Hypothesis Tests One Sample Means

A hypothesis test will allow me to decide if the claim is true or not! How can I tell if they really are underweight? A government agency has received numerous complaints that a particular restaurant has been selling underweight hamburgers. The restaurant advertises that it’s patties are “a quarter pound” (4 ounces). A hypothesis test will allow me to decide if the claim is true or not! Take a sample & find x. But how do I know if this x is one that I expect to happen or is it one that is unlikely to happen?

Steps for doing a hypothesis test “Since the p-value < (>) a, I reject (fail to reject) the H0. There is (is not) sufficient evidence to suggest that Ha (in context).” Assumptions Write hypotheses & define parameter Calculate the test statistic & p-value Write a statement in the context of the problem. H0: m = 12 vs Ha: m (<, >, or ≠) 12

Assumptions for t-inference Have an SRS from population (or randomly assigned treatments) s unknown Normal (or approx. normal) distribution Given Large sample size Check graph of data Use only one of these methods to check normality

Formulas: s unknown: m t =

Calculating p-values For t-test statistic – Use tcdf(lb, ub, df) Follow the same guidelines given previously based on the type of test

Draw & shade a curve & calculate the p-value: 1) right-tail test t = 1.6; n = 20 2) two-tail test t = 2.3; n = 25 P-value = .0630 P-value = (.0152)2 = .0304

Example 1: Bottles of a popular cola are supposed to contain 300 mL of cola. There is some variation from bottle to bottle. An inspector, who suspects that the bottler is under-filling, measures the contents of six randomly selected bottles. Is there sufficient evidence that the bottler is under-filling the bottles? Use a = .1 299.4 297.7 298.9 300.2 297 301

What are your hypothesis statements? Is there a key word? SRS? I have an SRS of bottles Normal? How do you know? Since the boxplot is approximately symmetrical with no outliers, the sampling distribution is approximately normally distributed Do you know s? s is unknown What are your hypothesis statements? Is there a key word? H0: m = 300 where m is the true mean amount Ha: m < 300 of cola in bottles p-value =.0880 a = .1 Plug values into formula. Compare your p-value to a & make decision Since p-value < a, I reject the null hypothesis. Write conclusion in context in terms of Ha. There is sufficient evidence to suggest that the true mean cola in the bottles is less than 300 mL.

(See Data in Power Point Notes.) Example 2: The Degree of Reading Power (DRP) is a test of the reading ability of children. Here are DRP scores for a random sample of 44 third-grade students in a suburban district: (See Data in Power Point Notes.) At the a = .1, is there sufficient evidence to suggest that this district’s third graders reading ability is different than the national mean of 34? 40 26 39 14 42 18 25 43 46 27 19 47 19 26 35 34 15 44 40 38 31 46 52 25 35 35 33 29 34 41 49 28 52 47 35 48 22 33 41 51 27 14 54 45

I have an SRS of third-graders Normal? How do you know? Since the sample size is large, the sampling distribution is approximately normally distributed OR Since the histogram is unimodal with no outliers, the sampling distribution is approximately normally distributed Do you know s? What are your hypothesis statements? Is there a key word? s is unknown H0: m = 34 where m is the true mean reading Ha: m ≠ 34 ability of the district’s third-graders Plug values into formula. p-value = tcdf(.6467,1E99,43)=.2606(2)=.5212 Use tcdf to calculate p-value. a = .1

Compare your p-value to a & make decision Conclusion: Since p-value > a, I fail to reject the null hypothesis. There is not sufficient evidence to suggest that the true mean reading ability of the district’s third-graders is different than the national mean of 34. Write conclusion in context in terms of Ha. A type II error – We decide that the true mean reading ability is not different from the national average when it really is different. What type of error could you potentially have made with this decision? State it in context.

What do you notice about the hypothesized mean? What confidence level should you use so that the results match this hypothesis test? 90% Compute the interval. What do you notice about the hypothesized mean? (32.255, 37.927)

Example 3: The Wall Street Journal (January 27, 1994) reported that based on sales in a chain of Midwestern grocery stores, President’s Choice Chocolate Chip Cookies were selling at a mean rate of $1323 per week. Suppose a random sample of 30 weeks in 1995 in the same stores showed that the cookies were selling at the average rate of $1208 with standard deviation of $275. Does this indicate that the sales of the cookies is lower than the earlier figure?

What is the potential error in context? Assume: Have an SRS of weeks Distribution of sales is approximately normal due to large sample size s unknown H0: m = 1323 where m is the true mean cookie sales Ha: m < 1323 per week Since p-value < a of 0.05, I reject the null hypothesis. There is sufficient evidence to suggest that the sales of cookies are lower than the earlier figure. What is the potential error in context? What is a consequence of that error?

Example 3 Continued: President’s Choice Chocolate Chip Cookies were selling at a mean rate of $1323 per week. Suppose a random sample of 30 weeks in 1995 in the same stores showed that the cookies were selling at the average rate of $1208 with standard deviation of $275. Compute a 90% confidence interval for the mean weekly sales rate. CI = ($1122.70, $1293.30) Based on this interval, is the mean weekly sales rate statistically less than the reported $1323?