There are 9 problems in this homework. Please write down your name and student ID, and show your work clearly and in details. Hints: It may help you to sketch the distribution in some questions.
1. A continuous random variable has the probability density function
f(x)=−1x+1, 0≤x≤2 2
a. Does this variable follow a uniform distribution? Why or Why not?b. Show that f(x) is a legitimate probability density function for a continuous random variable. c. Determine the probability the random variable will assume a value greater than 1.
2. Let x be a normal random variable with μ=8 and σ=5. a. Find P (x < 10.55).b. Find P (x > 6.8).c. Find P(9.25 < x ≤ 13).
d. Find P(−12 < x < 3.9).e. Find x0 such that P(x < x0) = 0.123.f. Find x0 such that P(x > x0) = 0.2296.g. Find x0 such that P(4.3 < x < x0) = 0.542. h. Find x0 such that P (x0 < x < 11.25) = 0.351.
3. Suppose that daily high temperatures in Columbia during April are approximately normally distributed with a mean of 65 degrees and a standard deviation of 5 degrees.
a. What is the probability that a randomly selected day will have a high temperature greater than 73?
b. On a randomly selected day in April, what is the probability that the high temperature, in Columbia, will be between 57 and 71 degrees?
c. Suppose 11.9% of April days in Columbia have a high temperature higher than x0, what is x0?
d. Find k such that the probability that the high temperature for a randomly selected day in April will be between 54 and k is .8569.
4. The time it takes to drive from home to school is normally distributed with mean 35 minutes and variance 9 minutes squared.
a. If you leave home 41 minutes before class starts, find the probability you will be late, i.e. P (X > 41).
b. What time should you leave the house if you want to be on time 99% of the time? That is, find x0 such that P(X < x0) = .99.
5. Suppose x is a binomial random variable with p = 0.4 and n = 25.a. Would it be appropriate to approximate the probability of x with a normal distribution?b. Assuming that a normal distribution provides an adequate approximation to the distribution of x,
what are the mean and variance of the approximating normal distribution?c. Use the tables for the binomial cumulative probability to find the exact value of P (8 < x ≤ 14). d. Use normal approximation to find P (8 < x ≤ 14).e. Compare your result from part c with part d. Comment.
6. A coffee shop is trying to estimate the amount of inventory it needs to keep on hand of a few popular menu items. They have an average of 400 customers per day. They know that 12% of all customers order a muffin with their coffee. (n = 400).
a. Write an expression using the binomial distribution for the probability that they will sell at least 50 muffins in a day.
b. Is the normal approximation appropriate in this situation?c. Approximate the probability of part(a) using the normal approximation.d. What is the probability that between 40 and 55 muffins (inclusive) are sold in a day?
e. How many muffins would they need to have available for purchase to ensure that on any given day the probability of running out is less than .0505? That is, find x0 such that P(X > x0) = .0505.
7. An inspection of a certain piece of machinery takes place every day between the hours of operation 8am to 6pm. The time of inspection can occur any time during this time period with exactly equal likelihood. (Hint: uniformly distributed from 8 (8am) to 18 (6pm))
a. Find the mean time this will occur, along with its variance and standard deviation.b. Find the probability the inspection will take place between 12 and 4.c. Find the time that represents the 90th percentile for this random variable.d. Find a k value such that the probability that the inspection takes place between 10am and k is .4.
8. Suppose that GPA of high school students in Columbia MO follows an approximately normal distribution with a mean GPA of 3.15 and a standard deviation of 0.4.
a. What is the probability that a random student who has a GPA higher than 3.5?
b. If you take a random sample of 15 students what is the probability that the average of the sample will be between 3 and 3.3?
9. Suppose that a random variable is uniformly distributed between 5 and 20 minutes.a. What is the mean of the random variable?b. What is the standard deviation of the random variable?c. Now, suppose a random sample of size n = 30 is selected from this distribution. What is the probability
that the mean of the sample will be between 11 and 14.5? (Hint: using the central limit theorem)