Suppose that 60% of people in the population support raising taxes to pay for wetlands preservation. Suppose you randomly sample 100 people and ask them whether they support raising taxes for this purpose. If you divide the number of people in your sample who support raising taxes by the total number of people asked (100), you obtain a random variable, let’s call it p^(p hat). Use the central limit theorem to come up with the approximate probability that p hat is within 0.1 of the true proportion 0.6.

Suppose that the test score of a student taking the final of a probability course is a random variable with mean 80. (a) Give an upper bound for the probability that a student's test score will exceed 90. P{score > 90} < (b) Suppose that we know, in addition, that the variance of students' test scores on the final is 30. What can you say about the probability that a student will score between 70 and 90 (do not use the central limit theorem)? P{70 < score < 90} ? (c) How many students would have to take the final to ensure with a probability of at least 0.95 that the class average would be within 3 of 80 (do not use the central limit theorem)? n = (d) If you use the central limit theorem in (c), what is your estimate for the number of students needed? п —

The number of clients arriving at a bank machine is Poisson distributed with an average of 2 per minute. For a 5-minute period, Find the expected value and standard deviation. What is the probability that 2 customers will arrive in a 5-minute period? What is the probability that no more than 2 customers will arrive in a 5-minute period? What are the underlying assumptions that allow us to consider this phenomenon as a Poisson distribution?

An elevator has a placard stating that the maximum capacity is 1344lb long dash—8 passengers.​ So, 8 adult male passengers can have a mean weight of up to 1344 divided by 8 equals 168 pounds.1344/8=168 pounds. If the elevator is loaded with 8 adult male​ passengers, find the probability that it is overloaded because they have a mean weight greater than 168lb.​ (Assume that weights of males are normally distributed with a mean of 171 lb and a standard deviation of 27 lb) Does this elevator appear to be​ safe? The probability the elevator is overloaded is____

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The average number of shoppers at a particular grocery store in one day is 505, and the standard deviation is 115. The number of shoppers is normally distributed. For a random day, what is the probability that there are between 200 and 400 shoppers at the grocery store? The answer should be typed as a decimal with 4 decimal places.

Suppose that the amount of time that students spend studying in the library in one sitting is normally distributed with mean 48 minutes and standard deviation 23 minutes. A researcher observed 14 students who entered the library to study. Round all answers to 4 decimal places where possible. If one randomly selected student is timed, find the probability that this student's time will be between 42 and 50 minutes. For the 14 students, find the probability that their average time studying is between 42 and 50 minutes. Find the probability that the randomly selected 14 students will have a total study time less than 714 minutes.