Confidence Intervals Chapter 10

Rate your confidence Name my age within 10 years? within 5 years? within 1 year? Shooting a basketball at a wading pool, will I make the basket? Shooting the ball at a large trash can, will I make the basket? Shooting the ball at a carnival, will I make the basket?

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

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

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 more preciseThe 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 the interval Using this method, ____% of the time the intervals constructed will contain or CAPTURE the true population parameter

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

Found from the confidence level 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 the statistic Margin of error

The 4-Step Process (from the Inference Toolbox) Step 1 (Population and parameter) Define the population and parameter you are investigating Step 2 (Conditions) Do we have biased data? If SRS, we’re good. Otherwise PWC (proceed with caution) Do we have independent sampling? If pop>10n, we’re good. Otherwise PWC. Do we have a normal distribution? If pop is normal or n>30 (CLT), we’re good. Otherwise, PWC.

The 4-Step Process (from the Inference Toolbox) Step 3 (Calculations) Find z* based on your confidence level. If you are not given a confidence level, use 95% Calculate CI. Step 4 (Interpretation) “With ___% confidence, we believe that the true mean is captured in the interval (lower, upper)”

The 4-Step Process Confidence Interval Step 1 (Population and parameter) Define the population and parameter you are investigating Step 2 (Conditions) SRS from population? Pop>10n? Pop is normal or n>30 (CLT)? If raw data is given, graph and see if distribution is normal  is known

The 4-Step Process Confidence Interval Step 3 (Calculations) Find z* based on your confidence level. If you are not given a confidence level, use 95% Calculate CI using Step 4 (Interpretation) “With ___% confidence, we believe that the true mean is captured in the interval (lower, upper)” Or…. “The methods used to construct the interval will capture the true mean ___% of the time”.

Statement: (memorize!!) We are ________% confident that the true mean context IS CAPTURED within the interval from ______ to ______. ( This means that using these methods, ____% of the time the intervals constructed will capture the true population mean.)

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 captured in the interval 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  = 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)  known We are 95% confident that the true mean potassium level is captured in the interval between 2.97 and % confidence interval?

99% confidence interval? Assumptions: Have an SRS of blood measurements Potassium level is normally distributed (given)  known We are 99% confident that the true mean potassium level is captured in the interval 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 (lower confidence level)  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 SHS students was taken and their mean SAT score was (Assume  = 105) What is a 95% confidence interval for the mean SAT scores of SHS students? We are 95% confident that the true mean SAT score for SHS students is captured in the interval 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  = 105) We are 95% confident that the true mean SAT score for SHS students is captured in the interval 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: Always round up to the nearest person!

The heights of SHS male students is normally distributed with  = 2.5 inches. How large a sample is necessary to be accurate within +.75 inches with a 95% confidence interval? n = or 43 students

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?

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

Graph examples of t- curves vs normal curve

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

How to find t* Use Table B for t distributions Look up confidence level at bottom & df on the sides 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! Need upper t* value with 5% is above – so 95% is below invT(p,df)

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

Assumptions for t-inference Have an SRS from population  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. t-procedures can be used with some skewness, as long as there are no outliers. Larger n can have more skewness.

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. (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 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  to do a z-interval – which is unrealistic in practice