Hypothesis Testing for means 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.