A college administers a student evaluation questionnaire for...

A college administers a student evaluation questionnaire for all its courses. For a random sample of 12 courses, the accompanying table and the data file Student Evaluation show both the average student ratings of the instructor (on a scale of 1 to 5), and the average expected grades of the students (on a scale of $\mathrm{A}=4$ to $\mathrm{F}=0$ ). $$ \begin{array}{ccccccccccccc} \hline \begin{array}{l} \text { Instructor } \\ \text { rating } \end{array} & 2.8 & 3.7 & 4.4 & 3.6 & 4.7 & 3.5 & 4.1 & 3.2 & 4.9 & 4.2 & 3.8 & 3.3 \\ \hline \begin{array}{l} \text { Expected } \\ \text { grade } \end{array} & 26 & 2.9 & 3.3 & 3.2 & 3.1 & 2.8 & 2.7 & 2.4 & 3.5 & 3.0 & 3.4 & 2.5 \\ \hline \end{array} $$ a. Find the sample correlation between instructor ratings and expected grades. b. Test, at the $10 \%$ significance level, the hypothesis that the population correlation coefficient is zero against the alternative that it is positive.

A college administers a student evaluation questionnaire for all its courses. For a random sample of 12 courses, the accompanying table and the data file Student Evaluation show both the average student ratings of the instructor (on a scale of 1 to 5), and the average expected grades of the students (on a scale of $\mathrm{A}=4$ to $\mathrm{F}=0$ ).
$$
\begin{array}{ccccccccccccc}
\hline \begin{array}{l}
\text { Instructor } \\
\text { rating }
\end{array} & 2.8 & 3.7 & 4.4 & 3.6 & 4.7 & 3.5 & 4.1 & 3.2 & 4.9 & 4.2 & 3.8 & 3.3 \\
\hline \begin{array}{l}
\text { Expected } \\
\text { grade }
\end{array} & 26 & 2.9 & 3.3 & 3.2 & 3.1 & 2.8 & 2.7 & 2.4 & 3.5 & 3.0 & 3.4 & 2.5 \\
\hline
\end{array}
$$
a. Find the sample correlation between instructor ratings and expected grades.
b. Test, at the $10 \%$ significance level, the hypothesis that the population correlation coefficient is zero against the alternative that it is positive.

Key Concepts

Pearson Correlation Coefficient

This statistical measure quantifies the strength and direction of the linear relationship between two continuous variables. It produces a value between -1 and 1, indicating perfect negative or positive linear relationships at the extremes, with values near zero implying weak or no linear association.

Sample Correlation

The sample correlation is an estimate of the true population correlation, calculated from a set of sample data. It is used to assess the association between two variables in the sample, serving as a basis for further statistical inference about the population.

Hypothesis Testing for Correlation

This involves evaluating whether the observed sample correlation could have arisen by chance if the true population correlation were zero. The test sets up a null hypothesis that there is no linear relationship between the variables (correlation coefficient equals zero) and compares the observed correlation against this hypothesis.

Test Statistic and t-Distribution

In testing the significance of the Pearson correlation coefficient, the test statistic is derived from the sample correlation and typically follows a t-distribution with n-2 degrees of freedom (where n is the sample size). This allows for determining whether the observed correlation is statistically significant at a chosen significance level.

Transcript

00:01 Okay, this question says, the following table gives the recorded grades for grade 10 students on a midterm test and final exam in a calculus course.

00:16 You can see the grades there for midterm test and then those for the final exam.

00:22 Then the first question says, calculate the rank correlation coefficient.

00:27 You must find the rank correlation coefficient.

00:30 They are okay.

00:31 Now then b says test the now hypothesis that p is greater than p is equal to zero against the alternative that says p is greater than zero use the zero comma alpha is called zero comma zero two five there okay now let's run straight to the uh let's delete this question and see how right so we are saying for us to find the rank correlation coefficient list this is the formula that we we use there right it's r right is equal to 1 minus 6 then multiplied by the summation of the difference between the rank squared that's d there, let me say d, d, i there.

01:19 Right.

01:20 Then over n brackets n to the power of n squared there minus 1.

01:31 Okay some they write n's cubed minus n there all right now we have to find d right we can find n is the number of tests that i there is 10 but d we don't have d so we just need to find d there which is the differences between the ranks the ranks the exam and the test there all right so now let's we have to rank so the table that i've included here is for the ranks right the first test there the lowest which is number one my lowest was zero this student got a zero in the test day so it was a zero and that's why he's ranked number one and the second one we got 72 so this is why he was ranked number two right then the number let's see let's see number nine number 10 number 10 we got 98 percent there okay so that's why it's number 10.

02:37 That's the highest score.

02:39 So from the list.

02:40 Okay, now i also ranked the exam, right? this way.

02:45 Now, i have to find the differences.

02:48 4 minus 4 is 0.

02:50 10 minus 2.

02:51 It's 8.

02:52 7 minus 8 minus 1.

02:54 2 minus 3 is minus 1...