A Statistics Course at a Large University Is Taught in Each

question 58

Essay

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 over 40% of students in the population scored a pass grade in semester 1, where a pass grade is 50% to 64%?
(b) Find the p-value of the test and briefly explain how to use it to test the hypotheses.

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Definitions:

Null Hypothesis

A statistical hypothesis that indicates no effect or no difference in the context of scientific experiments, assuming the observation results from chance.

Control

The use of standardized procedures and conditions in scientific experiments to ensure that changes observed in an experiment are due to the manipulation of the independent variable.