2003 Free Response Question #2 A simple random sample of adults living in a suburb of a large city was selected. The age and annual income of each adult in the sample were recorded. The resulting data are summarized in the table below. What is the probability that a person chosen at random from those in the sample will be in the age category? What is the probability that a person chosen at random from those in this sample whose incomes are over $50,000 will be in the age category? Show your work. Based on your answers to parts (a) and (b), is annual income independent of age category for those in this sample? Explain. Age$25,000- $35,000 $35,000- $50,000 Over $50,000 Total Over Total

12.1:Tests for the Mean of a Population

T-Statistic There is a different t- distribution for each sample size We specify a particular t distribution by giving its degrees of freedom = n-1. Notation: t(k) = t- distribution with k degrees of freedom.

Facts about t-distributions Density curve is similar in shape to standard normal curve. Spread is a bit greater than that of the standard normal dist. As d.o.f. k increase, the t(k) density curve approaches N(0,1) more closely.

T-distribution critical values. Find the p-value from Table C for a t- distribution with n = 20 and t = Ha: mean > 5. Find the p-value from Table C for a t- distribution with n = 37 and t = Ha: two-sided.

One-sample T Procedures Draw an SRS of size n from a population having unknown mean. A level C confidence interval for is: To test the hypothesis based on an SRS of size n, compute the one-sample t statistic

Cola Problem Cola makers test new recipes for loss of sweetness during storage. Trained tasters selected from an SRS rate the sweetness before and after storage. Here are the sweetness losses found by 10 tasters for one new cola recipe: Are these good data evidence that the cola lost sweetness? (Sweetness loss = sweetness before- sweetness after storage)

Matched Pairs t-procedures Recall: In a matched pairs design, subjects are matched in pairs and each treatment is given to one subject in each pair, or we can perform before-and-after observations on the same subjects. Taste-test example is a matched pairs study in which the same 10 tasters rated before-and-after sweetness. To compare the responses to the two treatments in a matched pairs design, apply the one-sample t procedures to the observed differences. The parameter in a matched pairs t procedure is the mean difference in the responses to the 2 treatments within matched pairs of subjects in the entire population.

Floral Scents and Learning We hear that listening to Mozart improves students’ performance on tests. Perhaps pleasant odors have a similar effect. To test this idea, 21 subjects chosen from an SRS worked a pencil- and-paper maze while wearing a mask; the mask was either unscented or had a floral scent. The response variable is their average time on 3 trials. Each subjected worked the maze with both masks, in random order. The following table gives the subjects’ average times with both masks.

Analysis: Subtract the scented time from the unscented time for each subject. The 21 differences form a single sample. First subject: 7.37 s slower wearing the scented mask, so difference was -. Shorter times = better performance, therefore + differences = student did better.

The developer of a new filter for filter-tipped cigarettes claims that it leaves less nicotine in the smoke than does the current filter. Because cigarette brands differ in a number of ways, he tests each filter on one cigarette of each of nine brands and records the difference in nicotine content (current filter – new filter). The mean difference is x-bar = 1.32 milligrams (mg), and the standard deviation of the differences is s = 2.35 mg. 1. What is the parameter being measured? 2. State Ho and Ha for this study in both symbols and words. 3. Describe the population of interest for which inference is being performed. 4. What conditions are required to carry out the significance test? Discuss the validity of each. 5. Determine the test statistic and the P-value. Show your work. 6. What do you conclude? 7. Construct a 90% confidence interval for the mean amount of additional nicotine removed by the new filter is. Interpret the interval.