AP Stats Unit 11 Review1. Old World cuckoos, famous as the voice in the “cuckoo clock,” are brood parasites. They lay their eggs in the nests of other bird species. The “host” birds then raise the young cuckoo, their own young perishing in the process. Biologists wondered whether the size of the cuckoo eggs might vary with the species being parasitized, especially as some of the target birds are much smaller than the cuckoo. The data are egg lengths of cuckoo eggs (in mm) laid in the nests of Wrens, Tree Pipits, and Pied Wagtails. These can be considered to be random samples of all cuckoo eggs laid in the nests of the given species.

a. Is there a significant difference in the mean cuckoo egg length of eggs laid in Wren and Tree Pipit nests?

The samples are random and are collected independently. The normal probability plots are very roughly linear, meeting or normality requirement.

Tree Pipit Wren

Wren Tree Pipit Pied Wagtail19.85 21.05 21.0520.05 21.85 21.8520.25 22.05 21.8520.85 22.45 21.8520.85 22.65 22.0520.85 23.25 22.4521.05 23.25 22.6521.05 23.25 23.0521.05 23.45 23.0521.25 23.45 23.2521.45 23.65 23.4522.05 23.85 24.0522.05 24.05 24.0522.05 24.05 24.0522.25 24.05 24.85

Reject H0, . A test statistic this extreme will occur by chance alone less than 1% of the time.

We have strong evidence that the mean length of cuckoo eggs laid in Wren nests is different from the mean of those laid in Tree Pipit nests.

b. Is there a significant difference in the mean cuckoo egg length of eggs laid in Tree Pipit and Pied Wagtail nests?

The samples are random and are collected independently. The normal probability plots are very roughly linear, meeting or normality requirement. Tree Pipit Pied Wagtail

Fail to reject H0, . A test statistic this extreme will occur by chance alone 68% of the time.

We lack strong evidence that the mean length of cuckoo eggs laid in Tree Pipit nests is different from the mean of those laid in Pied Wagtail nests.

c. What is the mean length of cuckoo eggs laid in Wren nests? Make a 95% confidence interval.

We will make a 1-sample t-interval for means.

Our assumptions have been met above.

We are 95% confident that the true mean length of cuckoo eggs laid in Wren nests is between 20.6 and 21.4 mm. In repeated random sampling, this method captures the true mean 95% of the time.

2. An experiment on the side effects of pain relievers assigned arthritis patients to one of several over-the-counter pain medications. Of the 440 patients who took one brand of pain reliever, 31 suffered some “adverse symptom.” Does the experiment provide strong evidence that fewer than 10% of patients who take this medication have adverse symptoms? Test at a 1% significance level.

The data came from an experiment, and presumably they randomly assigned to treatment.

Fail to reject H0, , a test statistic this small may occur by chance alone more than 1% of the

time.

We lack strong evidence that the true proportion of adverse symptoms is less than 10%.

3. In the 2001 regular baseball season, the World Series Champion Arizona Diamondbacks played 81 games at home and 81 games away. They won 48 of their home games and 44 of the games played away. We can consider these games as samples from potentially large populations of games played at home and away. Most people think that it is easier to win at home than away. Perform a significance test to determine whether this is true for the Arizona Diamondbacks.

We are given samples of large populations, in effect.

Since the 2 samples are the same size the products are the same for n1 and n2..

Fail to reject H0, . A test statistic as large as this may occur by chance alone 26% of the time.

We lack evidence of a higher proportion of wins at home games for the Arizona Diamondbacks.

4. The composition of the earth’s atmosphere may have changed over time. One attempt to discover the nature of the atmosphere long ago studies the gas trapped in bubbles inside ancient amber. Amber is tree resin that has hardened and been trapped in rocks. The gas in bubbles within amber should be a sample of the atmosphere at the time the amber was formed. Measurement on specimens of amber from the late Cretaceous era (75 to 95 million years ago) gives these percentages of nitrogen:

These values are quite different from the present 78.1% of nitrogen in the atmosphere, but are these differences significant? Assume (this is not yet agreed on by experts) that these observations are an SRS from the Cretaceous atmosphere.

We are given data that is from a sample of gases trapped in the late Cretaceous era. While not random, this may be representative. The sample size of 9 is small, but the data is not strongly skewed. We rely on the robustness of the t-test and continue.

(No shading is visible on a graph of the t-distribution.)

Reject H0, . A test statistic this extreme will occur by chance alone well less than 1% of the time.

We have strong evidence that the mean percentage of nitrogen trapped in amber during the Cretaceous era are different from the 78.1% today.

63.4 65.0 64.4 63.3 54.8 64.5 60.8 49.1 51.0