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Lecture_14 (short) - Class 14 Rafael Mendoza-Arriaga Rafael...

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Class 14 Rafael Mendoza-Arriaga Rafael Mendoza McCombs Elementary Business Statistics – Class 14
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Confidence Intervals when the population standard deviation , σ , is unknown Rafael Mendoza McCombs Elementary Business Statistics – Class 14
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Inference for the Mean of a Population, σ unknown Inference when σ is known : Tests and confidence intervals for the population mean μ are based on the sample mean ¯ x of an SRS. Confidence Intervals : ¯ x ± z σ n In practice, we usually do not know σ (population standard deviation). Substitute sample standard deviation s for population standard deviation σ . s = 1 n 1 Σ n i =1 ( x i ¯ x ) 2 Rafael Mendoza McCombs Elementary Business Statistics – Class 14
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t distributions If σ is known , we calculate z = ¯ x μ σ n This statistic has the standard Normal distribution , N (0 , 1) , and we use NORMINV function to calculate z . If σ is unknown , we calculate t = ¯ x μ s n This statistic has the t distribution with n 1 degrees of freedom , and we use TINV function. Assumption: Population distribution is Normal. The confidence intervals produced using t-distributions are robust to violations of normality in the population especially if the sample size is large. Rafael Mendoza McCombs Elementary Business Statistics – Class 14
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Properties of the t distributions The density curve of the t -distribution is symmetric and bell-shaped. The curve is more spread out than the standard normal curve (due to substituting s for σ ). k = n 1 is called the degrees of freedom. There is a di ff erent t ( k ) distribution for each integer, k . Normal vs. t -distribution Rafael Mendoza McCombs Elementary Business Statistics – Class 14
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Properties of the t distributions - continued Q.2 As the degrees of freedom (k) increase, the t distribution moves . . . . . . the Standard Normal distribution. (a) farther away from (b) closer to Normal vs. t -distribution Rafael Mendoza McCombs Elementary Business Statistics – Class 14
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Properties of the t distributions - continued Q.2 As the degrees of freedom (k) increase, the t distribution moves . . . . . . the Standard Normal distribution.
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