Exam
Name___________________________________

SHORT ANSWER.  Write the word or phrase that best completes each statement or answers the question.
1) All 423 Wisconsin public schools were all given a rating by the Wisconsin Department of Public Instruction based on several variables. The mean rating reported was 71.5 and the standard deviation was 4.87. To do a follow-up study a random sample of 40 schools was selected. In this sample, the mean rating was 70.9. One of the researchers is alarmed, thinking the report may have been mistaken. Do you think this sample result is unusually low? Explain.	1)  _____________

2) Wildlife scientists studying a certain species of frogs know that past records indicate the adults should weigh an average of 118 grams with a standard deviation of 14 grams. The researchers collect a random sample of 50 adult frogs and weigh them. In their sample the mean weight was only 110 grams. One of the scientists is alarmed, fearing that environmental changes may be adversely affecting the frogs. Do you think this sample result is unusually low? Explain.	2)  _____________

3) Herpetologists (snake specialists) found that a certain species of reticulated python has an average length of 20.5 feet with a standard deviation of 2.3 feet. The scientists collect a random sample of 30 adult pythons and measure their lengths. In their sample the mean length was 19.5 feet long. One of the herpetologists fears that pollution might be affecting the natural growth of the pythons. Do you think this sample result is unusually small? Explain.	3)  _____________

4) The average composite ACT score for Ohio students who took the test in 2003 was 21.4. Assume that the standard deviation is 1.05. In a random sample of 25 students who took the exam in 2003, what is the  probability that the average composite ACT score is 22 or more? (Make sure to identify the sampling distribution you use and check all necessary conditions.)	4)  _____________

1) We have a random sample of schools drawn from a much larger population. With a sample of size 40 the CLT says that the approximate sampling model for sample means will be N(71.5, 0.77). A sample mean rating of only 70.9 is about 0.78 standard deviations below what we expect, which is not a very unusual result.
2) We have a random sample of frogs drawn from a much larger population. With a sample of size 50 the CLT says that the approximate sampling model for sample means will be N(118, 1.98). A sample mean of only 110 grams is about 4 standard deviations below what we expect, a very unusual result.
3) We have a random sample of adult pythons drawn from a much larger population. With a sample size of 30, the CLT says that the approximate sampling model for sample means will be  A sample mean of only 19.5 feet is about 2.38 standard deviations below what we expect. The sample mean of 19.5 feet is unusually small.
4) Check the conditions
1. Random sampling condition We have been told that this is a random sample.
2. Independence assumption It's reasonable to think that the scores of the 25 students are mutually independent.
3. 10% condition 25 students is certainly less than 10% of all students who took the exam.
We're assuming that the model for composite ACT scores has mean µ = 21.4 and standard deviation  Since the sample size is large enough and the distribution of ACT scores is most likely unimodal and symmetric, CLT allows us to describe the sampling distribution of  with a Normal model with mean 21.4 and SD = () =  = 0.21.
An average score of 22 is z =  = 2.86 SDs above the mean.
P( > 22) = P(Z > 2.86) = 0.0021, so the probability that the average composite ACT score for a sample of 25 randomly selected students is 22 or more is 0.0021.