Chapter Seven Point Estimation and Confidence Intervals

Key Concepts Inferential Statistics = Statistics used to make inferences about population parameters based on point estimates from a sample. Point estimate = The statistic we get from our sample ◦ For example, a sample mean. Population parameter = The actual value in the population of whatever we get a point estimate of ◦ For example, the population mean.

Key Concepts The value of our point estimate and the value of a population parameter are probably not exactly the same. However, we can use probability to find the range of values that the population parameter probably falls within… based on our point estimate. This range is a confidence interval.

Confidence Intervals Confidence Interval = the range of values in which we can be “X percent” confident that the population parameter falls within. “X percent” depends on the alpha level we set (95% confident, 99% confident, etc.)

Confidence Interval: Population Mean With information from a sample, we can use it to create a confidence interval for a population mean. ◦ Note: This is only for samples of n=100 or greater. Confidence Interval for Population Mean:

Confidence Interval: Population Mean With information from a sample, we can use it to create a confidence interval for a population mean. ◦ Note: This is only for samples of n=100 or greater. Confidence Interval for Population Mean: Sample Mean

Confidence Interval: Population Mean With information from a sample, we can use it to create a confidence interval for a population mean. ◦ Note: This is only for samples of n=100 or greater. Confidence Interval for Population Mean: Confidence interval is a range above and below the sample mean, so we use this “plus and minus” sign.

Confidence Interval: Population Mean With information from a sample, we can use it to create a confidence interval for a population mean. ◦ Note: This is only for samples of n=100 or greater. Confidence Interval for Population Mean: How wide the interval is in standard deviation units (“Z-scores”), times our estimate of the standard error of the mean.

Confidence Interval: Population Mean What we need to calculate a confidence interval for a population mean: 1.The sample’s mean 2.The sample’s standard deviation 3.The sample size 4.Our alpha level (which we pick, and then we figure out the Z-score it matches) Popular Alpha Levels and critical Z-scores (2-tailed): α =.10, Z = 1.65 α =.05; Z = 1.96 α =.01, Z = 2.58

Confidence Interval: Population Mean Sample is taken to find average age of first sexual intercourse (loss of virginity) 100 random people are asked when they had sex for the first time ◦ Average age = years ◦ Standard deviation = 3.57 years Based on this sample, what’s the average age of first sexual intercourse in the population?

Confidence Interval: Population Mean Average age = years Standard deviation = 3.57 years Sample size = 100 people

Confidence Interval: Population Mean Step One: Choose an alpha level ◦ α =.05, associated with 95% confidence..05 is the probability that the true population mean will be outside our confidence interval (Type I Error) ◦ At this alpha level, Z =  In other words, we expect that the real population mean will be within 1.96 standard deviations above/below our sample mean. Step Two: Plug in numbers!

Confidence Interval: Population Mean Average age = years Standard deviation = 3.57 years Sample size = 100 people

Confidence Interval: Population Mean

Confidence Interval: Population Mean

Confidence Interval: Population Mean Once we calculate the confidence interval, we can use it to find the confidence limits (the top and bottom edges of the confidence interval). Lower Limit = – 0.71 = years Upper Limit = = years

Confidence Interval: Population Mean “We can say with 95% confidence that the average age of first sexual intercourse in the population is between and years.” What can we do to be more confident that we know the population mean?

Confidence Interval: Population Proportion Another population parameter we can estimate using a confidence interval is a proportion. ◦ Proportion of people in the population who will vote for a political candidate ◦ Proportion of people in the population who favor the death penalty

Confidence Interval: Population Proportion Sample proportion = These are calculated in a similar way (with a sample size of 100 or more): Confidence Interval for Population Proportion:

Confidence Interval: Population Proportion Example: 300 people are polled at random, and a proportion of.45 say that they will vote for a candidate for governor. What’s the interval that we can be 99% confident the population proportion falls within?

Confidence Interval: Population Proportion We need to know: ◦ Sample proportion (.45) ◦ Sample size (300) ◦ The alpha level we set (.01) and the critical Z score it corresponds to. Confidence Interval for Population Proportion =

Confidence Interval: Population Proportion Confidence Interval for Population Proportion

Confidence Interval: Population Proportion

Now that we have our confidence interval, we figure out our confidence limits: Lower Limit: =.38 Upper limit: =.52 “We can be 99% confident that the proportion of people who will vote for the candidate for governor is between.38 and.52.”

Confidence Intervals Note that if we’ve got a sample size less than 100, it’s not appropriate to base our confidence intervals (or our hypothesis tests) on the Z distribution. We need to use the t distribution instead (“Student’s t-distribution”). The t-distribution also looks like a bell curve, but how “steep” it is depends on your sample size. ◦ Degrees of freedom (df) = n – 1 ◦ Refer to the t-distribution chart on page 676. Our formulas for calculating “t-scores” and confidence intervals with the t-distribution are exactly the same. We just plug in a t-score instead of a z-score.