A Statistics Course at a Large University Is Taught in Each

question 67

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. $$\begin{array}{l}\begin{array} { | c | c | c | c | c | } \hline { \text { Sample of semester } 1 \text { final marks } } \\\hline 65 & 45 & 53 & 76 & 53 \\\hline 85 & 55 & 63 & 85 & 77 \\\hline 45 & 57 & 55 & 60 & 83 \\\hline 96 & 83 & 55 & 52 & 67 \\\hline 82 & 64 & 62 & 88 & 71 \\\hline\end{array}&\begin{array} { | c | c | c | c | c | } \hline { \text { Sample of semester } 2 \text { final marks } } \\\hline 45 & 40 & 53 & 58 & 75 \\\hline 46 & 82 & 54 & 75 & 59 \\\hline 45 & 54 & 87 & 77 & 63 \\\hline 81 & 60 & 56 & 53 & 65 \\\hline 52 & 65 & 60 & 65 & 54 \\\hline\end{array}\end{array}$$ $$\begin{array}{l}\begin{array} { | l | c | } \hline { \text { Semester } 1 } \\\hline \text { Mean } & 65.48 \\\hline \text { Stan dard Error } & 2.679 \\\hline \text { Median } & 63 \\\hline \text { Mode } & 55 \\\hline \text { Standard Deviation } & 13.395 \\\hline \text { Sample Variance } & 179.43 \\\hline \text { Range } & 43 \\\hline \text { Minimum } & 45 \\\hline \text { Maximum } & 88 \\\hline \text { Sum } & 1637 \\\hline \text { Count } & 25 \\\hline\end{array}&\begin{array} { | l | c | } \hline { \text { Semester 2 } } \\\hline \text { Mean } & 60.96 \\\hline \text { Standard Error } & 2.5136 \\\hline \text { Median } & 59 \\\hline \text { Mode } & 54 \\\hline \text { Standard Deviation } & 12.568 \\\hline \text { Sample Variance } & 157.96 \\\hline \text { Range } & 47 \\\hline \text { Minimum } & 40 \\\hline \text { Maximum } & 87 \\\hline \text { Sum } & 1524 \\\hline \text { Count } & 25 \\\hline\end{array}\end{array}$$ Estimate a 95% confidence interval for the difference in the proportions of students who received a high distinction in semester 1 to semester 2.

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Definitions:

Sample Proportion

The sample proportion is a statistic that estimates the proportion of elements in a population that have a certain characteristic, based on a sample from that population.

Population Proportion

A measure that represents the fraction of members in a population that have a particular property or attribute.

Binomial Random Variable

A type of random variable that takes on a fixed number of trials, each with two possible outcomes.

Normal Curve

A symmetrical bell-shaped curve that describes the distribution of many types of data where most occurrences take place around the mean.