Confidence Intervals with Means

Rate your confidence Name my age within 10 years? Name my age within 10 years? within 5 years? within 5 years? within 1 year? within 1 year? Shooting a basketball at a wading pool, will make basket? 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 large trash can, will make basket? Shooting the ball at a carnival, 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. 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. Then, using the random number generator from a normal distribution, a sample mean will be generated. Can you determine the true number? Can you determine the true number?

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

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

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

Confidence level Is the success rate of the method used to construct an interval that contains that true mean 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 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 95% chance that m is contained in the confidence interval The probability that the interval contains m is 95% 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. The method used to construct the interval will produce intervals that contain m 95% of the time.

Found from the confidence level Found from the confidence level The upper z-score with probability p lying to its right under the standard normal curve The upper z-score with probability p lying to its right under the standard normal curve Confidence leveltail areaz* Critical value (z*).05 z*= z*= z*= % 95% 99%

Confidence interval for a population mean: estimate Critical value Standard deviation of thestatistic Standard deviation of the statistic Margin of error

Steps for doing a confidence interval: 1) Assumptions – SRS from populationSRS from population Sampling distribution is normal (or approximately normal)Sampling distribution is normal (or approximately normal)  Given (normal)  Large sample size (approximately normal)  Graph data (approximately normal) σ is known σ is known 2) 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 4__666391__ 4__666391__ 4__666391__ 4__666391__ The purpose of this applet is to understand how the intervals move but the population mean doesn’t. The purpose of this applet is to understand how the intervals move but the population mean doesn’t.

Assumptions: Have an SRS of blood measurements Potassium level is normally distributed (given)  known We are 90% confident that the true mean potassium level is between 3.01 and 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 Potassium level is normally distributed (given) s known We are 95% confident that the true mean potassium level is between 2.97 and % confidence interval?

99% confidence interval? Assumptions: Have an SRS of blood measurements Potassium level is normally distributed (given) s known We are 99% confident that the true mean potassium level is between 2.90 and 3.50.

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 z* smaller (lower confidence level) σ smaller σ smaller (less variation in the population) n larger 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 (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 and

Suppose that we have this random sample of SAT scores: 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 and

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: 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 = 2.5 inches. How large a sample is necessary to be accurate within +/-.75 inches with a 95% confidence interval? n = 43

Student t-Distribution In a randomized comparative experiment on the effects of calcium on blood pressure, researchers divided 54 healthy, white males at random into two groups, takes calcium or placebo. The paper reports a mean seated systolic blood pressure of with standard deviation of 9.3 for the placebo group. Assume systolic blood pressure is normally distributed. Can you find a z-interval for this problem? Why or why not? No, don’t know σ Only sample s

Gossett Story

Can you use s instead of σ when calculating a z-score (so that you can find the +/- 3 σ )? Can you use s instead of σ when calculating a z-score (so that you can find the +/- 3 σ )? Not exactly. Look at the two equations. Not exactly. Look at the two equations. Parameters are constant – don’t expect the shape to change, just shift based on changes in ẋ Statistics are variables – each sample s will cause the shape to change away from a normal distribution ẋ has a normal distribution

Do t-score bingo. Do t-score bingo.

Student’s t- distribution Developed by William Gosset Developed by William Gosset Continuous distribution Continuous distribution Unimodal, symmetrical, bell-shaped density curve Unimodal, symmetrical, bell-shaped density curve Above the horizontal axis Above the horizontal axis Area under the curve equals 1 Area under the curve equals 1 Based on degrees of freedom Based on degrees of freedom

T-Curves Basic properties of t-Curves Property 1: The total area under a t-curve equals 1. Property 1: The total area under a t-curve equals 1. Property 2: A t-curve extends indefinitely in both directions, approaching the horizontal axis asymptotically Property 2: A t-curve extends indefinitely in both directions, approaching the horizontal axis asymptotically Property 3: A t-curve is symmetric about 0. Property 3: A t-curve is symmetric about 0.

T-curves continued Property 4: As the number of degrees of Property 4: As the number of degrees of freedom becomes larger, t-curves look increasingly like the standard normal curve

How does t compare to normal? Shorter & more spread out Shorter & more spread out More area under the tails More area under the tails As n increases, t-distributions become more like a standard normal distribution As n increases, t-distributions become more like a standard normal distribution

Graph examples of t- curve vs normal curve

z – Score and t - Score

Confidence Interval Formula: estimate Critical value Standard deviation of statistic Margin of error

How to find Margin of error when σ is not available – find t* Use Table B for t distributions Use Table B for t distributions Look up confidence level at bottom & degrees of freedom (df) on the sides Look up confidence level at bottom & degrees of freedom (df) on the sides df = n – 1 df = n – 1 Find these t* 90% confidence when n = 5 95% confidence when n = 15 t* =2.132 t* =2.145 Can also use invT on the calculator! For 90% confidence level, 5% is above and 5% is below Need upper t* value with 5% above – so 0.95 is p value invT(p,df)

Assumptions for t-inference Have an SRS from population σ unknown Normal distribution – –Given – –Large sample size – –Check graph of data

For the Ex. 4: Find a 95% confidence interval for the true mean systolic blood pressure of the placebo group. Assumptions: Have an SRS of healthy, white males Systolic blood pressure is normally distributed (given).  is unknown We are 95% confident that the true mean systolic blood pressure is between and

Robust An inference procedure is ROBUST if the confidence level or p-value doesn’t change much if the assumptions are violated. For adequately sized samples (n≥30), t-procedures can be used with some skewness, as long as there are no outliers.

Ex. 5 – A medical researcher measured the pulse rate of a random sample of 20 adults and found a mean pulse rate of beats per minute with a standard deviation of 3.86 beats per minute. Assume pulse rate is normally distributed. Compute a 95% confidence interval for the true mean pulse rates of adults. (70.883, )

Another medical researcher claims that the true mean pulse rate for adults is 72 beats per minute. Does the evidence support or refute this? Explain. The 95% confidence interval contains the claim of 72 beats per minute. Therefore, there is no evidence to doubt the claim.

Ex. 6 – Consumer Reports tested 14 randomly selected brands of vanilla yogurt and found the following numbers of calories per serving: Compute a 98% confidence interval for the average calorie content per serving of vanilla yogurt. Boxplot shows approx normal distr. (126.16, )

A diet guide claims that you will get 120 calories from a serving of vanilla yogurt. What does this evidence indicate? Since 120 calories is not contained within the 98% confidence interval, the evidence suggest that the average calories per serving does not equal 120 calories. NOT EQUAL Note: confidence intervals tell us if something is NOT EQUAL – never less or greater than!

Some Cautions: The data MUST be a SRS from the population (must be random) The formula is not correct for more complex sampling designs, i.e., stratified, etc. No way to correct for bias in data

Cautions continued: Outliers can have a large effect on confidence interval Must know σ to do a z-interval – which is unrealistic in practice