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381 Inferences About the Mean-II (Small Samples) QSCI 381 – Lecture 22 (Larson and Farber, Sect 6.2) Small Samples and the t-Distribution 381 In many / most real situations, the population standard deviation will be unknown and the sample size will be smaller than 30. The methods of yesterday’s lecture cannot therefore be applied. If the sampling distribution is normally distributed (or approximately so), the sample mean, x , is t-distributed. Side Note – Stout and Statistics. 381 The t-distribution is often referred to as “Student’s” t-distribution. “Student” (W.S. Gossett) was an employee of the Guinness Brewing Company and was prohibited from publishing papers (after someone published trade secrets) so he published under the pseudonym Student for his publications to avoid detection of his publications by his employer! Properties of the t-distribution-I 381 1. 2. 3. 4. The t-distribution is bell-shaped and symmetric. The total area under the curve is 1. The mean, median and mode are equal to 0. The t-distribution is a family of curves (rather than one as was the case for the normal distribution). Each element of the family is determined by a parameter called the degrees of freedom (n-1 for the case of the sample mean) x t s/ n df=5 Density 381 Properties of the t-distribution-II df=2 -3.5 -1.5 0.5 2.5 Critical Values and the t-Distribution 381 Find the values of t such that there is a probability of 0.025 beyond t for d.f. = 1, 5, 10, 20, 100. I used the EXCEL Function:TINV(p,d.f) d.f. Critical t 1 12.71 5 2.57 10 2.23 20 2.09 100 1.98 Confidence Interval for the Population Mean-I 381 1. 2. 3. 4. 5. Find the sample statistics n, x , and d.f (=n-1). Calculate the sample standard deviation. Find the level of t that corresponds to the confidence level (this depends on the d.f). Find the maximum error of estimate E: s E tc n Construct a confidence interval using the formula: x E x E Confidence Interval for the Population Mean-II 381 Find a 95% confidence interval for the mean when x 50; s 10 for d.f.s of 1, 5, and 10 (i.e. n=2, 6 and 11). d.f. tc E Confidence interval 1 12.71 89.84 -39.85; 139.85 5 2.57 10.49 39.51; 60.49 10 2.23 6.72 43.28; 56.72 Is n30? 381 Use the normal distribution with Yes E zc No If is unknown use s instead Is the population normally, or approximately normally distributed? Yes Is known? Yes No Use t-distribution with E tc n s n No You cannot use the normal distribution or the t-distribution Use normal distribution with E zc n Examples 381 You take: 24 samples, the data are normally distributed, is known. 14 samples, the data are normally distributed, is unknown. 34 samples, the data are not normally distributed, is unknown. 12 samples; the data are not normally distributed, is unknown. Sample Sizes-I 381 The width of a confidence interval can be reduced by increasing the size of the sample on which the mean is based. Given a c-confidence level and a maximum error of estimate E, the minimum sample size needed to estimate is: 2 z n c E In EXCEL: =(STDEV(D2:D51)*NORMINV(0.995,0,1)/Eval)^ 2 Sample Sizes-II 381 1. 2. Find the sample size so that we are 99% confident that the actual mean density is within 1 unit of the sample mean density if the standard deviation is 10.66. The level of confidence is 99% so the critical value of z is 2.575. The needed sample size is: 2 2.575 x10.66 n 754 (rounded up) 1