A statistics course at a large university is taught in each semester. A student has noticed that the students in semester 1 and semester 2 are enrolled in different degrees. To investigate, the student takes a random sample of 25 students from semester 1 and 25 students from semester 2 and records their final marks (%) provided in the table below. Excel was used to generate descriptive statistics on each sample.
Assume that student final marks are normally distributed in each semester. $$\begin{array}{l}
\begin{array} { | c | c | c | c | c | }
\hline { \text { Sample of semester } 1 \text { final marks } } \
\hline 65 & 45 & 53 & 76 & 53 \
\hline 85 & 55 & 63 & 85 & 77 \
\hline 45 & 57 & 55 & 60 & 83 \
\hline 96 & 83 & 55 & 52 & 67 \
\hline 82 & 64 & 62 & 88 & 71 \
\hline
\end{array}&
\begin{array} { | c | c | c | c | c | }
\hline { \text { Sample of semester } 2 \text { final marks } } \
\hline 45 & 40 & 53 & 58 & 75 \
\hline 46 & 82 & 54 & 75 & 59 \
\hline 45 & 54 & 87 & 77 & 63 \
\hline 81 & 60 & 56 & 53 & 65 \
\hline 52 & 65 & 60 & 65 & 54 \
\hline
\end{array}
\end{array}$$ $$\begin{array}{l}
\begin{array} { | l | c | }
\hline { \text { Semester } 1 } \
\hline \text { Mean } & 65.48 \
\hline \text { Stan dard Error } & 2.679 \
\hline \text { Median } & 63 \
\hline \text { Mode } & 55 \
\hline \text { Standard Deviation } & 13.395 \
\hline \text { Sample Variance } & 179.43 \
\hline \text { Range } & 43 \
\hline \text { Minimum } & 45 \
\hline \text { Maximum } & 88 \
\hline \text { Sum } & 1637 \
\hline \text { Count } & 25 \
\hline
\end{array}&
\begin{array} { | l | c | }
\hline { \text { Semester 2 } } \
\hline \text { Mean } & 60.96 \
\hline \text { Standard Error } & 2.5136 \
\hline \text { Median } & 59 \
\hline \text { Mode } & 54 \
\hline \text { Standard Deviation } & 12.568 \
\hline \text { Sample Variance } & 157.96 \
\hline \text { Range } & 47 \
\hline \text { Minimum } & 40 \
\hline \text { Maximum } & 87 \
\hline \text { Sum } & 1524 \
\hline \text { Count } & 25 \
\hline
\end{array}
\end{array}$$ (a) Can we conclude at the 5% level of significance that semester 1 students have a higher proportion of high distinctions than semester 2 students, where a high distinction is a final mark greater than or equal to 85%?
(b) Find the p-value of the test, and explain how to use it to test the hypotheses.

Question

A statistics course at a large university is taught in each semester. A student has noticed that the students in semester 1 and semester 2 are enrolled in different degrees. To investigate, the student takes a random sample of 25 students from semester 1 and 25 students from semester 2 and records their final marks (%) provided in the table below. Excel was used to generate descriptive statistics on each sample.
Assume that student final marks are normally distributed in each semester. $$\begin{array}{l}
\begin{array} { | c | c | c | c | c | } 
\hline  { \text { Sample of semester } 1 \text { final marks } } \
\hline 65 & 45 & 53 & 76 & 53 \
\hline 85 & 55 & 63 & 85 & 77 \
\hline 45 & 57 & 55 & 60 & 83 \
\hline 96 & 83 & 55 & 52 & 67 \
\hline 82 & 64 & 62 & 88 & 71 \
\hline
\end{array}&
\begin{array} { | c | c | c | c | c | } 
\hline  { \text { Sample of semester } 2 \text { final marks } } \
\hline 45 & 40 & 53 & 58 & 75 \
\hline 46 & 82 & 54 & 75 & 59 \
\hline 45 & 54 & 87 & 77 & 63 \
\hline 81 & 60 & 56 & 53 & 65 \
\hline 52 & 65 & 60 & 65 & 54 \
\hline
\end{array}
\end{array}$$ $$\begin{array}{l}
\begin{array} { | l | c | } 
\hline  { \text { Semester } 1 } \
\hline \text { Mean } & 65.48 \
\hline \text { Stan dard Error } & 2.679 \
\hline \text { Median } & 63 \
\hline \text { Mode } & 55 \
\hline \text { Standard Deviation } & 13.395 \
\hline \text { Sample Variance } & 179.43 \
\hline \text { Range } & 43 \
\hline \text { Minimum } & 45 \
\hline \text { Maximum } & 88 \
\hline \text { Sum } & 1637 \
\hline \text { Count } & 25 \
\hline
\end{array}&
\begin{array} { | l | c | } 
\hline  { \text { Semester 2 } } \
\hline \text { Mean } & 60.96 \
\hline \text { Standard Error } & 2.5136 \
\hline \text { Median } & 59 \
\hline \text { Mode } & 54 \
\hline \text { Standard Deviation } & 12.568 \
\hline \text { Sample Variance } & 157.96 \
\hline \text { Range } & 47 \
\hline \text { Minimum } & 40 \
\hline \text { Maximum } & 87 \
\hline \text { Sum } & 1524 \
\hline \text { Count } & 25 \
\hline
\end{array}
\end{array}$$ (a) Can we conclude at the 5% level of significance that semester 1 students have a higher proportion of high distinctions than semester 2 students, where a high distinction is a final mark greater than or equal to 85%?
(b) Find the p-value of the test, and explain how to use it to test the hypotheses.

Quizplus · Accepted Answer

The Answer of A statistics course at a large university is taught in each semester. A student has noticed that the students in semester 1 and semester 2 are enrolled in different degrees. To investigate, the student takes a random sample of 25 students from semester 1 and 25 students from semester 2 and records their final marks (%) provided in the table below. Excel was used to generate descriptive statistics on each sample.
Assume that student final marks are normally distributed in each semester. $$\begin{array}{l}
\begin{array} { | c | c | c | c | c | } 
\hline  { \text { Sample of semester } 1 \text { final marks } } \
\hline 65 & 45 & 53 & 76 & 53 \
\hline 85 & 55 & 63 & 85 & 77 \
\hline 45 & 57 & 55 & 60 & 83 \
\hline 96 & 83 & 55 & 52 & 67 \
\hline 82 & 64 & 62 & 88 & 71 \
\hline
\end{array}&
\begin{array} { | c | c | c | c | c | } 
\hline  { \text { Sample of semester } 2 \text { final marks } } \
\hline 45 & 40 & 53 & 58 & 75 \
\hline 46 & 82 & 54 & 75 & 59 \
\hline 45 & 54 & 87 & 77 & 63 \
\hline 81 & 60 & 56 & 53 & 65 \
\hline 52 & 65 & 60 & 65 & 54 \
\hline
\end{array}
\end{array}$$ $$\begin{array}{l}
\begin{array} { | l | c | } 
\hline  { \text { Semester } 1 } \
\hline \text { Mean } & 65.48 \
\hline \text { Stan dard Error } & 2.679 \
\hline \text { Median } & 63 \
\hline \text { Mode } & 55 \
\hline \text { Standard Deviation } & 13.395 \
\hline \text { Sample Variance } & 179.43 \
\hline \text { Range } & 43 \
\hline \text { Minimum } & 45 \
\hline \text { Maximum } & 88 \
\hline \text { Sum } & 1637 \
\hline \text { Count } & 25 \
\hline
\end{array}&
\begin{array} { | l | c | } 
\hline  { \text { Semester 2 } } \
\hline \text { Mean } & 60.96 \
\hline \text { Standard Error } & 2.5136 \
\hline \text { Median } & 59 \
\hline \text { Mode } & 54 \
\hline \text { Standard Deviation } & 12.568 \
\hline \text { Sample Variance } & 157.96 \
\hline \text { Range } & 47 \
\hline \text { Minimum } & 40 \
\hline \text { Maximum } & 87 \
\hline \text { Sum } & 1524 \
\hline \text { Count } & 25 \
\hline
\end{array}
\end{array}$$ (a) Can we conclude at the 5% level of significance that semester 1 students have a higher proportion of high distinctions than semester 2 students, where a high distinction is a final mark greater than or equal to 85%?
(b) Find the p-value of the test, and explain how to use it to test the hypotheses.

A Statistics Course at a Large University Is Taught in Each