MATH 2000 TEST 4



MATH 2280 TEST 4 SOLUTIONS

1. A grocery store advertises that the mean wait at its checkout lines is 5 minutes or less. You survey 64 customers and find that they had a mean wait 6 minutes with a population standard deviation of 2.5 minutes. At [pic] should you reject the store’s claim?

You will reject the store’s claim if the sample mean is significantly greater than 5. This makes the test right-tailed. The picture for the rejection region is:[pic]

You will reject [pic] if [pic], where [pic]= invnorm(.95) = 1.645. You may also get [pic] from a list of critical values.

[pic]

Reject [pic] if [pic]

[pic]

Reject [pic], the wait time is more than five minutes.

[Or, if you use the p-value method

p = normalcdf(3.2,10000) = .00069. Reject [pic] because [pic]]

2. A grocery store advertises that the mean weight of its onions is 5 ounces. You buy 36 onions and find that their mean weight is 6 ounces with a population standard deviation of 3.5 ounces. At [pic]can you conclude that the mean weight is different from 5 ounces?

The question asks if the mean is different from 5, which makes the test two-tailed.

You find [pic] from the following picture. Or you may look it up in a table of values.

[pic]

[pic]= invnorm(.975) =1.96

[pic]

Reject [pic] if [pic]

[pic]

Do not reject [pic]

[Or, if you use the p-value method

p = 2*normalcdf(1.714 ,10000) = .0865. Do not eject [pic] because [pic]]

3. A professor believes that the mean time it should take students to complete a questionnaire is at least 30 minutes. The mean time for a sample of 18 students to complete the questionnaire was 27 minutes with a sample standard deviation of 5 minutes. At [pic] can you conclude that the mean time is less than 30 minutes?

Since the sample size is less than 30 you use a t-test with d.f. = 17.

[pic]

Reject [pic] if [pic]

[pic]

Reject [pic]

[Or, if you use the p-value method

p = tcdf((10000,(2,546, 17) = .0104. Reject [pic] because [pic]]

4. The King of a small European nation is willing to provide funding for state parks if it is determined that at least 30% of the population would make use of the facilities. In a sample of 124 citizens it was found that 34 said they would use the parks. Can the king claim at [pic] that fewer than 30% of citizens would use the parks?

[pic] ; [pic]; [pic]

[pic]

Reject [pic] if [pic]

[pic]

Do not reject [pic]

The King cannot claim (but probably will) that fewer than 30% of citizens will use the parks.

[Or, if you use the p-value method

p = normalcdf((10000,(.6271) = .2653 Do not reject [pic] because [pic]]

5. Chef Heet la Greece is concerned that the variance in the weights of his baked potatoes is too high. His supplier claims that the variance is 10. The good chef takes a sample of 23 potatoes and finds the variance to be 15. At [pic] should Monsieur la Greece reject the supplier’s claim?

Note that this is a right tailed test because the chef will only reject the supplier’s claim if the variance is too high.

[pic]

Reject [pic] if [pic]

[pic]

Reject [pic]

[Or, if you use the p-value method

p = [pic](33,10000,22) = .0619. Reject [pic] because [pic]]

6. A manufacturer of batteries wants to advertise that his batteries last longer than a competitor’s batteries. A sample of 100 batteries has a mean life of 7.1 hours with a standard deviation of 1.3 hours. A sample of 80 of the competitor’s batteries has a mean life of 6.9 hours with a standard deviation of 1.5 hours. At [pic] can the manufacturer claim that his batteries last longer? Use the p-value method.

Recall d.f. is the smaller of [pic]- 1 and [pic]- 1.

[pic]

[pic]

Reject [pic] if [pic]

[pic]

[pic] [I do not have a calculator handy!]

Do not reject [pic]

7. A researcher gathered the following data:

[pic]

Find the 99% confidence interval for [pic]. At [pic] can you claim that the population means are different?

For 99% confidence from Table F, [pic]= 2.779

[pic]

[pic]

[pic]

You cannot claim the means are different because 0 is in the confidence interval.

8. A yogurt producer is concerned that the sugar content is too variable in cups of yogurt. She tests a new stirring machine to see if it reduces the variance in the sugar content in cups of yogurt. Below is a summary of the data collected:

[pic]

At [pic] can she claim that[pic]? Use the p-value method.

[pic]

[pic]

On the TI83 p = Fcdf(F, 10000, dfN, dfD) for a right-tailed test and p =2* Fcdf(F, 10000, dfN, dfD) for a two-tailed test.

p = Fcdf(1.4, 10000, 40, 27) = .18064

Do not reject [pic] because [pic].

-----------------------

[pic]

Shaded area = [pic]= 0.05

Shaded area = [pic]= 0.025

Shaded area = [pic]= 0.025

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