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Chapter 23 Inferences About Means Sampling Distribution for the Mean Now that we know how to create confidence intervals and test hypotheses about proportions, it’d be nice to be able to do the same for means. Just as we did before, we will base both our confidence interval and our hypothesis test on the sampling distribution model. The Central Limit Theorem told us that the sampling distribution model for means is Normal with mean μ and standard deviation SD y n All we need is a random sample of quantitative data, but the true population standard deviation, σ is unknown, so we use standard error: s SE y n Example A nutrition laboratory tests 61 “reduced sodium” hot dogs, finding that the mean sodium content is 320 mg, with a standard deviation of 37 mg. Find a 95% confidence interval for the mean sodium content of this brand of hot dog. Answer: the mean is between 310.4 and 329.6 mg Student’s t distribution Unfortunately, the shape of the sampling model changes— the model is no longer Normal. So, what is the sampling model? The sampling model found has been known as Student’s t. The Student’s t-models form a whole family of related distributions that depend on a parameter known as degrees of freedom. We often denote degrees of freedom as df, and the model as tdf. As the degrees of freedom increase, the t-models look more and more like the Normal. In fact, the t-model with infinite degrees of freedom is exactly Normal. Assumptions and Conditions Independence Assumption: Randomization Condition 10% Condition Nearly normal condition A Confidence Interval for Means One-sample t-interval for the mean When the conditions are met, we are ready to find the confidence interval for the population mean, μ. The standardized sample mean t y SE y follows a Student’s t-model with n – 1 degrees of freedom. The confidence interval is y t SE y n1 where the standard error of the mean is SE y s n The critical value tn*1 depends on the particular confidence level, C, that you specify and on the number of degrees of freedom, n – 1, which we get from the sample size. A Test for the Mean One-sample t-test for the mean The assumptions and conditions for the one-sample t-test for the mean are the same as for the one-sample t-interval. We test the hypothesis H0: = 0 using the statistic y 0 tn1 SE y s SE y n The standard error of the sample mean is When the conditions are met and the null hypothesis is true, this statistic follows a Student’s t-model with n – 1 df. We use that model to obtain a P-value. Finding t-Values By Hand The Student’s t-model is different 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. Example One of the most important factors of auto safety is the weight of the vehicle. Insurance companies are interested in knowing the average weight of cars currently licensed. They believe it is 3000 pounds. To see if this estimate is correct, the company checked a random sample of 91 cars. For that group the mean weight was 2866 pounds, with a standard deviation of 531.5 pounds. Is this strong evidence that the mean weight of all cars is not 3000 pounds? Homework Assignments Page 609 – 613 Problem # 1, 7, 19, 33, 37, 43 a,