Confidence Intervals Chapter 8 Section 1 Confidence Interval

Confidence Intervals Chapter 8 Section 1

• Confidence Interval – – an interval obtained from sample data that tries to estimate the true population parameter like µ or p the population mean or population proportion. (It’s a range of plausible values for the parameter) – It gives the probability that the method used will produce an interval that contains the true parameter like µ or p. – A statistic like , the sample mean, or p hat is at its center and the level of confidence determines the length of its extending ‘arms’ its margin of error Note: remember any statistic can be substituted in where is for corresponding parameter.

One-sample z Interval for Proportions Estimator ± (critical value)*(standard dev. of statistic) Estimate ± Margin of Error

One-sample z Interval for Means ± z*(σ/√n) Estimator ± (critical value)*(standard dev of statistic) Estimator ± Margin of Error = sample mean σ = standard deviation of population z* = confidence level, called the upper critical value – tells us how many st. dev. away from the estimate we should go to get our desired level of confidence n = sample size Assumptions/Conditions: • SRS • Sample comes from a normally distributed population or n ≥ 30 because then we know sample comes from an approximately normal sampling distribution. If n < 30, use shape of the distribution of the sample to assess normality of population • N > 10 n or n < 0. 10 N (tells you that your equation for S. D. is okay to use)

One-sample t Interval for Means ± t*(sx/√n) Estimator ± (critical value)*(standard dev of statistic) Estimator ± Margin of Error = sample mean sx = standard deviation of the sample called the standard error t* = confidence level, called the upper critical value – tells us how many st. dev. away from the estimate we should go to get our desired level of confidence using the t distribution n = sample size Assumptions/Conditions: • SRS • Sample comes from a normally distributed population or n ≥ 30 because then we know sample comes from an approximately normal sampling distribution. If n < 30, use shape of the distribution of the sample to assess normality of population • N > 10 n or n < 0. 10 N (tells you that your equation for S. D. is okay to use)

The degrees of freedom for the t-distribution is df = n - 1. z-distribution t-distribution

Interpreting Confidence Level Once a particular confidence interval is calculated, it either includes the parameter (probability = 1) or doesn’t include the parameter (probability = 0).

Go to Fathom Activity

Properties of Confidence Intervals • As confidence goes up margin of error goes up and vice versa as confidence margin of error • If there is a lot of variability in the population, large σ, (or sample, large s) margin of error is large • As sample size, n, the margin of error , m,

Interpreting Confidence Level and Interval AP® Exam Tip On a given problem, you may be asked to interpret the confidence interval, the confidence level, or both. Be sure you understand the difference: • the confidence interval gives a set of plausible values for the parameter • the confidence level describes the overall capture rate of the method.

Stating your results: • Interpreting Confidence Level. C% of all possible samples of the same size from the same population give an interval that will capture the true population parameter (mean or proportion). • Interpreting Confidence Interval: We are C% confident that the interval from _____ to _____ captures the [parameter in context]. ” When interpreting a confidence interval, make sure that you are describing the parameter and not the statistic. • CANNOT SAY: the population mean, µ, will fall in this interval (90%) of the time

Assignment: Pages: 506 -509 Problems: 1 -9 odd 15, 17, 19, 21, 23 -26