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,,nyy yK. Compute the sample mean ()1/nyy y=++L n. Then sampling distribution of y is normal with mean µ and standard deviation /ynσσ=. 1Sampling Distribution of Means from a Normal Distribution Population distribution of and sampling distribution of yy Black curve: Distribution of the population Mean = µ Standard deviation = σ Red curve: Sampling distribution ofy Mean = µ Standard deviation = / nσ Empirical Rule applied to Sampling Distribution: 95% of the time yis within 2/nσof µ 2Confidence Interval for a Mean Equivalent statements from the Empirical Rule: 95% of the time µis within 2/nσof y 95% of the time µis in the interval (2/,2/ynyσσ−+)n This is a 95% Confidence Interval: ()2, 2yyyyσσ−+ Example: Egg weight data: y= 65.4, s = 5.17, n=54 We don’t knowσ, so we use s in it’s place. This is ok if n > 30. / 5.17 / 54 .70sn== 2 / 2(.70) 1.40sn== 2 / 65.4 2(.70) 65.4 1.40 (64.0,66.8)ysn±=±=±→ 3Interpretation of Confidence Interval We are 95% confident that the population mean µis in the interval (64.0, 66.8) in the following sense: The population meanµwill be in the interval ()2, 2yyyyσσ−+ whenever yis within 2/nσofµ. From the Empirical Rule applied to the sampling distribution ofy, we know this happens 95% of the time. 4Confidence Interval for a Mean Conditions for interval 1.96 /yσ± n to be exactly valid: 1. Population normal, mean = µ, std. dev. = σ and 2. 12,,nyy yK is a random sample from the population The interval 2/ys± n approximately valid if 1. Population approximately normal and 2. n > 30 5Test 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. 6Setup for Test of Hypothesis about a Mean You want to test if there is evidence in your data that the population mean, µ, is different from a hypothesized value, µ0. Null Hypothesis: 00:Hµµ= Alternative Hypothesis: 00:Hµµ≠ Test Statistic: 0()/yzyµσ=− Rejection Rule: Reject H0 if z < -2 or z > 2 (equivalently, |z| >2). (The value 2 is a rounding of 1.96) 7Margin of Error Sampling distribution of y and its application: 95% of the time y will be within 2/nσ of µ. Stated another way: 95% of the time, the absolute difference between yand µ will be less than2/nσ. The probable “margin of error” of yas an estimate of µ is 2/nσ. 8Margin of Error, “MOE” Margin of Error: MOE = 2/nσ=2yσ Relationship to confidence intervals: 95% of the time, µ will be between yMOE−and yMOE+ Relationship to tests of hypothesis: Reject the hypothesis that 0µis the true mean if 0||yµ− >MOE. 9Review: Inference about a population mean Confidence interval: ()2, 2yyyyσσ−+ Example: 95% C. I. for mean egg weight y= 65.4, s = 5.17, n = 54 (use s in place of σ because n > 30) MOE = 2ys = 1.40 263.9yys−= 9 266.8yys 1+= 95% C. I. (64.0, 66.8) ⇒ 10Review: Inference about a population mean Statistical test of hypothesis aboutµ: 00: :aHH0µµµ=≠µ Test statistic: 00/yyyznµµσσ−−== Reject H0 if |z| > 2. Example: Test whether mean is ≠ 65 MOE = 1.40 0||.4yMOEµ−=< Equivalently, 065.4 65 .4.568 2.704 .704/yznµσ−−== ==< Conclude no statistical evidence that µ differs from 0µ= 65. 11Type I and Type II errors of statistical test Type I error : Reject true H0 Type II error: Not reject false H0 Analogy of statistical test to criminal trial Test of Hypothesis State of Null Hypothesis Test Outcome H0 True H0 False Reject Type I error OK Not Reject OK Type II error Criminal Trial State of Defendant Test Outcome Defendant Innocent Defendant Guilty Reject Type I error OK Not Reject OK Type II error 12Error Probabilities in Hypothesis Testing Probability of Type I error α= .05 = P(Reject true H0) = P(Type I error) Probability of Type II error β = P(Not reject false H0) = P(Type II error) (depends on µ) β (at µ = µ0 ): 0/2||/aPz znαµµβσ−⎛⎞=<−⎜⎟⎝⎠ 13Computing Type II Error Probabilities Ex. Type II error of egg weight test 0: 65 : 65aHHµµ=≠ β (at µ =64) = ()/2|64 65| 11.96 1.96 1.42.704/Pz z Pz Pznασ−⎛⎞⎛⎞<− = <− =<−⎜⎟⎜⎟⎝⎠⎝⎠ = P(z < .54) = .7054 β (at µ =63) = ()/2|63 65| 21.96 1.96 2.84.704/Pz z Pz Pznασ−⎛⎞⎛⎞<− = <− = <−⎜⎟⎜⎟⎝⎠⎝⎠ = P(z < -.88) = .1895 β (at µ =62) = ()/2|62 65| 31.96 1.96 4.26.704/Pz z Pz Pznασ−⎛⎞⎛⎞<− = <− = <−⎜⎟⎜⎟⎝⎠⎝⎠ = P(z < -2.30) = .0108 β (at µ =61) = ()/2|61 65| 41.96 1.96 5.68.704/Pz z Pz Pznασ−⎛⎞⎛⎞<− = <− = <−⎜⎟⎜⎟⎝⎠⎝⎠ = P(z < -3.72) = .0001 14Power of Statistical Test Power = P(Reject false H0) = 1- β Example: Power of egg weight test Power (at µ =61) = 1- β (at µ = 61) = 1-.0001 = .9999 Power (at µ =62) = 1- β (at µ = 62) = 1-.0108 = .9892 Power (at µ =63) = 1- β (at µ =
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