Confidence Intervals with Means

Rate your confidence 0 - 100 Name my age within 10 years? Shooting a basketball at a wading pool, will make basket? Shooting the ball at a large trash can, will make basket? Shooting the ball at a carnival, will make basket?

What happens to your confidence as the interval gets smaller? The larger your confidence, the wider the interval.

Guess the number Teacher will have pre-entered a number into the memory of the calculator. Then, using the random number generator from a normal distribution, a sample mean will be generated. Can you determine the true number?

Point Estimate Use a single statistic based on sample data to estimate a population parameter Simplest approach But not always very precise due to variation in the sampling distribution

estimate + margin of error Confidence intervals Are used to estimate the unknown population mean Formula: estimate + margin of error

Margin of error Shows how accurate we believe our estimate is The smaller the margin of error, the more precise our estimate of the true parameter Formula:

Confidence level Is the success rate of the method used to construct an interval that contains that true mean Using this method, ____% of the time the intervals constructed will contain the true population parameter

What does it mean to be 95% confident? 95% chance that m is contained in the confidence interval The probability that the interval contains m is 95% The method used to construct the interval will produce intervals that contain m 95% of the time.

Critical value (z*) Found from the confidence level The upper z-score with probability p lying to its right under the standard normal curve Confidence level tail area z* .05 1.645 .025 1.96 .005 2.576 .05 z*=1.645 .025 .005 z*=1.96 z*=2.576 90% 95% 99%

Confidence interval for a population mean: Critical value Standard Error (SD of Sampling Distribution) estimate Margin of error

CI of Means - Steps: Assumptions – SRS from population Sample is independent – smaller than 10% of population Sampling distribution is normal (or approximately normal) Given (normal) Large sample size (approximately normal) Graph data (approximately normal) σ is known Calculate the interval

3) Write a statement about the interval in the context of the problem. Statement: (memorize!!) We are __________% confident that the true mean context lies within the interval _______ and ______.

Confidence Interval Applet http://bcs.whfreeman.com/tps4e/#628644__666391__ The purpose of this applet is to understand how the intervals move but the population mean doesn’t.

A test for the level of potassium in the blood is not perfectly precise. Suppose that repeated measurements for the same person on different days vary normally with σ = 0.2. A random sample of three has a mean of 3.2. What is a 90% confidence interval for the mean potassium level? Assumptions: Have an SRS of blood measurements Assume patient has time to regenerate blood for each sample so each sample is independent Potassium level is normally distributed (given) s known We are 90% confident that the true mean potassium level is between 3.01 and 3.39.

95% confidence interval? Assumptions: Assume same as previous We are 95% confident that the true mean potassium level is between 2.97 and 3.43.

99% confidence interval? Assumptions: Assume same as previous We are 99% confident that the true mean potassium level is between 2.90 and 3.50.

the interval gets wider as the confidence level increases What happens to the interval as the confidence level increases? the interval gets wider as the confidence level increases

How can you make the margin of error smaller? z* smaller (lower confidence level) σ smaller (less variation in the population) n larger (to cut the margin of error in half, n must be 4 times as big) Really cannot change!

A random sample of 50 CHS students was taken and their mean SAT score was 1250. (Assume σ = 105) What is a 95% confidence interval for the mean SAT scores of CHS students? We are 95% confident that the true mean SAT score for CHS students is between 1220.9 and 1279.1

Suppose that we have this random sample of SAT scores: 1130 1260 1090 1310 1420 1190 What is a 95% confidence interval for the true mean SAT score? (Assume s = 105) We are 95% confident that the true mean SAT score for CHS students is between 1115.1 and 1270.6.

Find a sample size: If a certain margin of error is wanted, then to find the sample size necessary for that margin of error use: Always round up to the nearest person!

The heights of CHS male students is normally distributed with σ = 2 The heights of CHS male students is normally distributed with σ = 2.5 inches. How large a sample is necessary to be accurate within +/- .75 inches with a 95% confidence interval? n = 43