STAT 120

Bastola

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 \[ statistic \pm 2\times SE \]

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.

- 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

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 here for the link

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

- 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

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

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

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

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

95% CI

99% CI

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!!

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

- 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.

For Florida lakes, what is the correlation between average mercury level (ppm) in fish taken from a lake and acidity (pH) of the lake?

Lange, Royals, and Connor, Transactions of the American Fisheries Society (1993)

Bootstrapping correlation parameter.

We are 90% confident that the true correlation between average mercury level and pH of Florida lakes is between -0.702 and -0.433.

- Please download the Class-Activity-10 template from moodle and go to class helper web page

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