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Inference about a Mean Sampling Distribution of Means : You want to know the mean of a population. But you realize the population is too large to actually compute the true mean. So you observe a sample from the population and use the sample mean as an estimate of the population mean. You realize the sample mean would change with another sample, so you decide to make statistical inference about the population mean based on information in the sample. The population from which you collect data is normally distributed with mean μ and standard deviation σ . Let y denote an observation from the population. Draw sample of size n 12 ,, n yy y K . Compute the sample mean ( ) 1 / n y =++ L n . Then sampling distribution of y is normal with mean µ and standard deviation / y n σσ = . 1

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Sampling Distribution of Means from a Normal Distribution Population distribution of and sampling distribution of y y Black curve: Distribution of the population M e a n = µ Standard deviation = σ Red curve: Sampling distribution of y M e a n = Standard deviation = / n Empirical Rule applied to Sampling Distribution: 95% of the time y is within 2/ n of 2
Confidence Interval for a Mean Equivalent statements from the Empirical Rule: 95% of the time µ is within 2/ n σ of y 95% of the time is in the interval (2 /,2 / yn y σσ −+ ) n This is a 95% Confidence Interval: ( ) 2, 2 yy Example: Egg weight data: y = 65.4, s = 5.17, n=54 We don’t know , so we use s in it’s place. This is ok if n > 30. / 5.17/ 54 .70 sn == 2 / 2(.70) 1.40 2 / 65.4 2(.70) 65.4 1.40 (64.0,66.8) ysn ±= ± 3

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Interpretation of Confidence Interval We are 95% confident that the population mean µ is in the interval (64.0, 66.8) in the following sense: The population mean will be in the interval ( ) 2, 2 yy σ −+ whenever y is within 2/ n of . From the Empirical Rule applied to the sampling distribution of y , we know this happens 95% of the time. 4
Confidence Interval for a Mean Conditions for interval 1.96 / y σ ± n to be exactly valid: 1. Population normal, mean = µ , std. dev. = and 2. 12 ,, n yy y K is a random sample from the population The interval 2/ ys ± n approximately valid if 1. Population approximately normal and 2. n > 30 5

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Test of Hypothesis about Mean Suppose the long term mean for the egg weights is known to be μ =65. The mean of your current sample of 54 egg weights is 65.4. Is there statistical evidence in the egg weight data that the population mean, μ , has changed? That is, is there statistical evidence that the sample mean 65.4 differs significantly from the hypothetical population mean of 65? The answer to this question is “No,” because the hypothetical mean 65 is contained in the 95% confidence interval: 65.4 – 2(.70) < 65 < 65.4 + 2(.70) Equivalently: |65.4 – 65|/.70 < 2. This is basically an example of a Test of Hypothesis . You are making a computation to check if the absolute difference between the observed mean and hypothetical mean , |65.4 – 65|, is greater than two standard errors of the mean, .70. The computation |65.4 – 65|/.70 is the value of a test statistic .
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