The average number of miles (in thousands) that a car's tire will function before needing replacement is 67 and the standard deviation is 14. Suppose that 46 randomly selected tires are tested. Round all answers to 4 decimal places where possible and assume a normal distribution. b. What is the distribution of ¯ x x¯ ? ¯ x x¯ ~ N ( , ) ? c. If a randomly selected individual tire is tested, find the probability that the number of miles (in thousands) before it will need replacement is between 67 and 68.8. d. For the 46 tires tested, find the probability that the average miles (in thousands) before need of replacement is between 67 and 68.8.
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
The average number of miles (in thousands) that a car's tire will
b. What is the distribution of
¯
x
x¯
?
¯
x
x¯
~ N ( , ) ?
c. If a randomly selected individual tire is tested, find the
d. For the 46 tires tested, find the probability that the average miles (in thousands) before need of replacement is between 67 and 68.8.
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