Statistics Notes on Confidence Intervals
Spring 2019

## Chapter 8

#### §8.1 & §8.2 Confidence Intervals for a Single Population Mean

##### Objectives (text)
• Calculate and interpret confidence intervals for estimating a population mean.
• Understand the roles of sample size, confidence level, and spread on the margin of error.
• Population vs Sample
• Parameter vs Statistic
• Point Estimates:
• The sample mean barx is an estimate for the population mean mu.
• The sample proportion p' is an estimate for the population proportion p.
• The sample standard deviation s is an estimate for the population standard deviation sigma.
• Interval Estimates:
• The average age of college students is between 21.24 and 29.56.
The average age of a college student is within 4.16 years of 25.4.
• The proportion of females attending college is between 43% and 63%.
The proportion of female college students is within 10% of 53%.
• Sampling Distribututions
aka a histogram of sample means,
or a histogram of sample proportions.
##### Anatomy of a Confidence Interval

"margin"=("right"-"left")/2

"point estimate"=("right"+"left")/2

OR

"left"="point estimate"-"margin"

"right"="point estimate"+"margin"

##### Examples: (StatKey)
• Average Body Temperature: mu

Interval estimate: 98.048^circ F < mu < 98.468^circ F   or   (98.048^circ F, 98.468^circ F)

Point estimate: barx = 98.258^circ F and EBM=0.210^circ F

mu = 98.258^circ F +- 0.210^circ F

• Average Restaurant Tip: mu

$3.478 < mu <$4.232   or   ($3.478,$4.232)

barx=$3.855 and EBM=$0.377

mu = $3.855 +-$0.377
Interpretation:
We are 95% confident that the average daily tip at this restaurant is between $3.48 and$4.23.

#### §8.3 Confidence Intervals for a Population Proportion

##### Objectives (text)
• Calculate and interpret confidence intervals for estimating a population proportion.
• Understand the roles of sample size, confidence level, and spread on the margin of error.
##### Examples: (StatKey)
• Proportion of peanuts in a can of mixed nuts: p

Interval estimate: 0.42 < p < 0.62   or   (0.42, 0.62)

Point estimate: hat p = p' = 0.52 and EBP=0.10

p = 0.52 +- 0.10

• Proportion of home game wins for a soccer team: p

0.492 < p < 0.675   or   (0.492, 0.675)

hat p = p' = 0.583 and EBP=0.091

p = 0.583 +- 0.091
Interpretation:
We are 95% confident that proportion of home game wins for this soccer team is between 49.2% and 67.5%.
##### Interpreting the Confidence Level

A realization of 50 confidence intervals attempting to capture that actual population mean.

The confidence level is NOT a chance of success!

Which of the following statements best describes a 95% confidence level?

1. 95% of people's body temperatures are in this interval.
2. There is a 95% chance that a randomly selected person will be in the interval.
3. There is a 95% chance that the average human body temperature will be in this interval.
4. 95% of similar intervals will contain the actual mean human body temperature.

##### Sample Size Matters

Notice that as the sample size increases, the standard deviation of the sampling distribution decreases.

Female College Students: x=48, n=100, and the standard error is about 0.049.

Female College Students: x=479 , n=1000, and the standard error is about 0.016.

Female College Students: x=3501 , n=7313, and the standard error is about 0.0058.