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Review • Confidence Intervals • Sample Size Confidence Intervals The Confidence Interval is expressed as: xE x z 2 x x z E is called the margin of error. For samples of size > 30, s x z 2 n 2 n Sample Size The sample size needed to estimate m so as to be (1-)*100 % confident that the sample mean does not differ from m more than E is: z 2 n E …round up 2 Small Samples What happens if n is small (n < 30)? Our formulas from the last section no longer apply. Small Samples What happens if n is small (n < 30)? Our formulas from the last section no longer apply. There are two main issues that arise for small samples: 1) no longer can be approximated by s 2) The CLT no longer holds. That is the distribution of the sampling means is not necessarily normal. t- distribution If we have a small sample (n < 30) and wish to construct a confidence interval for the mean we can use a t-statistic, provided the sample is drawn from a normally distributed population. t-distributions is unknown so we use s (the sample standard deviation) as a point estimate of . We convert the nonstandard t-distributed problem to a standard t-distributed problem through the use of the standard t-score xm t s n t-distributions • Mean 0 • Symmetric and bell-shaped • Shape depends upon the degrees of freedom, which is one less than the sample size. df = n-1 • Lower in center, higher tails than normal. • See Table inside front cover in text Example In n=15 and after some calculation /2=0.025, we use the table and n-1 = 14 degrees of freedom to deduce t0.025 = 2.145 Confidence Interval for the mean when is unknown and n is small The (1- )*100% confidence interval for the population mean m is x tn 1, 2 s n m x tn 1, 2 s n The margin of error E, is in this case E t n 1, 2 s n N.B. The sample is assumed to be drawn from a normal population. Confidence Intervals for a small sample population mean The Confidence Interval is expressed as: xE x t 2 s n The degrees of freedom is n-1. xm t s/ n Example The following are the heat producing capabilities of coal from a particular mine (in millions of calories per ton) 8,500 8,330 8,480 7,960 8,030 Construct a 99% confidence interval for the true mean heat capacity. Solution: sample mean is 8260.0 sample Std. Dev. is 251.9 degrees of freedom = 4 = 0.01 7741.4 m 8778.6 Confidence intervals for a population proportion The objective of many surveys is to determine the proportion, p, of the population that possess a particular attribute.. Example: Determine the fraction of Canadians who support gun control. If the size of the population is N, and X people have this attribute, then as we already know, p X N is the population proportion. Confidence intervals for a population proportion If the size of the population is N, and X people have this attribute, then as we already know, p X N is the population proportion. The idea here is to take a sample of size n, and count how many items in the sample have this attribute, call it x. Calculate the sample proportion, pˆ x n . We would like to use the sample proportion as an estimate for the population proportion. E z 2 pˆ qˆ n The confidence interval for the population proportion is : pˆ E p pˆ E