EQ: When is a result statistically significant? Hypothesis Testing EQ: When is a result statistically significant?

Goal of HT To assess the evidence provided by data about some claim concerning a population

Recall Parameter-define parameter Hypotheses- define the hypotheses (null and alternative) using the correct parameter. Assumptions- SRS? Normality (and independence)? Sigma known? Declare α Test Statistics- Determine which one to use and calculate it P-value- calculate the p-value and decide to reject or fail to reject Interpret- make an interpretation about significance using context

Example 1 In Illinois, a random sample of 85 eighth grade students has a mean score of 285. The standard deviation for all students who take the national mathematics assessment test is 30. This test result prompts a state school administrator to declare that the mean score for the state’s eight graders on the examination is more than 275. Is there evidence to support the administrator’s claim? (Use 95% confidence level).

Assumptions? SRS-yes, normal-yes (CLT), Indep-yes, Sigma known Example 1: Ctd µ is the mean score for eighth grade students on the national mathematics assessment test Assumptions? SRS-yes, normal-yes (CLT), Indep-yes, Sigma known Test Statistic-we use z-score Thus we reject the null hypothesis There is evidence to support the administrator’s claim that the mean test scores are greater than 275.

BUT WAIT!!!! Will this work for ALL cases?????? What about if we don’t know the population standard deviation???????

Example 2 Home Depot brand light bulbs state on the package “Average Life 1000Hr.” A class believe this number to be high and tested 60 randomly selected bulbs. For these 60 bulbs, the average bulb life was 970 hours and the standard deviation was 120 hours. Is there enough evidence to reject home depots claim? Set up Parameter and Hypotheses: µ=average life of Home Depot brand bulbs α= 0.05

Test Statistic We determine this by the parameter and conditions Since we don’t know sigma what would we use in place of it???? **** Remember use z if sigma is known To calculate the p-value we use the t-distribution (it’s tcdf in the calculator). Don’t forget you need degrees of freedom to calculate it.

Problem 2 continued….Test Statistic Based on the data, we found a test statistic of -1.936. How likely is it to get that value? Make a decision to reject/fail to reject based on p-values P-value Remember: This means there is a 2.88% chance of getting a test statistic less than or equal to the one we got if the actual life of the bulbs is 1000 hours. Decision: Reject the null since .0288 < .05 It appears that Home Depot’s claim is false, the mean life of the light bulbs is not 1000 hours

Example 3 In an advertisement, a pizza shop claims that its mean delivery time is less than 30 minutes. A random sample of 36 delivery times has a sample mean of 28.5 minutes and a standard deviation of 3.5 minutes. Is there enough evidence to support the claim?

2 Tail test: Example 4 µ= average number of sales per day A manager for a company reports an average of 150 sales per day. His boss suspects that this averages is not accurate. He selects 35 days and determines the number of sales each day. The sample mean is 143 daily sales with a standard deviation of 15 sales. At the .01 level, is there evidence to doubt the mangers claim? µ= average number of sales per day ****Recall-When the alternative is not equal, it is a 2 tail test Conditions: SRS, Normal (n>30) Since .0092<.01, reject the null The average number of sales does not appear to be 150. There is evidence to support the boss’ claim.

Tests for Proportions Test Statistic For 1 sample proportions Conditions SRS Population > 10n np and n(1-p) > 10 For confidence intervals use p-hat, not p! To find the p-value use normcdf.

1. Set up Hypotheses: 2. Test Statistic 1 proportion z-test In the book, Life in America’s Small Cities, GS Thomas reports that in 1990, 22.1% of all 16-19 year olds in Key West, Florida were high school dropouts. In 1995, a random sample of 193 people in this Key West age group showed 32 were dropouts. Does this indicate the proportion of high school dropouts in Key West is less than 22.1% Test at 5% significance. 1. Set up Hypotheses: 2. Test Statistic 1 proportion z-test 193(.221) = 42.7 193(.779) =150.3 Conditions: SRS Pop > 1,930

3. Make a decision to reject/fail to reject based on p-values Since .0327 < .05, Reject the null 4. Interpret There is not enough evidence to find that the percent of dropouts in Key West is 22.1%. The percent appears to be lower.

1. Hypotheses: 2. Test Statistic 1 proportion z-test 493(.214) = 106 In Computer Studies of the Humanities and Verbal Behavior, D Wishart and SV Leach found that 21.4% of passages in Plato’s Dialogues follow a particular syllable pattern. An owner of an antiquities store in Athens claims to have a manuscript that is a part of Dialogues. A random sample of 493 passages from the manuscript showed that 136 exhibited Plato’s syllable pattern. Do the data indicate this manuscript is part of Plato’s Dialogues? 1. Hypotheses: 2. Test Statistic 1 proportion z-test 493(.214) = 106 493(.786) = 387 Conditions: SRS Pop > 4,930

3. Make a decision to reject/fail to reject based on p-values Since .00078 < .05, Reject the null 4. Interpret There is not enough evidence to find that the percent of passages with the pattern is 21.4%. It appears that the manuscript was not written by Plato

2005 AP FRQ Some boxes of a certain brand of breakfast cereal include a voucher for a free video rental inside the box. The company that makes the cereal claims that a voucher can be found in 20 percent of the boxes. However, based on their experiences eating this cereal at home, a group of students believes that the proportion of boxes with vouchers is less than 0.2. This group of students purchased 65 boxes of the cereal to investigate the company’s claim. The students found a total of 11 vouchers for free video rentals in the 65 boxes. Suppose it is reasonable to assume that the 65 boxes purchased by the students are a random sample of all boxes of this cereal. Based on this sample, is there support for the students’ belief that the proportion of boxes with vouchers is less than 0.2? Provided statistical evidence to support your answer.

1. Set up Hypotheses: 2. Test Statistic 1 proportion z-test 65(.169) = 11 65(.831) = 54 Conditions: SRS Pop > 650

3. Make a decision to reject/fail to reject based on p-values Since .2676 > .05, Fail to reject the null 4. Interpret There is not enough evidence to find that the proportion of boxes containing the vouchers differs from 20%.