AP Statistics Chapter 10 Notes

Confidence Interval Statistical Inference: Methods for drawing conclusions about a population based on sample data. Statistical Inference: Methods for drawing conclusions about a population based on sample data. Level C Confidence Interval (2 parts) Level C Confidence Interval (2 parts) 1. Confidence interval calculated from the data. 1. Confidence interval calculated from the data. Estimate ± margin of error Estimate ± margin of error 2. Confidence level – gives the probability that the interval will capture the true parameter value in repeated samples. (most often 95%) 2. Confidence level – gives the probability that the interval will capture the true parameter value in repeated samples. (most often 95%)

Conditions for constructing a CI (for μ) Data must come from an SRS. Data must come from an SRS. Independence: N > 10n Independence: N > 10n Sampling distribution of is approx Normal Sampling distribution of is approx Normal

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

Confidence Interval for a Population Mean Choose an SRS of size n from a population having an unknown mean μ and known standard deviation σ. A level C confidence interval for μ is… Choose an SRS of size n from a population having an unknown mean μ and known standard deviation σ. A level C confidence interval for μ is…

Steps for Constructing a CI 1. Identify the population and parameter of interest. 1. Identify the population and parameter of interest. 2. Verify that all conditions are met. 2. Verify that all conditions are met. 3. Do confidence interval calculations. 3. Do confidence interval calculations. 4. Interpret the results in context. 4. Interpret the results in context.

Example Suppose that the standard deviation of heart rate for all 18yr old males is 10 bpm. A random sample of year-old males yields a mean of 72 beats per minute. Suppose that the standard deviation of heart rate for all 18yr old males is 10 bpm. A random sample of year-old males yields a mean of 72 beats per minute. (a) Construct and interpret a 95% confidence interval for the mean heart rate μ. (a) Construct and interpret a 95% confidence interval for the mean heart rate μ. (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.

Interpretation We are 95% confident that the true mean heart rate of all 18 year old males is between bpm and bpm. We are 95% confident that the true mean heart rate of all 18 year old males is between bpm and bpm. What does 95% confidence mean? What does 95% confidence mean? 95% of the samples taken from the population will yield an interval which contains the true population mean heart rate. 95% of the samples taken from the population will yield an interval which contains the true population mean heart rate.

Margin of Error Margin of Error gets smaller when… Margin of Error gets smaller when… z * gets smaller. (lower z * = less confident) z * gets smaller. (lower z * = less confident) σ gets smaller. (not easy to do in reality) σ gets smaller. (not easy to do in reality) n gets larger. n gets larger. Using the heart rate example, what would my sample size need to be if I want a 95% confidence interval with a margin of error, m, of only 1 beat per minute? Using the heart rate example, what would my sample size need to be if I want a 95% confidence interval with a margin of error, m, of only 1 beat per minute?

Interval for unknown σ If we don’t know σ, (we usually don’t), we can estimate σ by using s, the sample standard deviation. If we don’t know σ, (we usually don’t), we can estimate σ by using s, the sample standard deviation. is called the standard error of the sample mean. is called the standard error of the sample mean. Known σ  z distribution (Standard Normal) Known σ  z distribution (Standard Normal) Never changes Never changes Unknown σ  t distribution (t(k)) Unknown σ  t distribution (t(k)) Changes based on its degrees of freedom k = n - 1 Changes based on its degrees of freedom k = n - 1

One Sample t-interval A level C confidence interval for μ is A level C confidence interval for μ is t * is the critical value for the t(n – 1) distribution. t * is the critical value for the t(n – 1) distribution.

Paired t Procedures Used to compare the responses to the two treatments in a matched pairs design or to the before and after measurements on the same subjects. Used to compare the responses to the two treatments in a matched pairs design or to the before and after measurements on the same subjects. The parameter μ d in a paired t procedure is the mean difference in response. The parameter μ d in a paired t procedure is the mean difference in response. Robust: accurate even when conditions are not met. Robust: accurate even when conditions are not met. t procedures are not robust against outliers but are robust against Non-Normality. t procedures are not robust against outliers but are robust against Non-Normality.

Confidence Interval for p Conditions: Conditions: SRS SRS Independence: N > 10n Independence: N > 10n and are > 10. and are > 10. Confidence interval for unknown p. Confidence interval for unknown p.

Finding sample size To find the sample size needed for a desired C and m… To find the sample size needed for a desired C and m… p * is a guessed value for p-hat. If you have no educated guess, then say p * =.5. p * is a guessed value for p-hat. If you have no educated guess, then say p * =.5.

Reminders The margin of error only accounts for random sampling error. Non-response, undercoverage, and response bias must still be considered. The margin of error only accounts for random sampling error. Non-response, undercoverage, and response bias must still be considered. Random sampling: allows us to generalize the results to a larger population. Random sampling: allows us to generalize the results to a larger population. Random assignment: allows us to investigate treatment effects. Random assignment: allows us to investigate treatment effects.

Confidence Interval Summary 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 > 10n. (b) Independence: N > 10n. (c) Normality: (c) Normality: For p: and are > 10. For p: and are > 10. For μ: Look for large n. (Central Limit Theorem) For μ: Look for large n. (Central Limit Theorem) If n is small, look to see if the data were sampled from a Normal population. At last resort, look at the sample data to make sure that there are no outliers or strong skewness. If n is small, look to see if the data were sampled from a Normal population. At last resort, look at the sample data to make sure that there are no outliers or strong skewness.

Summary Continued 3. Calculate the confidence interval. 3. Calculate the confidence interval. Estimate ± margin of error Estimate ± margin of error

Summary Continued 4. Interpret the interval in context. 4. Interpret the interval in context. We are ____% confident that the true population mean/proportion of ____________ falls between (, ). We are ____% confident that the true population mean/proportion of ____________ falls between (, ). If you are asked to interpret the confidence level… If you are asked to interpret the confidence level… ______% of the samples taken from the population yield an interval which contains the true population mean/proportion. ______% of the samples taken from the population yield an interval which contains the true population mean/proportion.