AP StatisticsAP Statistics

Chapter 10 NotesChapter 10 Notes

Confidence IntervalConfidence Interval

Statistical Inference: Methods for drawing Statistical Inference: Methods for drawing conclusions about a population based on conclusions about a population based on sample data.sample data.

Level C Confidence Interval (2 parts)Level C Confidence Interval (2 parts) 1. Confidence interval calculated from the 1. Confidence interval calculated from the

data.data. Estimate ± margin of errorEstimate ± margin of error

2. Confidence level – gives the probability that 2. Confidence level – gives the probability that the interval will capture the true parameter the interval will capture the true parameter value in repeated samples. (most often 95%)value in repeated samples. (most often 95%)

Conditions for constructing Conditions for constructing a CI (for a CI (for μμ))

Data must come from an SRS.Data must come from an SRS.

Independence: N Independence: N >> 10n 10n

Sampling distribution of is approx Sampling distribution of is approx Normal Normal

Critical ValuesCritical Values

Values (zValues (z**) that mark off a specified ) that mark off a specified area under the Normal curve.area under the Normal curve.

Confidence Interval for a Confidence Interval for a Population MeanPopulation Mean

Choose an SRS of size n from a Choose an SRS of size n from a population having an unknown mean population having an unknown mean μμ and known standard deviation and known standard deviation σσ. . A level C confidence interval for A level C confidence interval for μμ is…is…

Steps for Constructing a Steps for Constructing a CICI

1. Identify the population and 1. Identify the population and parameter of interest.parameter of interest.

2. Verify that all conditions are met.2. Verify that all conditions are met.

3. Do confidence interval 3. Do confidence interval calculations.calculations.

4. Interpret the results in context.4. Interpret the results in context.

ExampleExample

Suppose that the standard deviation of Suppose that the standard deviation of heart rate for all 18yr old males is 10 heart rate for all 18yr old males is 10 bpm. A random sample of 50 18-year-old bpm. A random sample of 50 18-year-old males yields a mean of 72 beats per males yields a mean of 72 beats per minute.minute.

(a) Construct and interpret a 95% (a) Construct and interpret a 95% confidence interval for the mean heart confidence interval for the mean heart rate rate μμ..

(b) Construct and interpret a 90% CI.(b) Construct and interpret a 90% CI. (c) Construct and interpret a 99% CI.(c) Construct and interpret a 99% CI.

InterpretationInterpretation

We are 95% confident that the true We are 95% confident that the true mean heart rate of all 18 year old males mean heart rate of all 18 year old males is between 69.23 bpm and 74.77 bpm.is between 69.23 bpm and 74.77 bpm.

What does 95% confidence mean?What does 95% confidence mean? 95% of the samples taken from the 95% of the samples taken from the

population will yield an interval which population will yield an interval which contains the true population mean heart contains the true population mean heart rate.rate.

Margin of ErrorMargin of Error

Margin of Error gets smaller when…Margin of Error gets smaller when… zz** gets smaller. (lower z gets smaller. (lower z** = less confident) = less confident) σσ gets smaller. (not easy to do in reality) gets smaller. (not easy to do in reality) n gets larger.n gets larger.

Using the heart rate example, what Using the heart rate example, what would my sample size need to be if I would my sample size need to be if I want a 95% confidence interval with a want a 95% confidence interval with a margin of error, m, of only 1 beat per margin of error, m, of only 1 beat per minute?minute?

Interval for unknown Interval for unknown σσ If we don’t know If we don’t know σσ, (we usually don’t), , (we usually don’t),

we can estimate we can estimate σσ by using s, the sample by using s, the sample standard deviation.standard deviation.

is called the is called the standard errorstandard error of the of the sample mean .sample mean .

Known Known σσ z distribution (Standard z distribution (Standard Normal)Normal) Never changesNever changes

Unknown Unknown σσ t distribution (t(k)) t distribution (t(k)) Changes based on its degrees of freedom k = Changes based on its degrees of freedom k =

n - 1n - 1

One Sample t-intervalOne Sample t-interval

A level C confidence interval for A level C confidence interval for μμ is is

tt** is the critical value for the t(n – 1) is the critical value for the t(n – 1) distribution.distribution.

Paired t ProceduresPaired t Procedures

Used to compare the responses to the Used to compare the responses to the two treatments in a matched pairs two treatments in a matched pairs design or to the before and after design or to the before and after measurements on the same subjects.measurements on the same subjects. The parameter The parameter μμdd in a paired t procedure is in a paired t procedure is

the mean difference in response.the mean difference in response. Robust: accurate even when conditions Robust: accurate even when conditions

are not met.are not met. t procedures are not robust against outliers t procedures are not robust against outliers

but are robust against Non-Normality.but are robust against Non-Normality.

Confidence Interval for pConfidence Interval for p

Conditions:Conditions: SRSSRS Independence: N Independence: N >> 10n 10n and are and are >> 10. 10.

Confidence interval for unknown p.Confidence interval for unknown p.

Finding sample sizeFinding sample size

To find the sample size needed for a To find the sample size needed for a desired C and m…desired C and m…

pp** is a guessed value for p-hat. If you is a guessed value for p-hat. If you have no educated guess, then say phave no educated guess, then say p** = .5.= .5.

RemindersReminders

The margin of error only accounts for The margin of error only accounts for random sampling error. Non-response, random sampling error. Non-response, undercoverage, and response bias must undercoverage, and response bias must still be considered.still be considered.

Random sampling: allows us to generalize Random sampling: allows us to generalize the results to a larger population.the results to a larger population.

Random assignment: allows us to Random assignment: allows us to investigate treatment effects.investigate treatment effects.

Confidence Interval Confidence Interval SummarySummary

1. State the population and the parameter.1. State the population and the parameter. 2. Explain how each condition is/isn’t met.2. Explain how each condition is/isn’t met. (a) SRS(a) SRS (b) Independence: N (b) Independence: N >> 10n. 10n. (c) Normality:(c) Normality:

For p: and are For p: and are >> 10. 10. For For μμ: Look for large n. (Central Limit Theorem): Look for large n. (Central Limit Theorem) If n is small, look to see if the data were If n is small, look to see if the data were

sampled from a Normal population. At last sampled from a Normal population. At last resort, look at the sample data to make sure resort, look at the sample data to make sure that there are no outliers or strong skewness.that there are no outliers or strong skewness.

Summary ContinuedSummary Continued

3. Calculate the confidence interval.3. Calculate the confidence interval. Estimate ± margin of errorEstimate ± margin of error

Summary ContinuedSummary Continued

4. Interpret the interval in context.4. Interpret the interval in context. We are ____% confident that the true We are ____% confident that the true

population population mean/proportionmean/proportion of of ____________ falls between ( , ).____________ falls between ( , ).

If you are asked to interpret the If you are asked to interpret the confidence confidence levellevel…… ______% of the samples taken from the ______% of the samples taken from the

population yield an interval which contains population yield an interval which contains the true population the true population mean/proportion.mean/proportion.