Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 23 Inference About Means

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Gosset’s t William S. Gosset, an employee of the Guinness Brewery in Dublin, Ireland, worked long and hard to find out what the sampling model was. The sampling model that Gosset found has been known as ________________. The Student’s t-models form a whole ________ of related distributions that depend on a parameter known as __________________________. We often denote degrees of freedom as ___, and the model as ____.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What Does This Mean for Means? A practical sampling distribution model for means When the conditions are met, the standardized sample mean follows a Student’s t-model with ______ degrees of freedom. We estimate the standard error with

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What Does This Mean for Means? (cont.) Student’s t-models are __________, _________, and ______________, just like the Normal. But t-models with only a few degrees of freedom have ___________________ than the Normal.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What Does This Mean for Means? (cont.) As the degrees of freedom ____________, the t-models look more and more like the Normal. In fact, the t-model with infinite degrees of freedom is exactly Normal.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Finding t-Values By Hand The Student’s t-model is ____________ for each value of degrees of freedom. Because of this, Statistics books usually have one table of t-model critical values for selected confidence levels.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Finding t-Values By Hand (cont.) Alternatively, we could use technology to find t critical values for any number of degrees of freedom and any confidence level you need. Use your calculator to find the following. a)P(t > 1.645) df = 12b) P(t < 0.53) df = 23 c)t-values for 90% CId) t-values for 95% CI df = 6 df = 35

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Assumptions and Conditions Independence Assumption: Randomization Condition: The data arise from a _________________ or suitably _________ _________________. Randomly sampled data (particularly from an SRS) are ideal. 10% Condition: When a sample is drawn __________________, the sample should be no more than 10% of the population.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Assumptions and Conditions (cont.) Normal Population Assumption: We can never be certain that the data are from a population that follows a Normal model, but we can check the Nearly Normal Condition: The data come from a distribution that is ____________ and ________________. Check this condition by making _____________ or __________________________.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Assumptions and Conditions (cont.) Nearly Normal Condition: The smaller the sample size (___________), the more closely the data should follow a Normal model. For moderate sample sizes (_________________), the t works well as long as the data are unimodal and reasonably symmetric. For larger sample sizes, the t methods are safe to use even if the data are skewed.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide A nutrition laboratory tests 40 “reduced sodium” hot dogs, finding that the mean sodium content is 310 mg, with a standard deviation of 36 mg. a) Find a 95% confidence interval for the mean sodium content of this brand of hot dog. b) What assumptions have you made in this inference? Are the appropriate conditions satisfied? c) Explain clearly what your interval means.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide More Cautions About Interpreting Confidence Intervals Remember that interpretation of your confidence interval is key. What NOT to say: “90% of all the vehicles on Triphammer Road drive at a speed between 29.5 and 32.5 mph.” The confidence interval is about the _______ not the _______________________. “We are 90% confident that a randomly selected vehicle will have a speed between 29.5 and 32.5 mph.” Again, the confidence interval is about the ________ not the _______________________.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide More Cautions About Interpreting Confidence Intervals (cont.) What NOT to say: “The mean speed of the vehicles is 31.0 mph 90% of the time.” The true mean does not ______ —it’s the confidence interval that would be different had we gotten a different sample. “90% of all samples will have mean speeds between 29.5 and 32.5 mph.” The interval we calculate does not __________________ for every other interval—it is no more (or less) likely to be correct than any other interval.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Assignment Pg. 541 #1, 5, 8, 9, 12, 14 Slide

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 23 Inference About Means (2)

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide One-Sample t-test for the Mean The conditions for the one-sample t-test for the mean are the same as for the one-sample t-interval. We test the hypothesis H 0 :  =  0 using the statistic The standard error of the sample mean is When the conditions are met and the null hypothesis is true, this statistic follows a ________________________ ___________. We use that model to obtain a _________.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Ex. 2In 1960 census results indicated that the age at which American men first married had a mean of 23.3 years. It is widely suspected that young people today are waiting to get married. A random sample of 40 men married at and average age of 24.2 years with a standard deviation of 5.3 years. a) Are the assumptions satisfied? b) Test this claim. c) What does the p-value mean in context? Slide

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Ex. 3 Psychology experiments sometimes involve testing the ability of rats to navigate mazes. The mazes are classified according to difficulty, as measured by the mean length of time it takes rats to find the food at the end. One researcher needs a maze that will take rats an average of about one minute to solve. He test one maze on several rats, collecting the data that follow. a)Plot the data. Do you think the conditions for inference are satisfied? Explain. b)Test the hypothesis that the mean completion time for this maze is 60 seconds. What is your conclusion? c)Eliminate the outlier, and test the hypothesis again. What is your conclusion? d)Do you think this maze meets the “one-minute average” requirement? Explain. Slide Time (sec)

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Sample Size To find the sample size needed for a particular confidence level with a particular margin of error (ME), solve this equation for n: The problem with using the equation above is that we don’t know most of the values. We can overcome this: We can use s from a small pilot study. We can use z* in place of the necessary t value.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Ex. 4 Suppose we want a margin of error of 3 minutes and we think the standard deviation is 5 minutes based on a pilot study. a)Using α = 0.05 for a one tail test, what is the sample size we should use initially? b)Using degrees of freedom based on this sample size recalculate, the sample size again. Slide

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Sample Size (cont.) Sample size calculations are _______ exact. The margin of error you find after collecting the data _____________________the one you used to find n. The sample size formula depends on quantities you won’t have until you ___________________, but using it is an important first step. Before you collect data, it’s always a good idea to know whether the sample size is ____________ to give you a good chance of being able to tell you what you want to know.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What have we learned? Statistical inference for means relies on the same concepts as for proportions—only the _________ and the _______ have changed. The reasoning of inference, the need to verify that the appropriate ___________________, and the proper interpretation of _________________ and __________ all remain the same regardless of whether we are investigating means or proportions.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Assignment Pg. 544 #21, 23, 26, 28, 30, 33, 35 Slide