Transcript Lecture16
Review • Confidence Intervals • Sample Size Estimator and Point Estimate An estimator is a “sample statistic” (such as the sample mean, or sample standard deviation) used to approximate a population parameter. A Point Estimate is a single value or point used to approximate a population parameter. A point estimator may be biased or unbiased. Central Limit Theorem Take ANY random variable X and compute m and s for this variable. If samples of size n are randomly selected from the population, then: 1) For large n, the distribution of the sample means, x will be approximately a normal distribution, 2) The mean of the sample means will be the population mean m and 3) The standard deviation of the sample means will be s n Confidence Intervals The Confidence Interval is expressed as: xE x z 2s x x z E is called the margin of error. For samples of size > 30, s x z 2 n s 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 2s n E …round up 2 Practice Problems • #7.11 page 329 • #7.19 page 331 • #7.21 page 331 Small Samples What happens if and n is small (n < 30)? Our formulas from the last section no longer apply. Small Samples What happens if and 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) s 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 s is unknown so we use s (the sample standard deviation) as a point estimate of s. 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, t0.025 = 2.145 Confidence Interval for the mean when s 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 tn1, 2 s n The margin of error E, is in this case E tn 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. 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. Therefore at the (1-a)*100 % level of confidence, the Error estimate of the population proportion is E z 2 pˆ qˆ n At the (1-a)*100 % level of confidence, the confidence interval for the population proportion is : pˆ E p pˆ E Determining Sample Size In calculating the confidence interval for the population proportion we used E z 2 pˆ qˆ n Perhaps we might be interested in knowing how large a sample we should use if we are willing to accept a margin of error E with a degree of confidence of 1-. Determining Sample Size If we already have an idea of the proportion (either through a pilot study, or previous results) one can use z / 2 n 2 E pˆ qˆ 2 If we have no idea of what the proportion is then we use z / 2 0.25 n 2 E 2 In Class Exercises • #7.31, 7.36, 7.41 Pages 341, 342 • #7.50, 7.57 on page 349, 350 • #7.75, 7.78 Pages 356, 357 Shortcut for finding z/2 • Recall that as n the Student’s Tdistribution approaches the normal distribution. • Look at T-table inside front cover, the last row represents the values of tn1, /2, as n becomes large which is essentially z/2. • Therefore, for some common values of we are able to find z/2 quite quickly. • z0.025 1.960, z0.10 1.282 Homework • Review Chapter 7.3-7.5 • Read Chapters 8.1-8.3 • Quiz on Tuesday: Chapter 5 23