Question

Statistics 200: Lab Activity for Section 3.3

Constructing Bootstrap Confidence Intervals - Learning objectives:

• Describe how to select a bootstrap sample to compute a bootstrap statistic
• Recognize that a bootstrap distribution tends to be centered at the value of the original statistic
• Use technology to create a bootstrap distribution
• Estimate the standard error of a statistic from a bootstrap distribution
• Construct a 95% confidence interval for a parameter based on a sample statistic and the standard error from a bootstrap distribution.

Activity 1: Create a bootstrap confidence interval for the average number times adults laugh in a day
Let’s say we conducted a very small survey to estimate the average number of times adults laugh in a day. We asked 6 people to record how many times they laughed one day and got the following data values:

         

17, 22, 4, 30, 8, 42

Our goal in activities 1, and 2 today is to carefully construct a 95% bootstrap confidence interval for the mean number of laughs per day.

1. What is the population parameter we are trying to estimate?

2. Calculate the sample estimate for this parameter. Use correct notation!

At the front of the lab are stacks of cut index cards. You are to use them to create your own bootstrap samples. Before you go up to get them, figure out how you can use them to complete this task.
1. How many index cards will you need?

2. What do you put on the cards?

3. How do you use them to create a bootstrap sample?

4. What statistic should you compute from each bootstrap sample?

5. Use your cards to create three bootstrap samples and give them below.
a.
b.
c.

6. What were the bootstrap statistics calculated from each of your three bootstrap samples?
a.
b.
c.

7. For each set of values below, determine whether it is a possible bootstrap sample from the original sample of laughs per day. If not, state why not.

22, 42, 30, 8, 17

30, 8, 30, 22, 42,17

9, 22, 4, 17, 31, 8

30, 22, 8, 4, 17, 42

Activity 2: Use Statkey to make bootstrap CI for laughter
As you can tell from Activity 1, creating bootstrap samples by hand takes a long time. That’s why we have software create bootstrap samples for us! StatKey can create as many bootstrap samples and statistics as we want very quickly. In this activity we will use StatKey to do just this to create a bootstrap distribution for the number of laughs.
Open up Statkey and select Bootstrap confidence interval for a mean, median, Std.Dev. You’ll need to enter in the data for the laughing adults. To do this, click ‘Edit Data’, erase what’s in there, and enter the values from the original sample in Activity 1.
You can choose if you want to include a header row (name of variable) or not. Just be sure to select the correct option below your data.

1. Create a bootstrap distribution of 5,000 bootstrap statistics.
a. what is the shape of the distribution?

b. what is the standard error?


2. Use the standard error from part (b) above and the sample mean from activity 1, to create a 95% confidence interval for the population parameter.


3. Interpret your interval estimate in context.



4. What is something we could do to make the interval narrower?


Activity 3: Mood of the nation
Gallup conducted a nationwide poll from January 4-8, 2017 to gauge public opinion on the question ‘will America be better off in 2020?’    1,032 people took the survey. Exact numbers were not available, but assume that 416 people in the survey identified as democrats, of which 58 thought that America will be better off in 2020. Assume 502 survey participants identified as republican, and of these 427 thought America will be better off in 2020.
Our goal in this activity is to calculate a 95% confidence interval for the difference in proportions that think America will be better off when comparing republicans to democrats. Let group 1 be republicans and group 2 be democrats.

1. What population parameter are we trying to build an interval estimate for? Use correct notation


2. Calculate the sample estimate for this quantity. Use correct notation.

3. Now we use Statkey to build a bootstrap confidence interval. Create a bootstrap distribution using at least 5,000 bootstrap samples.

What is the standard error for the original sample statistic?

4. What is the center of your bootstrap distribution? How does it compare to the sample estimate from the original sample?



5. Compute the 95% confidence interval for the population parameter.

6. Interpret the confidence interval you computed in question 5.

Answer 1