9.2 Notes

Tests Involving µ

Procedure for Hypothesis Testing when σ is known

1) Establish the ______ Hypothesis (____)

2) Establish the ____________ Hypothesis (_______)

3) Use _____ to determine location of critical region (____________, ________ ________, or ___________)

4) Use ___ and __________ to determine the critical value(s) for the critical region

5) Using Z-Test in calc., calculate the statistical value based on the sample

6) Based on statistical value, draw your conclusions

1) Reject H0 - (We are ?% confident that H1, therefore …)

2) Fail to reject H0 - (At the ?% level of significance the evidence is not strong enough to imply H1, therefore …)

Ex. 1 The St. Louis Zoo wishes to obtain eggs of a rare Mississippi river turtle. The zoo will hatch the eggs and raise the turtles as an exhibit of a rare and endangered species. Past research has shown that lengths of turtle eggs are normally distributed, and lengths of the rare turtle eggs have μ = 7.50 cm with σ = 1.5 cm. Then mean lengths of all other turtle eggs in the area are longer than 7.50 cm with similar deviation. A biologist found an abandoned nest of 36 eggs that have a mean length = 7.74 cm. Because a lot of effort goes into removing and incubating the eggs, the biologist is a little concerned that the eggs may be from another species that lays larger eggs. Should the biologist bother to remove and incubate the eggs? Test at the 1% level of significance.

x

Ex. 2 A research meteorologist has been studying wind patterns over the Pacific Ocean. Based on these studies, a new route is proposed for commercial airlines going from San Francisco to Honolulu. The new route is intended to take advantage of existing wind patterns to reduce flying time. It is known that for the old route the distribution of flying times for a large four-engine jet has mean μ = 5.25 hours with standard deviation σ = 0.6 hour. Thirty-six flights on the new route have yielded a mean flying time of 4.90 hours. Should the company adopt the new flight path? Test at the 5% level of significance.

Ex. 3 A machine makes twist-off caps for bottles. The machine is adjusted to make caps of diameter 1.85 cm. Production records show that when the machine is so adjusted, it will make caps with mean diameter 1.85 cm and with standard deviation σ = 0.05 cm. During production, an inspector checks the diameters of caps to see if the machine has slipped out of adjustment. A random sample of 64 caps is taken. If the mean diameter for this sample is 1.87 cm, does this indicate that the machine has slipped out of adjustment and needs correction? Test at α = 0.01.

Procedure for Hypothesis Testing if σ is unknown

Same as for when σ is known except:

1. To find the critical region value (tc), stop at d.f. = _______, where n = ____________________.

2. Use ___________ in calc to find t-value. May require putting data into L1

Ex. 1 H.J. Heinz, Campbell Soup, Kellogg, Hershey Foods, Quaker Oats, and so on are important food producers. How profitable are such food companies? Let x be a random variable that represents annual profit as a percentage of assets for the nation’s largest food companies. Assume that x has a distribution that is approximately normal with mean μ = 6.6 %. Suppose that recent financial reports for randomly selected national food companies gave the following x values:

6.6 9.1 3.3 2.5 8.4 5.1 4.8 3.4

Use a 5% level of significance to test the claim that the population average annual profits as a percentage of assets for large food companies has dropped.

Day 1 AssignmentP. 426 #1-4, 7, 8, 11, 13

Day 2 AssignmentP. 426 #5, 6, 10, 14, 16, 20