Math  /  Data & Statistics

QuestionElvin collected the scores of a random sample 41 students on the first exam in a certain class and their corresponding scores on the second exam in that class. Here is computer output on the sample data:
Summary statistics \begin{tabular}{lrrrr} Variable & n & Mean & StDev & SE Mean \\ \hlinex=x= first exam score & 41 & 59.5 & 19.7 & 3.1 \\ y=y= second exam score & 41 & 59.4 & 21.7 & 3.4 \end{tabular}
Regression: second exam score vs. first exam score \begin{tabular}{lrr} Predictor & Coef & SE Coef \\ \hline Constant & 6.985 & 6.65 \\ First exam score & 0.881 & 0.11 \\ S =13.2=13.2 & R-sq =62.89%=62.89 \% & \end{tabular}
Assume that all conditions for inference have been met. Which of these is an appropriate test statistic for testing the null hypothesis that the population slope in this setting is 0 ?
Choose 1 answer: (A) t=6.9856.65t=\frac{6.985}{6.65} (B) t=0.8810.1141t=\frac{0.881}{\frac{0.11}{\sqrt{41}}} (C) t=0.8810.11t=\frac{0.881}{0.11} (D) t=59.43.4t=\frac{59.4}{3.4} (ㄷ) t=6.9856.6541t=\frac{6.985}{\frac{6.65}{\sqrt{41}}}

Studdy Solution

STEP 1

What is this asking? We're checking if there's a *real* relationship between scores on the first and second exams, specifically if a higher score on the first exam *actually* predicts a higher score on the second! Watch out! Don't get lost in the tables!
We're looking for a specific test statistic related to the *slope*, not the intercept or the means of the exams.

STEP 2

1. Understand the Regression Output
2. Calculate the Test Statistic

STEP 3

The table labeled "Regression: second exam score vs. first exam score" gives us the important information about the relationship between the two exam scores.
The "Coef" column gives us the estimated values for the intercept and the slope of the regression line.
The "SE Coef" column gives us the standard error of these estimates.

STEP 4

We're interested in the *slope*, which tells us how much the second exam score is predicted to change for every one-point increase in the first exam score.
The estimated slope is β1=0.881\beta_1 = 0.881, and its standard error is SE(β1)=0.11SE(\beta_1) = 0.11.

STEP 5

To test if the *true* population slope is zero (meaning no relationship between the exam scores), we use a *t*-test.
The formula for the test statistic is: t=Estimated SlopeHypothesized SlopeSE(Estimated Slope)t = \frac{\text{Estimated Slope} - \text{Hypothesized Slope}}{SE(\text{Estimated Slope})}

STEP 6

In our case, the estimated slope is β1=0.881\beta_1 = 0.881, the hypothesized slope (under the null hypothesis) is 00, and the standard error of the estimated slope is SE(β1)=0.11SE(\beta_1) = 0.11.
Plugging these values into the formula, we get: t=0.88100.11=0.8810.11t = \frac{0.881 - 0}{0.11} = \frac{0.881}{0.11}

STEP 7

The correct answer is (C), t=0.8810.11t = \frac{0.881}{0.11}.
This calculation gives us the test statistic to check if the slope of the relationship between first and second exam scores is *significantly* different from zero!

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