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Inference and Confidence Intervals

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Outline Inferring a population mean: Constructing confidence intervals Examining the difference between two means for the same person Confidence intervals for the difference between groups Confidence intervals for medians Summary

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Constructing Confidence Intervals Population μ (pronounced mu) Estimate with sample mean ( ), the plug-in principle But with sampling error. Estimating the region which will usually include the population mean. t 0.05 is usually about 2, but you need to look it up the t table. Need to know df. df = n - 1 for this test. Here 94 - 1 = 93.

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Example: Newton's (1998) Hostility Data The mean on arrival for the 94 prisoners was 28.3 with a standard deviation of 8.0. df = n-1 so 93, or about 90 for the t table

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t distribution with df = 93 and 2.5% in each tail

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What does having 95% CI of 28.3 ± 1.6 mean? We expect that about 95% of the time when a confidence interval is made that the population mean (μ) will be within the interval created. This allows us to be fairly confident that the confidence interval we calculate contains the population mean. It is not that there is a 95% probability that μ is within the interval. This is a tricky concept. Confidence intervals are a fundamental tool for the frequentist statistician. In the long run, you should be right (i.e., μ with the interval) about 95% of the time.

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Plotting the precision of the estimate (confidence intervals) and the spread of the distribution (standard deviations). Both are in units of the original variable (here in years).

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Examining the Difference between Two Means for the Same Person Difference in means ± t 0.05 times standard error Standard error of difference using estimate of the standard deviation of the difference. Just calculate a variable for the difference, and perform the calculations as you did before.

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Brewed Awakenings http://mybrewedawakening.com/ Table 5.1 Data from 10 participants comparing how much they like two different types of coffee. FRESHi INSTANTiDIFFi DIFFi - (DIFFi - ) 2 5 3 2 1 1 4 3 1 0 0 6 5 1 0 0 3 4 -1 -2 4 4 4 0 -1 1 5 3 2 1 1 6 3 3 2 4 3 3 0 -1 1 5 3 2 1 1 4 4 0 -1 1 Sum 45 35 10 0 14 Mean 4.5 3.5 1.0 0 1.56* * This 1.56 is not the actual mean. It is the variance of the variable DIFFi, the sum of squares divided by the number of cases minus one (14/9). We calculated it this way because it can be used in later calculations. ________________________________________________________________________

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Chapter 11- Confidence Intervals for Univariate Data Math 22 Introductory Statistics.

Chapter 11- Confidence Intervals for Univariate Data Math 22 Introductory Statistics.

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