Question 1

The website Mattress Advisor claims that 20% of college students pull all-nighters each semester.^[2] Do you think this claim is accurate?  What information would be helpful to answer this  question?

Question 2

In a previous activity, we looked at the percentage of college students who had pulled all-nighters. In the study, 34 out of 253 students said they had an all-nighter in  the last semester.^[3]

1. Calculate the point estimate ([latex]\hat{p}[/latex]) for the proportion of college students who  had pulled all-nighters.
2. Looking at the point estimate alone, do you think the claim by Mattress Advisor is accurate?
3. Why should we create a confidence interval if we already have an estimate  for the population proportion?

Question 3

We will assume that random sampling is used and that the sample is less than 10%  of the population. Verify the sample size condition so that we may assume that the  sampling distribution of the sample proportions is approximately normal.

Question 4

Rather than using a simple point estimate, let’s calculate confidence intervals for a  population proportion using the DCMP Inference for a Population Proportion tool at https://dcmathpathways.shinyapps.io/Inference_prop/ for the study, where 34 out of  253 students said they had all-nighters in the last semester.^[4]

1. Change the “Enter Data” box to “Number of Successes.”
2. Input the sample size ([latex]n[/latex]) and the number of successes ([latex]x[/latex]), which in this case is  the number of students who pulled all-nighters.
3. Specify appropriate labels for success/failure by checking the appropriate boxes and typing in labels. For example, we can label success as “All-nighters” and  failure as “No all-nighters.”
4. Slide the confidence level to the desired level. The default is 95%
5. View the z critical value by selecting the box “Show z-score for Margin of Error.”
6. The confidence interval will appear to the right along with the point estimate,  standard error, margin of error, and z critical value (labeled as “z-score”).

1. What is the standard error for the proportion of college students who had all nighters?
2. Assume the level of confidence is 95%. What is the z critical value, [latex]z^{*}[/latex], that  corresponds to the confidence level?
3. Calculate the margin of error for the sample proportion of college students  who had all-nighters from the equation for the margin of error ([latex]E[/latex]):[latex]E = z^{*} \bullet (standard~error)[/latex]Notice how the values in the web tool match the calculated value from the  equation.

Question 5

Now let’s estimate the proportion of college students who had all-nighters by  creating a 95% confidence interval.

1. Begin by writing the confidence interval in both [latex]\pm[/latex] format and interval notation  form (lower bound, upper bound).
2. Represent the confidence interval on the following number line. Include the  point estimate, upper bound, and lower bound.
   

Question 6

Use the DCMP tool to calculate the 99% confidence interval for the proportion of  college students who had all-nighters.

1. Write the 99% confidence interval using interval notation form (lower bound,  upper bound).
2. How is this confidence interval different from the 95% confidence interval?
3. Looking at the point estimate, standard error, z critical value, and margin of  error, what do you think contributed to the change from the 95% confidence  interval to the 99% confidence interval?
4. What do you think would happen to the length of the interval if we changed it  to 90% confidence?

Question 7

Based on your answers to Questions 4 and 5, do you think there is evidence to  support the claim from Mattress Advisor? Explain.