The mileage (in thousands of miles) that car ownersget with a certain kind of radial tire is a random variablehaving an exponential distribution with θ = 40. Find theprobabilities that one of these tires will last(a) at least 20,000 miles;(b) at most 30,000 miles.

Assume that women's weights are normally distributed with a mean given by μ=143 lb and a standard deviation given by 29 lb.(a)  If 1 woman is randomly selected, find the probabity that her weight is between 113 lb and 172 lb(b)  If 3 women are randomly selected, find the probability that they have a mean weight between 113 lb and 172 lb(c)  If 73 women are randomly selected, find the probability that they have a mean weight between 113 lb and 172 lb

Students arrive one at a time, completely at random, to an advice clinic at a rate of10 per hour. Students take on average 5 minutes of advice but there is wide variation inthe time they need; this variation may be well modeled by the exponential distribution.b.  Again assuming one advisor, what is the probability that there are more than tenstudents in the clinic at a random point in time? What is the probability that the timespent in the clinic exceeds 30 minutes?.

Assume that IQ scores are normally distributed with μ = 100 and σ = 15 What IQ score would be needed to score in the top 2% of IQ scores?

The life X (in hours) of a battery in constant use is a random variable with exponential density. What is the probability that the battery will last more than 12 hours (h) if the average life is 8 h?

Benford's Law claims that numbers chosen from very large data files tend to have "1" as the first nonzero digit disproportionately often. In fact, research has shown that if you randomly draw a number from a very large data file, the probability of getting a number with "1" as the leading digit is about 0.301. Suppose you are an auditor for a very large corporation. The revenue report involves millions of numbers in a large computer file. Let us say you took a random sample of n = 250 numerical entries from the file and r = 60 of the entries had a first nonzero digit of 1. Let p represent the population proportion of all numbers in the corporate file that have a first nonzero digit of 1. Test the claim that p is less than 0.301 by using α = 0.01. What does the area of the sampling distribution corresponding to your P-value look like?         a. The area in the right tail of the standard normal curve.   b. The area not including the right tail of the standard normal curve.…