Instructions: This is a close book exam. Anyone who cheats in the exam shall receive a grade of F. Please provide complete solutions for full credit. Good luck!

1 (for all students in class). In a study of hypnotic suggestion, 5 male volunteers participated in a two-phase experimental session. In the first phase, respiration was measured while the subject was awake and at rest. In the second phase, the subject was told to imagine that he was performing muscular work, and respiration was measured again. Hypnosis was induced between the first and second phases; thus, the suggestion to imagine muscular work was “hypnotic suggestion” for these subjects. The accompanying

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John Pauzke, president of Cereals Unlimited, Inc., wants to be very certain that the mean weight μ of packages satisfies the package label weight of 16 ounces. The packages are filled by a machine that is set to fill each package to a specified weight. However, the machine has random variability measured by σ2. John would like to have strong evidence that the mean package weight is above 16 ounces. George Williams, quality control manager, advises him to examine a random sample of 25 packages of cereal. From his past experience, George knew that the weight of the cereal packages follows a normal distribution with standard deviation 0.4 ounce. At the significance level α

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Please derive the general formula for sample size calculation based on the Type I and II error rates first.

Solution:

(1)

[pic]

[pic]. [pic].

[pic].

[pic]

Hence, the rejection region is [pic].

(2)

[pic]

[pic]

(3)

[pic]

[pic][pic].

[pic]

Hence, about 25 packages of cereal should be sampled to achieve a power of 80% when (=16.2 ounces.

3a (for all except AMS PhD students). Inference on one population mean when the population is normal, and the population variance is known. Let [pic], be a random sample from the given normal population. Please prove that 1) [pic]. 2) [pic].

Solution: (1)

[pic]

Thus, [pic]

(2)

[pic]

Thus, [pic]

3b (for AMS PhD students ONLY). For a random sample from any population for which the mean and variance exist. Please prove that 1) The sample mean and sample variance are unbiased estimators of the population mean and variance respectively. 2) When the population is normal, we have learned that the sample mean and the sample variance, are indeed, independent. Please prove this for n = 2. That is, for a random sample of size 2 only.

Solution:

(1) [pic]

[pic]

(2) When n=2, [pic],