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# Normal distribution, probability, confidence levels

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1). In a normal distribution with a standard deviation of 5.0, the probability that an observation selected at random exceeds 21 is 0.14 (14%).

a). Find the mean of the distribution.
b). Find the value of below which 4 percent of the values in the distribution lie.

2). The company's manager calculates that it requires 11 months on average to complete the publication process of a book from manuscript to the finished product, with a standard deviation of 2.4 months. He believes that the normal distribution well describes the distribution of publication times. Out of 19 books he will handle this year, approximately how many will complete the process in less than a year?

3). On average, 15 percent of those enrolled in a sign language training course wil have to repeat the course. If the current class size is 20, what is the probability that:
a) Exactly 4 will have to repeat the course?
b) Fewer than 3 will have to repeat the course?
c) More than 5 will have to repeat the course?
(Use table 3 in the appendix)

4) Joe Jones supervises the packaging of college textbooks for Jones Publishers. He knows that the number of boxes he will need depends partly on the size of books. All Jones books use the same size paper, but may have differing number of pages. After pulling shipment records for the last 5 years, Joe derived the following set of probabilities:

# of pages 100 300 500 700 900 1100
Probability 0.05 0.10 0.25 0.25 0.20 0.15

a) If Joe bases his box purchase on an expected length of 600 pages, will he have enough boxes?
b) If all 700-page books are edited down to 500 pages (and the same edited pages ratio holds for all sized books), what expected number of pages should he use?

5). A recent study by the EPA has determined that the amount of contaminants in Minnesota lakes (in parts per million) is normally distributed with mean 64 ppm and variance 17.6. Suppose 35 lakes are randomly selected and sampled. What is the probability that the sample average amount of contaminants is

a) Above 72 ppm?
b) Between 64 and 72 ppm?
c) Exactly 64 ppm?
d) Above 94 ppm? _
e) If, in our sample, we found x = 100 ppm, would you feel confident in the study conducted by the EPA? Explain briefly.

6). A ferry carries 25 passengers. The weight of each passenger has a normal distribution with mean 168 pounds and variance 361 pounds squared. Safety regulations state that for this particular ferry, the total weight of passengers on the boat should not exceed 4, 250 pounds more than 5 percent of the time. As a service to the ferry owners, find

a) The probability that the total weight of passengers on the ferry will exceed 4, 250 pounds.
b)The 95th percentile of the distribution of the total weight of passengers on the ferry.
c) Is the ferry complying with regulations?

7) Given that a random variable, X, has a normal distribution with means 6.4 and standard deviation 2.7, find

a) P(4.0 < x < 5.0).
b) P(x > 2.0).
c) P(x < 7.2).
d) P((x < 3.0) or (x > 9.0)).

8). Given the following sample sizes and t values used to construct confidence intervals, find the corresponding confidence levels:

a) n = 27; t = +2.056
b)n = 5; t = +2.132
c)n = 18; t = +2.898

9). If our goal is to accept a null hypothesis that (u = population mean) u = 36.5 with 96 percent certainty when it's true,and our sample size is 50, diagram the acceptance and rejection regions for the following alternative hypotheses:

a). u =/ (not equal) 36.5
b) u > 36.5
c) u < 36.5