Question 1

According to the Centers for Disease Control and  Prevention (CDC), without protection from the sun’s ultraviolet (UV) rays, damage can be done to  your skin in as little as 15 minutes. Over time, this  damage can increase the risk of developing skin cancer; thus, it is important to always practice “sun  safety,” such as wearing sunscreen and  sunglasses and taking breaks in the shade.^[1]

As part of the National College Health Assessment^[2]  survey, researchers asked a random sample of college students:

• When you are outdoors in the sun, how often do you wear sunscreen?
• When you are outdoors in the sun, how often do you wear sunglasses to protect  yourself from UV exposure?

Credit: iStock/grafxart8888

1. If you had to guess, what proportion of college students do you think always wear sunscreen while outdoors in the sun?
2. What proportion of college students do you think always wear sunglasses to  protect themselves from UV rays?

Question 2

Out of 38,442 responses, 2,693 college students stated that they always wear  sunscreen when they are outside in the sun.

The printout below contains the results from the DCMP Inference for a Population  Proportion tool. The number of “Successes” refers to the 2,693 college students who stated they always wear sunscreen.

1. When constructing confidence intervals for a proportion, remember that it is  important to first confirm that the following conditions are met:
   • Random sampling is used.
   • The sample is less than 10% of the population.
   • The sample is large enough that [latex]n\hat{p} \geq 10[/latex] and [latex]n(1 − \hat{p}) \geq 10[/latex].Although it is not specified above, we can assume that the sample is less  than 10% of the population.
     Confirm that the other two conditions are met.
2. Using the previous results from the DCMP Data Analysis Tool, identify the  point-estimate, standard error, margin of error, confidence level, and lower  and upper bounds of the confidence interval for estimating the population  proportion of American college students who always wear sunscreen while  outdoors in the sun. Complete the following table.

   Point-Estimate
   Standard Error
   Margin of Error
   Confidence Level
   Lower Bound
   Upper Bound

3. A friend reads the output and interprets the confidence interval as the  following:“There is a 95% chance that the population proportion of college students  who always wear sunscreen is between 0.0675 and 0.07261, or 6.75% and  7.261%.”

   Is this a correct interpretation of the confidence interval? Explain. If it is not a  correct interpretation, what is the correct interpretation?

4. As part of their Healthy People 2020 initiative, the CDC set a target goal of  increasing the percentage of young people who regularly wear sunscreen to  11.2%. Given the calculated confidence interval and its correct interpretation, does it appear the proportion of college students who always wear  sunscreen hits this goal?

Question 3

Out of 38,281 responses, 5,954 college students stated that they always wear  sunglasses to protect themselves from UV rays.

1. Use the tool (https://dcmathpathways.shinyapps.io/Inference_prop/) to  calculate a 95% confidence interval for estimating the population proportion  of college students who always wear sunglasses to protect themselves from  UV rays.
   • Under the “Enter Data” drop-down menu, select “Number of Successes.”
   • Enter 38,281 as the sample size and 5,954 as the number of successes.
   • Complete the following table with the appropriate calculations from the  output.

     Point-Estimate
     Standard Error
     Margin of Error
     Confidence Level
     Lower Bound
     Upper Bound

2. Interpret the 95% confidence interval that you constructed.
3. Using the same previous figure from the CDC (i.e., 11.2%), does it appear  more than 11.2% of college students always wear sunglasses to protect  themselves from UV rays? Explain.
4. Without using the DCMP Data Analysis Tool or a calculator, determine  whether a 99% confidence interval calculated from these sample data would  result in a narrower or wider interval. Explain.

Question 4

Reflect back on your responses to Question 1. Do the confidence intervals you  constructed in Questions 2 and 3 align with your initial guesses? Explain.