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Confidence Intervals. Chapter 10. Rate your confidence 0 - 100. Name my age within 10 years? within 5 years? within 1 year? Shooting a basketball at a wading pool, will make basket? Shooting the ball at a large trash can, will make basket? - PowerPoint PPT Presentation

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  • Confidence Intervals

    Chapter 10

  • Rate your confidence0 - 100Name my age within 10 years?within 5 years?within 1 year?

    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.

    Simulation

  • Point 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

  • Confidence intervalsAre used to estimate the unknown population meanFormula:

    estimate + margin of error

  • Margin of errorShows how accurate we believe our estimate isThe smaller the margin of error, the more precise our estimate of the true parameterFormula:

  • Confidence levelIs the success rate of the method used to construct the interval

    Using this method, ____% of the time the intervals constructed will contain the true population parameter

  • Critical value (z*)Found from the confidence levelThe upper z-score with probability p lying to its right under the standard normal curve

    Confidence leveltail areaz*.051.645.0251.96.0052.576z*=1.645z*=1.96z*=2.57690%95%99%

  • What 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.

  • Confidence interval for a population mean:estimateCritical valueStandard deviation of the statisticMargin of error

  • Steps for doing a confidence interval:Assumptions SRS from populationSampling distribution is normal (or approximately normal)Given (normal)Large sample size (approximately normal)Graph data (approximately normal) s is knownCalculate the intervalWrite 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 ______.

  • Assumptions:Have an SRS of blood measurementsPotassium level is normally distributed (given)s known

    We are 90% confident that the true mean potassium level is between 3.01 and 3.39.

    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 s = 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 measurementsPotassium level is normally distributed (given)s known

    We are 95% confident that the true mean potassium level is between 2.97 and 3.43.

    95% confidence interval?

  • 99% confidence interval?

    Assumptions:Have an SRS of blood measurementsPotassium 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

  • Critical value (z*)Found from the confidence levelThe upper z-score with probability p lying to its right under the standard normal curve

    Confidence leveltail areaz*.051.645.0251.96.0052.576z*=1.645z*=1.96z*=2.57690%95%99%

  • How 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!

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

  • 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 BGHS male students is normally distributed with s = 2.5 inches. How large a sample is necessary to be accurate within + .75 inches with a 95% confidence interval?n = 43Homework pg. 632-633 7-11, 13, 15

  • t- distributionDeveloped by William GossetContinuous distributionUnimodal, symmetrical, bell-shaped density curveAbove the horizontal axisArea under the curve equals 1Based on degrees of freedom

  • Graph examples of t- curves vs normal curve

  • How does t compare to normal?Shorter & more spread outMore area under the tailsAs n increases, t-distributions become more like a standard normal distribution

  • How to find t*Use Table B for t distributionsLook up confidence level at bottom & df on the sidesdf = n 1

    Find these t*90% confidence when n = 595% confidence when n = 15t* =2.132t* =2.145Can also use invT on the calculator!

    Need upper t* value with 5% is above so 95% is below

    invT(p,df)

  • Formula:estimateCritical valueStandard deviation of statisticMargin of error

  • Assumptions for t-inferenceHave an SRS from population s unknownNormal distributionGivenLarge sample sizeCheck graph of data

  • RobustAn inference procedure is ROBUST if the confidence level or p-value doesnt 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.

  • Outliers are always a concern, but they are even more of a concern for confidence intervals using the t-distributionSample mean is not resistant; hence the sample mean is larger or smaller (drawn toward the outlier) (small numbers of n in t-distribution!)Sample standard deviation is not resistant; hence the sample standard deviation is largerConfidence intervals are much wider with an outlier includedOptions: Make sure data is not a typo (data entry error)Increase sample size beyond 30 observations

  • A medical researcher measured the pulse rate of a random sample of 20 adults and found a mean pulse rate of 72.69 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.We are 95% confident that the true mean pulse rates of adults is between 70.883 and 74.497 beat per minute.

  • 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.

  • Consumer Reports tested 14 randomly selected brands of vanilla yogurt and found the following numbers of calories per serving:16020022023012018014013017019080120100170Compute a 98% confidence interval for the average calorie content per serving of vanilla yogurt.We are 98% confident that the true mean calorie content per serving is between 126.16 and 189.56 calories.

  • 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.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 populationThe 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 s to do a z-interval which is unrealistic in practice

  • Homework:

    10.27, 28, 29 Pg.648-649

    *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*