Question

1. The random variable x has a normal distribution with standard deviation 23. It is known that the probability that x exceeds 151 is .90. Find the mean muμ of the probability distribution.

2. In an assessment of the quality of the electrical power​ system, one measure of quality is the degree to which voltage fluctuations cause light flicker in the system. The perception of light flicker x when the system is set at 380kV was measured periodically​ (over 10-minute​ intervals). For transformers supplying heavy industry​ plants, suppose the light flicker distribution was found to follow​ (approximately) a normal distribution with muμequals=2.9​% and sigmaσequals=0.50. If the perception of light flicker exceeds 3.4​%, the transformer is shut down and the system is reset. How likely is it for a transformer supplying a heavy industry plant to be shut down due to light​ flicker?

3. Consider a country​ where, to become​ president, a candidate must win 273 of the total of 539 votes in an electoral college.​ Let's suppose that the largest region gets 55 votes. Assuming that a candidate wins this​ region, the number of additional electoral college votes can be approximated by a normal distribution with muμequals=245.5 and sigmaσequals=52.9 votes. Given that the candidate wins the largest​ region, what are the chances that he or she becomes the next​ president?

Answer 1

1)

Solution :

standard deviation = = 23

x = 151

Using standard normal table,

P(Z > z) = 0.90

1 - P(Z < z) = 0.90

P(Z < z) = 1 - 0.90

P(Z < -1.28) = 0.10

z = -1.28

Using z-score formula,

   = (x - z * ) = (151 - (-1.28 * 23) = 151 + 18.2 = 170.2

The mean μ of the probability distribution is 170.2