2Rate 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?
3What happens to your confidence as the interval gets smaller? The larger your confidence, the wider the interval.
4Point EstimateUse a single statistic based on sample data to estimate a population parameterSimplest approachBut not always very precise due to variation in the sampling distribution
5estimate + margin of error Confidence intervalsAre used to estimate the unknown population meanFormula:estimate + margin of error
6Margin of error Shows how accurate we believe our estimate is The smaller the margin of error, the more precise our estimate of the true parameterFormula:
7Confidence levelIs the success rate of the method used to construct the intervalUsing this method, ____% of the time the intervals constructed will contain the true population parameter
8What does it mean to be 95% confident? 95% chance that m is contained in the confidence intervalThe 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.
9Critical value (z*) z*=1.645 z*=1.96 z*=2.576 .05 .025 .005 Found from the confidence levelThe upper z-score with probability p lying to its right under the standard normal curveConfidence level tail area z*.05z*=1.645.025.005z*=1.96z*=2.57690%95%99%
10Confidence interval for a population mean: Standard deviation of the statisticCritical valueestimateMargin of error
12Steps for doing a z-interval for means: Assumptions –SRS from populationSample is < 10% of the populationIndependence among data values is plausibleSampling distribution is normal (or approximately normal)Given (normal)Large sample size (n>30)Graph data (unimodal and relatively symmetric)s is knownCalculate the intervalWrite a conclusion about the interval in the context of the problem.
13Conclusion:We are ________% confident that the true mean context lies within the interval ______ and ______.
14The NAEP (National Assessment of Educational Progress) includes a short test of quantitative skills, covering basic arithmetic and the ability to apply it. The standard deviation of the test is 60. Suppose a random sample of 50 young adult men are taken from a large population. If the sample mean of their scores is 265, what is a 95% confidence interval for the true mean score for young adult men on this test?What about a 90% confidence interval?
15A 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 s = A random sample of three has a mean of What is a 90% confidence interval for the mean potassium level?Assumptions:Have an SRS of blood measurementsPotassium level is normally distributed (given)s knownWe are 90% confident that the true mean potassium level is between 3.01 and 3.39.
16Have an SRS of blood measurements 95% confidence interval?Assumptions:Have an SRS of blood measurementsPotassium level is normally distributed (given)s knownWe are 95% confident that the true mean potassium level is between 2.97 and 3.43.
1799% confidence interval?Assumptions:Have an SRS of blood measurementsPotassium level is normally distributed (given)s knownWe are 99% confident that the true mean potassium level is between 2.90 and 3.50.
18the 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
19How can you make the margin of error smaller? z* smaller(lower confidence level)s 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!
20A random sample of 50 PWSH students was taken and their mean SAT score was (Assume s = 105) What is a 95% confidence interval for the mean SAT scores of PWSH students?We are 95% confident that the true mean SAT score for PWSH students is between and
21Suppose 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 PWSH students is between and
22Find 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!
23The heights of PWSH male students is normally distributed with s = 2 The heights of PWSH male students is normally distributed with s = 2.5 inches. How large a sample is necessary to be accurate within inches with a 95% confidence interval?n = 43
24In 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?
25Student’s t- distribution Developed by William GossetContinuous distributionUnimodal, symmetrical, bell-shaped density curveAbove the horizontal axisArea under the curve equals 1Based on degrees of freedom
26t- curves vs normal curve Y1: normalpdf(x)Y2: tpdf(x,2)Y3:tpdf(x,5) use the -0Change Y3:tpdf(x,30)Window: x = [-4,4] scl =1Y=[0,.5] scl =1
27How does t compare to normal? Shorter & more spread outMore area under the tailsAs n increases, t-distributions become more like a standard normal distribution
28How to find t* Use Table for t distributions Can also use invT on the calculator!Need upper t* value with 5% is above – so 95% is belowinvT(p,df)Use Table for t distributionsLook up confidence level at bottom & df on the sidesdf = n – 1Find these t*90% confidence when n = 595% confidence when n = 15t* =2.132t* =2.145
29Formula: Margin of error Standard deviation of statistic Critical valueestimateMargin of error
30Assumptions for t-inference s unknownHave an SRS from populationSample is < 10% of the populationIndependence among data values is plausibleSampling distribution is normal (or approximately normal.Given (population normal)Graph data (unimodal and relatively symmetric with no outliers) or large sample size
31For 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 males27 white males (placebo group) is <10% of white malesWe assume blood pressures are independentSystolic blood pressure is normally distributed (given).s is unknown, so we will construct a t-intervalWe are 95% confident that the true mean systolic blood pressure of healthy white males is between and
32RobustAn 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.
33Ex. 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, )
34Another 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.
35Ex. 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, )
36A 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.
37Some 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
38Cautions continued:Outliers can have a large effect on confidence intervalMust know s to do a z-interval – which is unrealistic in practice