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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 I make the basket? Shooting the ball at a large trash can, will I make the 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 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 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 or CAPTURE the true population parameter

  • 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.

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

    Confidence leveltail areaz*.051.645.0251.96.0052.576Critical value (z*)z*=1.645z*=1.96z*=2.57690%95%99%

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

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

  • The 4-Step ProcessConfidence 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 measurementsPotassium level is normally distributed (given)s known

    We are 90% confident that the true mean potassium level is captured in the interval 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 captured in the interval 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 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)

    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 SHS 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 SHS students?

    We are 95% confident that the true mean SAT score for SHS students is captured in the interval between 1220.9 and 1279.1

  • Suppose that we have this random sample of SAT scores: 1130 1260 1090 1310 1420 1190What 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 SHS students is captured in the interval 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 SHS 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 = 42.68 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 114.9 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?

  • Students 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

  • 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). s is unknown

    We are 95% confident that the true mean systolic blood pressure is between 111.22 and 118.58.

  • 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.

  • Ex. 5 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.(70.883, 74.497)

  • 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:16020022023012018014013017019080120100170Compute a 98% confidence interval for the average calorie content per serving of vanilla yogurt.(126.16, 189.56)

  • 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

    *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*