AP Statistics Section 6.2

TESTS OF SIGNIFICANCE

USES/PURPOSES: 1. What are these tests used for?

- To assess…

2. What do these tests compare?

3. How are the results of these tests expressed?

4. What are the results a probability of?

5. What are the 4 components of a Test of Significance? a.

b.

c.

d.

In Ch. 6, we will be looking at testing a population mean () Example:

Recall Tim and his obsessive weight measuring. He measures his weight once a week, every week. On Tim’s drivers license, it says that he is 187. However that driver’s license is a few years old, and Tim wants to know if his weight has changed since then. So Tim decides to take his sample of 4 measurements from before and compare their average to what it says on his license. Measurements (in lbs.): 190.5, 189, 195.5, 187. Assume =3, like before.

Tim notices that his average weight for these 4 measurements is 190.5. Now Tim really thinks he has gained weight.

1. If Tim’s real weight was 187, then his average for the 4 measurements should be around what value?

2. Since his average is not around this value, Tim has a reason to believe that he gained weight. Do you think Tim has gained weight? Why or why not?

3. What is the question of interest? What are we looking to find out?

4. Is the number 187 a parameter or a statistic? What variable does the number 187 represent?

5. Is the number 190.5 a parameter or a statistic? What variable does the number 190.5 represent?

6. What is our sample size?

7. What type of problem (chapter 5) are we looking at here?

8. Tim’s friend Bob reminds him that sometimes you just get a bad sample; it could be high or low. Bob suggests doing some probability calculations. What is the probability that Tim would get a sample with an average that high or higher if he really was 187? Assume =3, like before. (use probability notation) COMPONENTS: (page 454 – 462 in the book) 1. Hypotheses What do the hypotheses always describe?

What types of symbols do the hypotheses always use?

Name: Null Hypothesis Symbol:

What is it?

FORM:

Example:

If the null hypothesis is true, what do we expect from the estimate?

Name: Alternative Hypothesis Symbol:

What is it?

FORM: Two types of alternative hypotheses:

Examples: 2. Tests Statistic What does the test statistic measure?

What is this test statistic used for?

What type of a variable is this test statistic and what type of distribution does it have?

Formula (for the test statistic of a population mean):

Z =

Example:

3. P-Value Definition:

The smaller the p-value…

What is true about our evidence as the sample mean (x-bar) gets further from the population mean (μ)?

How do we use the test statistic to find the P-Value? What distribution do we use? 4. Conclusion

Statistical Significance Definition:

What does statistical significance compare?

What is a significance level? What symbol do we use to denote this level?

What does a significance level of 0.05 or 0.01 indicate about the null hypothesis?

If a significance level is not given, what level do we use?

Conclusions:

What are the two conclusions that we can have (be careful how they are stated!)

Our conclusion must always be a ______in terms of ______.

*Tests of Significance Summary: bottom of page 459 to top of page 450 in the book* Tests of Significance Example:

Over his career, a pitcher for a baseball team has thrown his fastball at an average speed of 90 mph and a standard deviation of 2.2 mph. Some ESPN analysts have been saying recently that his average fastball speed is getting slower because he is getting older. The pitcher doesn’t believe them and wants to test and see if this is really the case. He takes a sample of 10 of his pitches, recorded below:

85, 87, 92, 90, 82, 85, 86, 86, 88, 81

Test to see if his average fastball speed is getting slower if  = 0.05.

1. Hypotheses:

Ho:

Ha:

2. Test Statistic:

3. P-Value:

4. Conclusion: AP Statistics Section 6.2 - Hypothesis Test for Mean

1. State the appropriate null and alternative hypotheses for each situation. a. A tenant group thinks that the apartments are smaller than the advertised 1250 square feet.

b. Larry switches to new motor oil that is advertised to increase his current gas mileage, which is 32 mpg. He wants to determine if his mileage has increased.

c. A manufacturer measures the diameter in a sample of motors to determine whether the mean diameter has changed from the goal of 5 mm.

d. A researcher thinks that a loud noise will cause mice to complete a maze faster than the previous average of 18 seconds.

e. A sociologist suspects that at least one third of male students will name math as their favorite subject.

2. A commercial aircraft manufacturer buys rivets. A rivet is considered bad if its shear strength is less than 925 lbs. A random sample of 50 rivets is selected and the mean strength is calculated to be 921.18 with  = 18. Test the hypothesis if  = .05.

3. The following question was asked of the Student Affairs Office at a college: “How far does the average community-college student commute to college daily?” The office replied: “No more than 9 miles.” If we take a random sample of 50 students and find a mean of 10.22 miles, test the hypothesis at a significance level of 0.05 with  = 5. 4. It has been claimed that the mean weight of women students at a college is 54.4 kg. A professor sets out to show the mean is not 54.4 kg. In a random sample of 100 weights, the mean is 53.75. Is this sufficient evidence for the professor to reject the statement? Use  = .01 and  = 5.4.

5. The manager at Air Express feels that the weights of packages shipped recently are less than in the past. Records show that in the past packages have had a mean weight of 36.7 lb. and a standard deviation of 14.2 lb. A random sample of last month’s shipping records yielded a mean weight of 32.1 lb for 64 packages. Is this sufficient evidence to reject the null hypothesis in favor of the manager’s claim? Use  = 0.01.

6. The estimated U.S. intake of trans-fatty acids is 8 g per day. Consider a research project involving 150 individuals in which their daily intake of trans-fatty acids was measured. Suppose the sample mean was 12.5 g. Assuming that  = 8.0, test the research hypothesis that  > 8 at  = 0.05.

7. The average stay in days for nongovernment not-for-profit hospitals is given to be 7.0 days. A sample of 40 such hospitals was selected to test the hypothesis that the average stay different than the national average. The sample mean equals 6.1 days. Is this sufficient evidence to reject the null hypothesis? Use  = 0.05 and  = 1.5 days. AP Statistics Section 6.2B – Hypothesis Test for Mean

1. A manufacturer claims that a new brand of air-conditioning unit uses only 6.5 kilowatts of electricity per day. A consumer agency believes the true figure is higher and runs a test on a sample of size 50. If the sample mean is 7.0 kilowatts and we know the population standard deviation is .4, should the manufacturer’s claim be rejected at a significance level of 5%? Of 1%?

2. A local chamber of commerce claims that the mean family income level in a city is $12,250. An economist runs a hypothesis test, using a sample of 135 families, and find s mean of $11,500 with a standard deviation of $3180. Should the $12,250 claim be rejected at a 5% level of significance?

3. Someone comments: “Because only a minority of high school students take the SAT, the scores overestimate the ability of typical high school seniors. The mean SAT math score is 475, but I think that if all seniors took the test, the mean would be no more than 450.” You test the person’s claim by giving the test to an SRS of 500 seniors from California. The mean is 461. Assuming the standard deviation is 100, is there evidence at the .01 level of significance to support the person’s claim?

4. Do middle-aged male executives have different average blood pressure than the general population? The National Center for Health Statistics reports that the mean blood pressure for males 35 to 44 years of age is 128 and the standard deviation in this population is 15. The medical director of a company looks at the medical records of 72 executives in this age group and finds that the mean blood pressure is 126.07. Is this evidence that executive blood pressures differ from the national average?