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16_Confidence_Intervals - Confidence Intervals Confidence...

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Unformatted text preview: Confidence Intervals Confidence In pairs: In Randomly select one sheet of paper from the Randomly envelope envelope Calculate 68% confidence intervals from your Calculate sample of 5 values sample (See end of slide show for help) Confidence Intervals Confidence Confidence intervals are ways to say something Confidence about population values from sample data about E.g. what percent of the population of MSU students E.g. smoke? What is the mean amount that MSU students have spent on parking tickets this year? have Because we only have partial information, we Because cannot give an exact answer about the population population We can only narrow our range to a specified We interval interval Confidence Interval Confidence A confidence interval is an range of values confidence computed from sample data that is almost sure to cover the true population number sure Today we will be talking about forming Today confidence intervals for proportions and and means means CI = margin of error CI We have seen confidence intervals already: with We the margin of error from a survey the Pew Research Center for the People and the Press, Pew 2003 2003 survey of 1,515 U.S. adults survey 54% agree that gay and lesbian couples could be 54% good parents good Margin of error 3 percentage points This is a 95% confidence interval Another example… Another Boxers or briefs? Some might worry that briefs might be too warm and lower Some sperm count, but… sperm “Mean scrotal temperature plus or minus [one] standard Mean deviation was 33.8 +/- .8 degrees C in and 33.6 +/- 1.1 degrees C in the brief group, respectively [body temp is 37 C]. There were no significant differences between the groups.” Munkelwitz and Gilbert (1998) Journal of Urology Urology 68% confidence intervals for the two groups overlap considerably considerably CI for proportions CI Gives a range of likely values for the Gives population proportion population What proportion of MSU students smoke? Standard Error Standard The standard deviation for the possible sample The proportions is called the standard error standard Standard error=Sqrt(true proportion*(1-true Standard proportion)/n) proportion)/n) It is important to distinguish this from the It standard deviation of the sample we draw standard The s.e. of the mean tells us about the shape of a The hypothetical group of sample means hypothetical The s.d. of the sample tells us about the spread of the The sample sample Rule for Sample Proportions Rule Numerous samples of the same size: Numerous frequency curve made from proportions from frequency the various samples will be approximately bell-shaped bell-shaped mean = the true proportion from the mean population population standard deviation of this curve (otherwise standard known as the standard error)=Sqrt(true proportion*(1-true proportion)/n) proportion*(1-true There is a similar rule for means There http://onlinestatbook.com/stat_sim/sampling_d Applying to CI’s Applying Because drawing a large number of sample Because proportions produces a bell shaped curve: proportions In 95% of all samples, the sample In proportion will fall within 2 s.e. of the mean, which is the true proportion of the population population Formula for CI’s for proportion Formula Formula for a 95% confidence interval for a Formula population proportion: population Sample proportion +/- 2*s.e. Where s.e.=sqrt(sample proportion*(1-sample Where proportion)/n) proportion)/n) Some notes Some Actually, should be 1.96, not 2 So our answers may differ a little from excel Using the results from Chapter 8, Note that we Using can multiply the s.e. by different values to obtain different sized (i.e. other than 95%) confidence intervals intervals Sample proportion +/- 3*s.e. creates 99.7% Sample confidence intervals confidence Sample proportion +/- 1.645*s.e. creates 90% Sample confidence intervals confidence Why? Some notes (con.) Some In chapter 4, we used an approximate In measure for creating confidence intervals measure Sample proportion +/- 1/sqrt(n) This is an ok estimate; it is exact when the This sample proportion=.5 sample Now we can find this with greater Now accuracy accuracy Example: Drug Usage Example: Annual American high school study 1993 17,000 seniors % who used the following in the past year: LSD: 6.8 Marijuana: 26.0 Alcohol: 76.0 Compute 95 % confidence intervals Do more than ½ of seniors use alcohol? Sample proportion +/- 2*s.e. Where s.d.=sqrt(sample proportion*(1-sample proportion)/n) LSD: .068+/-2*sqrt(.068*(1-.068)/17,000)=(.064,.072) Marijuana: .26+/-2*sqrt(.26*(1-.26)/17,000)=(.25, .27) Alcohol: .76+/-2*sqrt(.76*(1-.76)/17,000)=(.75, .77) Notice that we have converted % to proportions; Notice sometimes figures will be given in proportions, other times in percentages times Confidence intervals for means Confidence What is the mean amount that MSU students What spend on parking tickets? spend Rule for Sample Means Rule The Rule for Sample Means can only be The applied iif there is a large random sample (>=30) or f the population of measurements is bell the shaped (IQ’s, male heights, etc) shaped So we can only construct CI’s if we have a So sample of greater than 30 or if the population of measurements is bell shaped shaped Confidence Intervals for a Mean Confidence 95% Confidence intervals for means: Sample value+/- 2 * s.e. Where s.e.=population standard deviation/sqrt(n) We usually have to estimate the s.e. with the sample We s.d., so s.e.=sample standard deviation/sqrt(n) s.d., Note that the s.e. differs for proportions and means CI for Means and Proportions CI A generic formula for the construction of 95% generic confidence intervals is: confidence Sample value +/- 2 * measure of variability Sample Where the measure of variability for means is the standard error of the means, and the standard measure of variability for proportions is the standard error for proportions standard We will also discuss a measure of variability for We two means two Confidence interval for difference in means means What if we want to compare the two means? That is, we want a confidence interval around mean1-mean2 Usually to see if it contains zero or if the entire interval is positive Usually or negative; that is, is it likely that there’s a real difference in the means in the population means 95% CI=mean1-mean2 +/-2*s.e. Where se= sqrt[sd12/n1 + sd22/n2] sd sd1 is the standard deviation for the first group, n1 is the number in first group (same for group 2) number Example: CI for 2 means Example: Housework hours per week, national survey Housework of men and women of Sex Sample Sample Size Mean Housework Hrs Hrs 32.6 32.6 18.1 18.1 Female Male 6764 6764 4252 4252 Standard Standard deviation Housework hrs hrs 18.2 18.2 12.9 12.9 95% CI for 2 means example 95% Mean1-mean2=32.6-18.1=14.5 S.e=sqrt[(sd1)2/N1+ (sd2)2/N2]=sqrt[18.22/6764 + /6764 12.92/4252)=.3 12.9 95% CI=difference in means +/- 2*se 14.5 +/- 2*.3=14.5 +/-.6 = (13.9, 15.1) Interpretation: The mean amount of weekly time that Interpretation: women spend on housework is between 13.9 and 15.1 hours more than men spend on it 15.1 Notes on CI for two means Notes The two means must come from independent The samples samples That is, the selection of elements in one sample can That NOT influence the selection of elements in another NOT E.g. can’t use scores collected at two points in time E.g. from the same people, or compare two groups that are naturally paired (e.g. husbands to wives) are Can compare two subgroups (e.g. men and women, Can blacks and whites) from a simple random sample, or from two separate simple random samples (e.g. a sample of men, a sample of women) Interpretation of CIs Interpretation The interpretation of 95% confidence The intervals is that in 100 samples, 95 of the means will fall within our interval means Similar interpretation for 90%, 99.7%, etc. It’s just that we will have to adjust our It’s formula: formula: Sample value+/-2*measure of variability 68% confidence intervals for height data data Draw 5 heights from the hat (one slip of paper with box labelled Draw “height”) “height”) Calculate 68% confidence intervals for the mean Calculate the mean Calculate s.e.=s.d./sqrt(5) Take mean + 1 s.e for the upper bound Take mean – 1 s.e. for the lower bound s.e.=s.d./sqrt(5) Remember that the s.d. is the square root of (the sum of the Remember squared deviations/n-1) (see page 138) squared Help each other out!! 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