Hypothesis Tests One Sample Means How can I

Hypothesis Tests One Sample Means

How can I tell ifhas they really A government agency are underweight? received numerous complaints A hypothesis test that will a particular restaurant has allow me to been selling underweight decide if the claim is true or not! hamburgers. restaurant Take The a sample & find x. advertises that it’s patties are “a quarter pound” (4 ounces). 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 1) Assumptions (fail to reject) the H 0. There is (is not) sufficient evidence to suggest thathypotheses Ha (in context). ” 2) Write & define parameter H 0: m = 12 vs Ha: m (<, >, or ≠) 12 3) Calculate the test statistic & p-value 4) Write a statement in the context of the problem.

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: t= m

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 P-value =. 0630 2) two-tail test t = 2. 3; n = 25 P-value = (. 0152)2 =. 0304

Example 1: Bottles of a popular cola are supposed to contain 300 m. L 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

• I have an SRS of bottles SRS? Normal? • Since the boxplot is approximately symmetrical with no outliers, the sampling distribution is approximately. How do you know? normally distributed Do you • s is unknown know s? What are your H 0: m = 300 where m is the true mean amount hypothesis statements? Is Ha: m < 300 of cola in bottles there a key word? p-value =. 0880 a =. 1 Plug p-values to Compare your into decision formula. a & make Since p-value < a, I reject the null hypothesis. Writethat conclusion in There is sufficient evidence to suggest the true context in terms of Ha. mean cola in the bottles is less than 300 m. L.

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?

• I have an SRS of third-graders SRS? Normal? • Since the sample size is large, the sampling distribution is How do you approximately normally distributed OR know? Do you • Since the histogram is unimodal withs? no outliers, the know What are your sampling distribution is approximately normally distributed hypothesis • s is unknown statements? Is a key word? H 0: m = 34 where m is the true mean there reading Ha: m ≠ 34 ability of the district’s third-graders Plug values into formula. p-value = tcdf(. 6467, 1 E 99, 43)=. 2606(2)=. 5212 Use tcdf to calculate p-value. a =. 1

Compare your p-value to a & make decision Since p-value > a, I fail to reject the null hypothesis. Conclusion: 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 What type of error could you average when it really is different. potentially have made with this decision? State it in context.

What confidence level should you use so that the results match this hypothesis test? 90% Compute the interval. (32. 255, 37. 927) What do you notice about the hypothesized mean?

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?

Assume: • Have an SRS of weeks • Distribution of sales is approximately normal due to large sample size • s unknown H 0: m = 1323 where m is the true mean cookie sales error in context? Ha: m < 1323 What is the per potential week What is a consequence of that error? 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.

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?