EQ: When to use t-tests? Hypothesis Testing

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

Outline of a test 1. Set up Hypotheses a. Only use parameters b. The null always has equality! 2. Calculate a test statistic. 3. Make a decision to reject/fail to reject based on p-values 4. Interpret the results

In Illinois, a random sample of 85 eighth grade students has a mean score of 282 with a standard deviation of 35 on a national mathematics assessment test. 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? Null Hypothesis:Alternative Hypothesis:

A manufacturer of sprinkler systems designed for fire protection claims that the average activating temperature is at least 135 degrees F. To test this claim, you randomly select a sample of 32 systems and find the mean activation temperature to be 133 degrees F with a standard deviation of 3.3 degrees F. Is there evidence to support the manufacturers claim? Null Hypothesis:Alternative Hypothesis:

A nutritionist claims that the mean tuna consumption by a person in the United States is 3.1 pounds per year. A random sample of 60 people in the U.S. shows that the mean tuna consumption by a person is 2.9 pounds per year with a standard deviation of 0.94 pounds. Is there evidence to support the nutritionists claim? Null Hypothesis:Alternative Hypothesis:

Example 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? 1. Set up Hypotheses: Null Hypothesis: Alternative Hypothesis:

Test Statistic Determined by the parameter and conditions 1 sample means The value of the parameter (mu or p) is from the null hypothesis Conditions 1.SRS 2.Normal Use z if sigma is known

2. Test Statistic Based on the data, we found a test statistic of How likely is it to get that value? 3. Make a decision to reject/fail to reject based on p-values 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. P-value

P-values The probability of obtaining a test statistic with a value as extreme or more extreme than the one produced from data The SMALLER the p-value, the more evidence there is to reject the null hypothesis A rejection level can be set a specific level denoted by α, if not stated use α=.05 From the example, the p-value was.0288 Decision: Reject the null since.0288 < Interpret the results It appears that Home Depot’s claim is false, the mean life of the light bulbs is not 1000 hours

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? 1. Hypotheses: 2. Test Statistic:1-sample mean t-test, 35 df Conditions:SRS, Normal (n>30) 3. Decision Reject the null

4. Interpret The average delivery time appears to be less than 30 minute. The company’s claim is correct.

2 Tail test 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? 1. Hypotheses: When the alternative is not equal, it is a 2 tail test

2. Test Statistic:1-sample mean t-test, 34 df Conditions:SRS, Normal (n>30) 3. Decision We must consider both directions when finding the p-value Since.0092<.01, reject the null

4. Interpret The average number of sales does not appear to be 150. There is evidence to support the boss’ claim.

Tennis balls produced by a particular machine do not have identical diameters because of variation in the manufacturing process. A machine was initially calibrated to produce balls with a mean of 3 inches and a standard deviation of.25 inches. A random sample of 90 balls produced a mean of 3.05 inches. Is there evidence that the diameter of the tennis balls is larger than it is supposed to be? 1. Hypotheses: 2. Test Statistic:1-sample mean z-test Conditions:SRS, Normal (n>30)

3. Decision Since.0289>.01, fail to reject the null 4. Interpret There is not enough evidence to find that the machine is making tennis balls with diameters that are larger than specified.