• Find a bootstrap confidence interval for a population parameter and difference in population parameters
• Describe what a bootstrap confidence interval means and use it make inference regarding the population

Jumping Frog Jubilee

Every year, bullfrogs compete in a jumping contest at the Calaveras County Jumping Frog Jubilee (a contest inspired by a short story by Mark Twain). One year, researchers recorded the jump distances of frogs entered in the contest.^[1]

The following are the jump distances (in meters) for a sample of [latex]15[/latex] bullfrogs.

0.1	0.4	0.6	0.8	1.3	1.5	1.6	1.7
1.8	1.8	1.9	1.9	1.9	2.0	2.2

A dotplot of the sample jump distances is shown here.

If we were interested in estimating the population mean jump distance (the mean jump distance for all frogs entered in the competition), it would not be appropriate to use a one-sample t confidence interval because the sample size is not greater than [latex]30[/latex] and, because the data distribution is skewed, it is not reasonable to think that the population jump distance distribution is approximately normal.

Bootstrap confidence intervals can be constructed for population parameters other than the population mean. Bootstrap confidence intervals provide an alternative to traditional intervals based on [latex]z[/latex] or [latex]t[/latex] distributions that can be used in situations where conditions necessary for the traditional methods are not met. 

To calculate a bootstrap confidence interval, let’s use the statistical tool: https://istats.shinyapps.io/Boot1samp/.

Step 1: For the “Enter Data” option, choose “Your Own.”
Step 2: For “Name of Variable,” type “Jump Distance.”
Step 3: Type the values from the sample or copy and paste them into the “Enter Observations” box. Separate the data values by spaces or commas. The values for the sample are: 0.1, 0.4, 0.6, 1.2, 0.8, 1.5, 1.6, 1.7, 1.8, 1.8, 1.9, 1.9, 1.9, 2.0, and 2.2.
Step 4: For the “Statistic of Interest” option, select “Median.”
Step 5: Click on “1,000” for “Select how many bootstrap samples you want to generate,” and then click on “Draw Bootstrap Sample(s).”
Step 6: To get the confidence interval, at the very top of the display, click on the Get Confidence Interval tab. Check to make sure that the confidence level is set to [latex]95\%[/latex]. The corresponding bootstrap confidence interval is on the right-hand side of the display.

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