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Chapter 5: Q 1. (page 246)
Why is probability theory important to statistics?
Probability theory help in predicting values and in the estimations in statistics.
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Expressing Confidence Intervals Example 2 showed how the statistics of n= 22 ands= 14.3 result in this 95% confidence interval estimate of \(\sigma \): 11.0 < \(\sigma \) < 20.4. That confidence interval can also be expressed as (11.0, 20.4), but it cannot be expressed as 15.7 \( \pm \) 4.7. Given that 15.7\( \pm \)4.7 results in values of 11.0 and 20.4, why is it wrong to express the confidence interval as 15.7\( \pm \)4.7?
Jurors. From 10 men and 8 women in a pool of potential jurors 12 are chosen at random to constitute a jury. Suppose that you observe the number of men who are chosen for the jury. Let A be the event that at least half of the 12 jurors are men, and let B be the event that at least half of the 8 women are on the jury.
Part (a) Determine the sample space for this experiment
Part (b) Find (A or B) ,(A & B) and (A &(not B)), listing all the outcomes for each of those three events
Part (c) Are events A and B are mutually exclusive.? are events A and (not B)? are events (not A) and (not B)? Explain.
Persons per Housing Unit. From the document American Housing Survey for the United States, published by the U.S. Census Bureau, we obtained the following frequency distribution for the number of persons per occupied housing unit, where we have used "7" in place of 鈥�7 or more.鈥� Frequencies are in millions of housing units.
| Person | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| Frequencies | 27.9 | 34.4 | 17.0 | 15.5 | 6.8 | 2.3 | 1.4 |
For a randomly selected housing unit, let Y denote the number of persons living in that unit.
a. Identify the possible values of the random variable Y.
b. Use random-variable notation to represent the event that a housing unit has exactly three persons living in it.
c. Determine P(Y = 3); interpret in terms of percentages.
d. Determine the probability distribution of Y.
e. Construct a probability histogram for Y.
Vitamin C and Aspirin A bottle contains a label stating that it contains Spring Valley pills with 500 mg of vitamin C, and another bottle contains a label stating that it contains Bayer pills with 325 mg of aspirin. When testing claims about the mean contents of the pills, which would have more serious implications: rejection of the Spring Valley vitamin C claim or rejection of the Bayer aspirin claim? Is it wise to use the same significance level for hypothesis tests about the mean amount of vitamin C and the mean amount of aspirin?
In Exercises 5.16-5.26, express your probability answers as a decimal rounded to three places.
Housing Units. The U.S. Census Bureau publishes data on housing units in American Housing Survey for the United States. The following table provides a frequency distribution for the number of rooms in U.S. housing units. The frequencies are in thousands.
A housing unit is selected at random. Find the probability that the housing unit obtained has
(a) four rooms.
(b) more than four rooms.
(c) one or two rooms.
(d) fewer than one room.
(e) one or more rooms.
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