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# Mean, Standard Deviation, Probability, Normal Distribution

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1) Adult American males have normally distributed heights with a mean of 5.8 feet and a standard deviation of 0.2 feet. What is the probability that a randomly chosen adult American male will have a height between 5.6 feet and 6.0 feet?
A. 0.6826
B. 0.5000
C. 0.9544
D. 0.7500

2) A jar contains 12 red jelly beans, 20 yellow jelly beans, and 16 orange jelly beans.
Suppose that each jelly bean has an equal chance of being picked from the jar.
If a jelly bean is selected at random from the jar, what is the probability that it is not red?

3) Which of the following statements is NOT true?
A. A probability must be less than or equal to 1.
B. If an event cannot possibly occur, then the probability of the event is a negative number.
C. If only two outcomes are possible for an experiment, then the sum of the probabilities of
the outcomes is equal to 1.
D. If events E and F are mutually exclusive events, then P(E Ç F) = 0.

4) In a certain manufacturing process, the probability of a type I defect is 0.09, the probability of a type II defect is 0.11, and the probability of having both types of defects is 0.03.
Find the probability that neither defect occurs.
A. 0.97
B. 0.77
C. 0.83
D. 0.80

5) Which of the following statements is NOT true?
A. The variance is the square root of the standard deviation.
B. The variance is a measure of the dispersion or spread of a distribution about its mean.
C. If all of the data values in a data set are identical, then the standard deviation is 0.
D. The variance must be a nonnegative number.

6) A contest has 20 finalists. One finalist is awarded first prize, another finalist is awarded
second prize, and another is awarded third prize. How many different ways could the prizes be awarded?

7) An advisory board of 5 students is to be chosen from a group of 12 students.
8 of the students are seniors and 4 of the students are juniors.
(a) In how many ways can the advisory board of 5 students be chosen from the group of 12 students?
(b) In how many ways can the 5-member advisory board be chosen from the group of 12
students, if 3 members must be seniors and 2 members must be juniors?
(c) If the 5-member advisory board is selected at random from the group of 12 students , what is the probability that the board consists of 3 seniors and 2 juniors?

8) According to a recent report, 0.65 is the probability that an American household is owner occupied. Six Americans households are randomly selected. Find the probability that exactly 4 of the 6 American households are owner-occupied.

https://brainmass.com/math/probability/mean-standard-deviation-probability-normal-distribution-588065

#### Solution Preview

1)Adult American males have normally distributed heights with a mean of 5.8 feet and a
standard deviation of 0.2 feet. What is the probability that a randomly chosen adult American male will have a height between 5.6 feet and 6.0 feet?
A. 0.6826
B. 0.5000
C. 0.9544
D. 0.7500
Since z=(x-mean)/standard deviation=(x-5.8)/0.2, P(5.6<X<6.0)=P((5.6-5.8)/0.2<Z<(6.0-5.8)/0.2))=P(-1<Z<1)=0.6826 from standard normal table.

2)A jar contains 12 red jelly beans, 20 yellow jelly beans, and 16 orange jelly beans.
Suppose that each jelly bean has an equal chance of being picked from the jar.
If a jelly bean is selected at random from the jar, what is the probability that it is not red?
First, there are totally 12+20+16=48 jelly beans. Among those jelly beans, there are 20+16=36 jelly beans which are not red. So P(not red)=36/48=0.75

3)Which of the following statements is NOT true?
A. A probability must be less than or equal to 1.
B. If an event cannot possibly occur, ...

#### Solution Summary

The solution gives detailed steps on solving 8 questions on basic statistics including the topic of mean, standard deviation, probability and normal distirbution.

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## Sampling Distribution, Mean and Standard Deviation

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1) A manufacturer of paper used for packaging requires a minimum strength of 20 pounds per square inch. To check on the quality of the paper, a random sample of 10 pieces of paper is selected each hour from the previous hour's production and a strength measurement is recorded for each. The standard deviation &#963; of the strength measurements, computed by pooling the sum of squares of deviations of many samples, is know to equal 2 pounds per square inch, and the strength measurements are normally distributed.

a) What is the approximate sampling distribution of the sample mean of n = 10 test pieces of paper?
b) If the mean of the population of strength measurements is 21 pounds per square inch, what is the approximate probability that, for a random sample of n = 10 test pieces of paper, ¯x < 20?
c) What value would you select for the mean paper strength &#956; in order that P (¯x < 20) be equal to .001?

2) Suppose a random sample of n = 25 observations is selected from a population that is normally distributed, with mean equal to 106 and standard deviation equal to 12?
a) Give the mean and standard deviation of the sampling distribution of the sample mean ¯x.
b) Find the probability that ¯x exceeds 110
c) Find the probability that the sample mean deviates from the population mean &#956; = 106 by no more than 4.

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