Statistical Inference Chapter 10: Intro to Inference Section 10.1 Estimating with Confidence "How good is your best guess?" "How confident are you in your method?" provides methods for about a from the. Nov 12 2:58 PM Apr 14 8:42 AM What are confidence intervals? General Form of a Confidence Interval According to the rule, will be within SD of the population mean, μ, in of all samples. So, if I'm and I reach out 2 SD away from me on both sides, I'm 95% sure that μ is within my grasp. Confidence Interval for a Population Mean, σ known What we are describing is a range of values that we can be reasonably sure contains the true population mean. This is a. Apr 14 8:49 AM Apr 14 9:04 AM 1
Example: 80% C.I. Confidence level -gives that the method used produces an that covers the ; ex. 90%, 95% -helps determine for computing confidence interval Critical value (z * or t * ) C.L. Tail Area to the right z * - z-value (or t-value) with probability, p, lying to the right under the standard curve (or t-curve) Apr 14 9:13 AM Apr 14 9:15 AM General Case Conditions for Confidence Interval of Population Mean, σ known 1) SRS: data come from an of the population 2) Normality: of is approximately Normal 3) Independence: individual observations are ; or N 10n (when sampling without ) *When the population is normal, this interval is. Otherwise, it is approximately correct for large sample sizes (N > 10n) Apr 14 9:16 AM Apr 14 9:20 AM 2
Facts about sampling distribution of 1) has a distribution ( ) 2) The of this normal sampling distribution is the same as the population mean, μ. 3) The standard deviation of is. What do we want from our confidence interval? Ideally, we want a high level of with a low. MOE = MOE gets smaller when: 1) gets smaller 2) gets smaller 3) gets larger Apr 14 8:46 AM Apr 14 9:19 AM How big of a sample size do I need to maintain confidence and reduce MOE? Choose your sample size based on your desired MOE, m : Solve for n Remember to round up to accomodate for fractional sample sizes. Nov 16 12:12 PM Apr 14 9:26 AM 3
Interpreting Confidence Intervals Data must come from SRS. The confidence interval is not accurate for more complex sampling designs. There is no correct method for inference when data is collected with bias. There is no rescue for badly produced data. Outliers can have a large effect on C.I.'s. Try to correct or remove outliers. If n is small and population is not normal, the true confidence level C. Only when n > 15, is the confidence level not greatly affected by non-normal populations. You must know σ in order to use the methods in this section. m only covers random sampling errors due to chance variation Don't misstate what confidence intervals mean Apr 14 9:33 AM Don't suggest that the parameter varies "There is a 95% chance that the true proportion is between 42.7% and 51.3%." Don't claim that other samples will agree with yours "In 95% of samples of U.S. adults the proportion who think marijuana should be decriminalized will be between 42.7% and 51.3%" 4
Don't forget it's the parameter, not the statistic "I'm 95% confident that ^p is between 42.1% and 61.7%" Don't be certain about the parameter "Between 42.1% and 61.7% of sea fans are infected" Don't claim to know too much "I'm 95% confident that between 42.1% and 61.7% of all the sea fans in the world are infected." Do take responsibility "I'm 95% confident that between 42.1% and 61.7% of all the sea fans in the Las Redes Reef are infected." 5
The 2000 Presidential Election A closely contested presidential election pitted George W. Bush against Al Gore in 2000. A poll taken immediately before the 2000 election showed that 51% of the sample intended to vote for Gore. The polling organization announced that they were 95% confident that the sample result was within +/-2 points of the true percent who favored Gore. a) Explain in plain language what "95% confident" means in this statement. b) The poll showed Gore leading. Yet the polling organization said the election was too close to call. Explain why. Discuss with a neighbor A student reads that a 95% confidence interval for the mean SAT Math score of California high school seniors is 452 to 470. Asked to explain the meaning of this interval, the student says, "95% of California high shcool seniors have SAT Math scores between 452 and 470." Is the student right? Justify your answer. c) On hearing of the poll, a nervous politician asked, "What is the probability that over half the voters prefer Gore?" A statistican replied that this question can't be answered from the poll results, and that it doesn't even make sense to talk about such a probability. Explain why. Mar 30 9:45 AM Mar 30 9:52 AM 6