Hypothesis Tests for Notes: Page 194 Hypothesis Tests for One Sample Means Notes: Page 194

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). How can I tell if they really are underweight? Take a sample & find x. expect unlikely But how do I know if this x is one that I expect to happen or is it one that is unlikely to happen? A hypothesis test will allow me to decide if the claim is true or not!

Steps for doing a hypothesis test 1)Given 2)Assumptions 3)Write hypotheses & define parameter 4)Calculate the test statistic & p-value 5)Write a statement in the context of the problem. H 0 :  = 12 vs H a :  (, or ≠) 12 “Since the p-value ) , I reject (fail to reject) the H 0. There is (is not) sufficient evidence to suggest that H a (in context).”

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

Formulas:  unknown: t = 

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  = Enter the data in L1 STAT, EDIT, L1 Enter the data in L1 STAT, EDIT, L1

I have an SRS of bottles Since the boxplot is approximately symmetrical with no outliers, the sampling distribution is approximately normally distributed  is unknown SRS? p-value =.0880  =.1 Normal? How do you know? Is it given? No Large sample size? No So, create a graph with the data using STAT PLOT, boxplot with outliers. H 0 :  = 300where  is the true mean amount H a :  < 300 of cola in bottles What are your hypothesis statements? Is there a key word? Plug values into formula. Do you know  ? Since p-value < , I reject the null hypothesis. There is sufficient evidence to suggest that the true mean cola in the bottles is less than 300 mL. Compare your p-value to  & make decision Write conclusion in context in terms of H a. STAT, CALC 1-Var Stats for mean and s Be sure to define the variable,  and use context.

Draw & shade a curve & calculate the p-value: 1) right-tail test t = 1.6; n = 20 2) two-tail testt = 2.3; n = 25 P-value =.0630 P-value = (.0152)2 = nd VARS tcdf(1.6, 10^99, 19) 2 nd VARS tcdf(2.3, 10^99, 24)

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: (data on note page) At the  =.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 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  is unknown SRS? p-value = tcdf(.6467,10^99,43)=.2606(2)=.5212  =.1 Do you know  ? Normal? How do you know? Use tcdf to calculate p-value. H 0 :  = 34where  is the true mean reading H a :  ≠ 34 ability of the district’s third-graders What are your hypothesis statements? Is there a key word? Plug values into formula.

A type II error – We decide that the true mean reading ability is not different from the national average when it really is different. Conclusion: Since p-value > , 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 H a. Compare your p-value to  & make decision What type of error could you 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. What do you notice about the hypothesized mean? (32.255, )

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  unknown H 0 :  = 1323 where  is the true mean cookie sales H a :  < 1323 per week Since p-value <  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 $275. Compute a 90% confidence interval for the mean weekly sales rate. CI = ($ , $ ) Based on this interval, is the mean weekly sales rate statistically less than the reported $1323?

Homework: Page Note: Tomorrow: Classwork P 193, Thursday: Quiz over Unit 12 Friday: Matched Pairs Testing Method