evidence at the 5% level of significance to conclude that there is a significant linear relationship between the Y intercept and the slope.
d.
There is sufficient evidence at the 5% level of significance to conclude that there is no significant linear relationship between X and Y.
Solution
To answer this question, we need to conduct a hypothesis test for the slope of the regression line. The null hypothesis is that the slope is equal to zero (which would mean there is no linear relationship between X and Y), and the alternative hypothesis is that the slope is not equal to zero (which would mean there is a significant linear relationship between X and Y).
The test statistic for this hypothesis test is calculated as the estimated slope divided by the standard error of the slope. In this case, the test statistic is 0.66 / 0.08 = 8.25.
Next, we compare this test statistic to a critical value from the t-distribution with n-2 degrees of freedom, where n is the number of pairs of observations. In this case, with 19 pairs of observations, we have 17 degrees of freedom. The critical value for a two-tailed test at the 5% level of significance with 17 degrees of freedom is approximately ±2.11.
Since our test statistic of 8.25 is greater than the critical value of 2.11, we reject the null hypothesis. Therefore, there is sufficient evidence at the 5% level of significance to conclude that there is a significant linear relationship between X and Y.
So, the correct answer is:
b. There is sufficient evidence at the 5% level of significance to conclude that there is a significant linear relationship between X and Y.
Similar Questions
A consumer's spending is widely believed to be a function of their income. To estimate this relationship, a university professor randomly selected 19 of his students and collected information on their spending (Y, in dollars) and income (X, in dollars) patterns in week 6 of the semester. Assuming a linear relationship between Y and X, the professor used the least-squares method and found that the Y intercept = 20.90 and the slope = 0.66. Based on this information, the slope should be interpreted as: a. For each increase of $1 in a student's weekly income, his/her mean value of weekly spending is estimated to increase by $0.66 (or 66 cents). b. For each increase of $1in a student's weekly spending, his/her mean value of weekly income is estimated to increase by $0.66 (or 66 cents). c. For each increase of $1 in a student's daily income, his/her mean value of daily spending is estimated to increase by 0.66 cents. d. For each increase of $1in a student's weekly income, his/her mean value of weekly spending is estimated to increase by $20.90.
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