Many undergraduate students are employed at the same time they are enrolled in school. In 2018, the National Center for Education Statistics reported that 43% of full-time students worked.[1] Being employed while in school can help a student pay for tuition, housing, and other expenses, but it can also be associated (either positively or negatively) with a student’s academic performance.
Question 3
1) Do you think working while in school has an overall positive or negative association with academic performance for full-time college students? Explain.
Credit: iStock/Six_Characters
A random sample of 15 employed full-time students at a large university was selected for a survey on employment. The following is the number of hours (in increasing order) worked per week for each of those 15 students:
2 5 7 10 16 19 19 22 23 26 27 27 31 40 50
Question 2
2) Go to the DCMP Explore Quantitative Data tool at https://dcmathpathways.shinyapps.io/EDA_quantitative/.
- Select “Your Own” under “Enter Data.”
- Enter a descriptive name for the variable (e.g., Hours Worked per Week). • Enter the previous observations into the tool.
a) Examine the histogram, boxplot, dotplot, and summary statistics of these data. Write a few sentences describing the features of the data distribution. Address the shape, center, spread, and outliers.
b) What is the sample mean hours worked per week? What is the appropriate symbol for this value?
c) What is the sample standard deviation? What is the appropriate symbol for this value?
d) Write a sentence interpreting the standard deviation in the context of the problem.
Question 3
3) Suppose the hours worked per week for all employed full-time students at this university varies according to an approximate normal distribution with mean µ and standard deviation σ.
Consider the standardized sample mean, or z-statistic:
[latex]z=\frac{\bar{x}-[mean\;of\;\bar{x}}{[std.\;deviation\;of\;\bar{x}}=\frac{\bar{x}-\mu}{\sigma/\sqrt{n}}[/latex]
a) Which of the following quantities in the z-statistic are parameters and which are statistics: [latex]\bar{x},\mu,\sigma[/latex]?
b) If we were to take repeated random samples of size [latex]n[/latex] = 15 from this population, which of the quantities in the z-statistic could change from sample to sample? There may be more than one correct answer.
a) [latex]\bar{x}[/latex]
b) [latex]\mu[/latex]
c) [latex]\sigma[/latex]
d) [latex]n[/latex]
c) What distribution does the z-statistic follow across many random samples?
Suppose we know that the mean hours worked per week for all employed full-time students at this university is [latex]\mu=19[/latex] hours, but we do not know the value of the population standard deviation, [latex]\sigma[/latex]. If we want to calculate the value of a standardized sample mean, however, we need to know [latex]\sigma[/latex]. So, instead of [latex]\sigma[/latex], let’s substitute in our best estimate for [latex]\sigma[/latex]: the sample standard deviation, [latex]s[/latex].
Recall that an estimate of the standard deviation of a statistic is called the standard error of that statistic. Since we estimate the standard deviation of the sample mean [latex]\sigma/\sqrt{n}[/latex] by substituting in [latex]s[/latex] for [latex]\sigma[/latex], the standard error of the sample mean is
[latex]SE(\bar{x})=\frac{s}{\sqrt{n}}[/latex]
Question 4
4) Now, consider the quantity
[latex]t=\frac{\bar{x}-[mean\;of\;\bar{x}]}{[std.\;error\;of\\;\bar{x}}=\frac{\bar{x}-\mu}{s/\sqrt{n}}[/latex]
This quantity is called a t-statistic.
(a) Which of the following quantities in the t-statistic are parameters and which are statistics: [latex]\bar{x},\mu,s[/latex]?
(b) If we were to take repeated random samples of size [latex]n[/latex] = 15 from this population, which of the quantities in the t-statistic could change from sample to sample? There may be more than one correct answer.
a) [latex]n\bar{x}[/latex]
b) [latex]\mu[/latex]
c) [latex]s[/latex]
d) [latex]n[/latex]
(c) Go to the DCMP t Distribution tool at https://dcmathpathways.shinyapps.io/tdist/. Under the Explore tab, select “Show Standard Normal Curve.” Compare the t Distributions for 3, 8, and 14 degrees of freedom. Explain what happens to the t Distribution curve as the degrees of freedom increase.
| t Distribution
When taking many, many random samples of size [latex]n[/latex] from a population distribution with mean [latex]\mu[/latex] and standard deviation [latex]\sigma[/latex], the t-statistic [latex]n[/latex] will follow a t Distribution with [latex]n-1[/latex] degrees of freedom if the population distribution is normal or if the population distribution is not too skewed and the sample size is large (e.g., [latex]n[/latex] ≥ 30). |
Since the t-statistic exhibits more sampling variability than the z-statistic, its distribution has slightly more variability than a standard normal distribution. However, as the sample size increases, there is less sampling variability associated with the standard error of the sample mean, so its distribution gets closer to a standard normal distribution.
A picture of the t Distribution for various degrees of freedom, along with the standard normal distribution for reference, is shown below.
Question 5
5) Assume the hours worked per week for all employed full-time students at this university varies according to an approximate normal distribution with mean [latex]\mu[/latex] = 19 hours.
a) Calculate the standard error of the sample mean for the observed sample data shown before Question 2. Write a sentence interpreting this value in the context of the problem.
b) Calculate the value of the t-statistic for the observed sample data shown before Question 2. Write a sentence interpreting this value in the context of the problem.
Question 6
6) Consider the t-statistic for Question 5, Part B.
a) What distribution did the t-statistic value come from? Explain why the conditions for using the t Distribution are met, and then sketch the distribution.
b) The probability of observing 0.77 or higher on a standard normal distribution is 0.2206. Would you predict the probability of observing 0.77 or higher on the distribution you specified in Part A to be less than, equal to, or larger than 0.2206? Explain.
c) Using the Find Probability tab in the DCMP t Distribution tool, calculate the probability of observing a t-statistic of 0.77 or higher for the distribution you specified in Part A. Does this match your prediction from Part B?
- U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics. (2020). College student employment. Retrieved from https://nces.ed.gov/programs/coe/pdf/coe_ssa.pdf ↵
