Normal distribtion and binomial distribution

Normal Distribution:

1. Find the value of z if the area under a standard normal curve
a) to the right of z is 0.3662
b) to the left of z is 0.1131
c) between 0 and z, with z > 0, is 0.4838
d) between -z and z, with z>0, is 0.9500

2. According to Chebyshev's theorem, the probability that any random variable assumes a value within 3 standard deviations of the mean is at least 8/9. If it is known that the probability distribution of a random variable X is normal with mean µ and variance &#417; ², what is the exact value of P(µ - 3&#417; ² < X < µ + 3&#417; ²) ?

3. The finished inside diameter of a piston ring is normally distributed with a mean of 10 centimeters and a standard deviation of 0.03 centimeter.
a) What proportion of rings will have inside diameters exceeding 10.075 centimeters?
b) What is the probability that a piston ring will have an inside diameter between 9.97 and 10.03 centimeters?
c) Below what value of inside diameter will 15% of the piston rings fall?

4. In the November 1990 issue of Chemical Engineering Progress, a study discussed the percent purity of oxygen from a certain supplier. Assume that the mean was 99.61 with a standard deviation of 0.08. Assume that the distribution of percent purity was approximately normal.
a) What percentage of the purity values would you expect to be between 99.5 and 99.7?
b) What purity value would you expect to exceed exactly 5 % of the population?

5. If a set of observations is normally distributed, what percent of these differ from the mean by
a) more than 1.3&#417; ?
b) less than 0.52&#417; ?

6. Given a continuous uniform distribution, show that
a) µ = (A + B / 2), and
b) &#417; ² = ((B - A)^2) / 12)

7. A bus arrives every 10 minutes at a bus stop. It is assumed that the waiting time for a particular individual is a random variable with a continuous uniform distribution.
a) What is the probability that the individual waits more than 7 minutes?

b) What is the probability that the individual waits between 2 and 7 minutes?

Normal Approximation to the Binomial:

8. A coin is tossed 400 times. Use the normal-curve approximation to find the probability of obtaining
a) between 185 and 210 heads inclusive
b) exactly 205 heads
c) less than 176 or more than 227 heads

9. A process yields 10% defective items. If 100 items are randomly selected from the process, what is the probability that the number of defectives
a) exceeds 13?
b) is less than 8?

10. Researchers at George Washington University and the National Institutes of Health claim that approximately 75 % of the people believe "tranquilizers work very well to make a person more calm and relaxed." Of the next 80 people interviewed, what is the probability that
a) at least 50 are of this opinion?
b) at most 56 are of this opinion?

11. A pharmaceutical company knows that approximately 5% of its birth-control pills have an ingredient that is below the minimum strength, thus rendering the pill ineffective. What is the probability that fewer than 10 in sample of 200 pills will be ineffective?

12. A commonly used practice of airline companies is to sell more tickets than actual seats to a particular flight because customers who buy tickets do not always show up for the flight. Suppose that the percentage of no-shows at flight time is 2%. For a particular flight with 197 seats, a total of 200 tickets was sold. What is the probability that the airline overbooked this flight?

See attached file for full problem description.

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Solution Summary

The solution is comprised of explanations of calculation of probability related to normal distribution and binomial distribution.