Understanding Confidence Intervals and Bootstrap

STAT 120

Bastola

Confidence Interval Recap

A confidence interval for a parameter is an interval computed from sample data by a method that will capture the parameter for a specified proportion of all samples.

\[CI = PE \pm ME\] 95 % CI \[ statistic \pm 2\times SE \]

Confidence level analogy and recap

Shooting Arrows:

Bow’s confidence level determines accuracy.
A “95% confident” bow:
- 95 arrows hit inside the bulls eye out of 100
- 5 arrows miss

Single Shot Principle:

Each arrow either hits or misses.
No in-between or partial accuracy.

Confidence level analogy with throwing darts

Conceptual Understanding: Repeated Sampling

The success rate (proportion of all samples whose intervals contain the parameter) is known as the confidence level
A 95% confidence interval will contain the true parameter for 95% of all samples

Repeated Sampling of 100 95% Confidence Intervals, Truth = Vertical Line

Example 1

A survey of 1,502 Americans in January 2012 found that 86% consider the economy a “top priority” for the president and congress. The standard error for this statistic is 0.01.

What is the 95% confidence interval for the true proportion of all Americans that considered the economy a “top priority” at that time?

(1). (0.85, 0.87)

(2). (0.84, 0.88)

(3). (0.82, 0.90)

Click for answer

The correct answer is (2).

Confidence Interval Interpretation

Which of the following is an appropriate interpretation for a 95% confidence interval:

A. “we are 95% sure the interval contains the parameter”

B. “there is a 95% chance the interval contains the parameter”

C. Both A and B

D. Neither A nor B

Click for answer

The correct answer is A.

Common Misintepretations

Misinterpretation 1: “A 95% confidence interval contains 95% of the data in the population”
Misinterpretation 2: “I am 95% sure that the mean of a sample will fall within a 95% confidence interval for the mean”
Misinterpretation 3: “The probability that the population parameter is in this particular 95% confidence interval is 0.95”
Correct: I am 95% sure that the mean of a population will fall within a 95% confidence interval for the mean

Example 2

A 98% confidence interval for mean pulse rate is 65 to 71. The interpretation “I am 98% sure that all students will have pulse rates between 65 and 71.” is

A. Correct

B. Incorrect

Click for answer

The correct answer is B.

Example 3

A 98% confidence interval for mean pulse rate is 65 to 71. The interpretation “I am 98% sure that the mean pulse rate for this sample of students will fall between 65 and 71” is

A. Correct

B. Incorrect

Click for answer

The correct answer is B.

Example 4

A 98% confidence interval for mean pulse rate is 65 to 71. The interpretation “I am 98% sure that the mean pulse rate for the population of all students will fall between 65 and 71” is

A. Correct

B. Incorrect

Click for answer

The correct answer is A.

Level of Confidence

Which is wider? a 99% confidence interval or a 95% confidence interval?

95% CI
99% CI

Click for answer

The correct answer is (b). We need a larger interval (range of likely parameter values) to have more confidence.

Recap: Sampling Distribution Vs Bootstrap Distribution

Sampling Distribution of a statistic

Take many samples from the population, compute the statistic for each sample
Shape: bell-shaped when n is large
Center: population parameter
Spread: called the SE of the statistic

Bootstrap Distribution of a statistic

Take many bootstrap samples from the original sample, compute the statistic for each bootstrap sample
Shape: bell-shaped when n is large
Center: original sample statistic!
Spread: called the bootstrap SE of the statistic

The standard errors from both approaches should be similar!!

Percentile Method Bootstrap

If the bootstrap distribution is approximately symmetric, a P% confidence interval equals the percentiles in the bootstrap distribution so that the proportion of bootstrap statistics between the percentiles equal P%.

Percentiles of a bootstrap distribution

The Magic of Bootstrapping

We can use bootstrapping to approximate the SE for many types of sample statistic!
- Mean, proportion, differences, correlation, slope
- Standard deviation, median
What should the bootstrap distribution look like?
- “smooth” (i.e. not a lot of spikey-ness)
- If using \(95\% ME = 2SE\), should be symmetric and bell-shaped.