Transcript Chapter21
What is a Confidence Interval? Sampling Distribution of the Sample Mean The statistic x estimates the population mean We want the sampling distribution to be centered at the value of the parameter and to have little variation Facts about sampling distribution of x x x n Notice that as n increases the sample to sample variability in x decreases If our sample comes from a normal distribution with mean and standard deviation then Z x has a standard normal distribution n Central Limit Theorem If we sample from a population with mean and standard deviation then x is approximately standard Z normal for large n n If n = 30 or larger, the central limit theorem will apply in almost all cases Example A population of soft drink cans has amounts of liquid following a normal distribution with 12 and 0.2 oz. What is the probability that a single can is between 11.9 and 12.1 oz. What is the probability that x is between 11.9 and 12.1 for n = 16 cans Example A population of trees have heights with a mean of 110 feet and a standard deviation of 20 feet A sample of 100 trees is selected Find x Find x Find P( x 108 feet) What about P( X 108) Sampling Distribution of the Sample Proportion Population Proportion # in population with characteristic p # in population Sample Proportion # in sample with characteristic pˆ n p̂ is a point estimate of p pˆ p pˆ p1 p n If we sample from a population with a proportion of p, then pˆ p is approximately standard Z p 1 p normal for large n n Example Suppose the president’s approval rating is 56% and we look at samples of size 100 Find p̂ Find p̂ Example A survey of 120 registered voters yields 60 who plan to vote for the republican candidate p = proportion of all voters who plan to vote for the republican candidate Calculate the point estimate for p Calculate the margin of error Can we calculate the variance of the sampling distribution Do you see where the margin of error comes from? Estimating Proportions with Confidence The population proportion p is an unknown parameter We wish to estimate p based on a sample p̂ is a statistic which estimates p We call p̂ a point estimate because its value is a point on the real line Unfortunately, for a continuous distribution the ˆ p is 0 because there is zero probability that p probability for any point Statisticians prefer interval estimates pˆ E E (error tolerance) depends on the sample size, how certain we want to be, and the amount of variability in the data pˆ (1 pˆ ) EZ n The degree of certainty (probability that we are correct) is known as the Level of Confidence (level of significance) is one minus the level of confidence Notice that increasing the level of confidence, decreases the (level of significance) probability of being incorrect and increases the width of the interval All confidence intervals are two-sided probabilities with a total area of Common Values for z for 90% confidence z for 95% confidence 2 1.645 2 z 1.96 2 for 99% confidence z 2.576 2 Example A survey of 1,200 registered voters yields 540 who plan to vote for the democratic candidate Find a 95% confidence interval for p We are 95% confident that the true proportion of voters who will vote for the democratic candidate is between 42.2% and 47.8%