Lecture 24

Interpreting Confidence

DATA 8

Fall 2018

Slides created by John DeNero (denero@berkeley.edu) and Ani Adhikari (adhikari@berkeley.edu)

Announcements

The Bootstrap

Key to Resampling

• From the original sample,
• draw at random
• with replacement
• as many values as the original sample contained

​

• The size of the new sample has to be the same as the original one, so that the two estimates are comparable

Why the Bootstrap Works

population

sample

resamples

All of these look pretty similar, most likely.

Why We Need the Bootstrap

population

sample

resamples

What we wish we could get

What we really get

Each line here is a confidence interval from a fresh sample from the population

95% Confidence Interval

• Interval of estimates of a parameter
• Based on random sampling
• 95% is called the confidence level
• Could be any percent between 0 and 100
• Higher level means wider intervals
• The confidence is in the process that generated the interval:
• It generates a “good” interval about 95% of the time.

​

(Demo)

Use Methods Appropriately

Can You Use a CI Like This?

By our calculation, an approximate 95% confidence interval for the average age of the mothers in the population is (26.9, 27.6) years.

​

True or False:

• About 95% of the mothers in the population were between 26.9 years and 27.6 years old.

​

Answer: False. We’re estimating that their average age is in this interval.

Is This What a CI Means?

An approximate 95% confidence interval for the average age of the mothers in the population is (26.9, 27.6) years.

​

True or False:

• There is a 0.95 probability that the average age of mothers in the population is in the range 26.9 to 27.6 years.

Answer: False. The average age of the mothers in the population is unknown but it’s a constant. It’s not random. No chances involved.

When Not to Use The Bootstrap

• If you’re trying to estimate very high or very low percentiles, or min and max
• If you’re trying to estimate any parameter that’s greatly affected by rare elements of the population
• If the probability distribution of your statistic is not roughly bell shaped (the shape of the empirical distribution will be a clue)
• If the original sample is very small

(Demo)

Confidence Intervals For Testing

Using a CI for Testing

• Null hypothesis: Population average = x
• Alternative hypothesis: Population average ≠ x
• Cutoff for P-value: p%
• Method:
• Construct a (100-p)% confidence interval for the population average
• If x is not in the interval, reject the null
• If x is in the interval, can’t reject the null

(Demo)