Example 1. Assume that the random variable X follows the Normal distribution with mean 75 and standard deviation 10. Use Python to
(a) Compute P(65 < X < 85) and interpret the findings
(b) Compute P(55 < X < 95) and interpret the findings
(c) Compute P(X > 100) and interpret the findings
Example 2. Assume that the random variable X follows the Normal distribution with mean µ and standard deviation σ. Compute
(a) P(µ − σ < X < µ + σ)
(b) P(µ − 2σ < X < µ + 2σ)
(c) P(µ − 3σ < X < µ + 3σ)
(Hint: convert X to standard normal distribution and use Python)
Example 3. For a standard normal Z, compute P(Z < −2)
Example 4. For a standard normal Z, find a point a such that P(Z < a) = 0.05
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How can you find out the following probabilities assuming that you know the mean and standard deviation? P(X <= µ+2σ) = ? P(µ <= X <= µ+3σ) = ? P(|X - µ| <= 3σ) = ? P(|X - µ| > 3σ) = ? Consider two situations. One in which you do not know the probability distribution and another when the distribution is Normal.
Suppose the random variable X follows a normal distribution with mean µ = 84 and standard deviation σ = 20. Calculate each of the following: P(X > 100) P(80 < X < 144) P(124 < X < 160) P(X < 50) P(X > X*) = .0062. What is the value of X*?
If X has a normal distribution with mean μ and standard deviation σ, and Z is the standard normal random variable whose cumulative distribution function is P(Z s Z)-0(Z), then which of the following statements is NOT correct? O E. All of the given statements are not correct
Consider a normal distribution with mean 25 and standard deviation 5. What is the probability a value selected at random from this distribution is greater than 25? (Round your answer to two decimal places.) Assume that x has a normal distribution with the specified mean and standard deviation. Find the indicated probability. (Round your answer to four decimal places.) μ = 14.9; σ = 3.5 P(10 ≤ x ≤ 26) = Need Help? Read It Assume that x has a...
A normal distribution has mean LaTeX: \mu=14μ = 14 and standard deviation LaTeX: \sigma=3σ = 3. Find and interpret the z-score for LaTeX: x=11x = 11.
= X- 4) A normal distribution has mean u = 65 and a population standard deviation o= 20. Find and interpret the z - Score for x = 64. u a) The z - score for x = 64 is 64-65 b) Interpret these results. (Explain): 5) A sample size 28 will be drawn from a population with mean 120 and standard deviation 21. a) Is it appropriate to use the normal distribution to find probabilities for x? yes or...
Recall from class that the standard normal random variable, Z, with mean of 0 and stan- dard deviation of 1, is the continuous random variable whose probability is determined by the distribution: a. Show that f(-2)-f(2) for all z. Thus, the PDF f(2) is symmetric about the y-axis. b. Use part a to show that the median of the standard normal random variable is also 0 c. Compute the mode of the standard normal random variable. Is is the same...
Question 18: a) Compute the moment generating function, MGF, of a normal random variable X with mean µ and standard deviation σ. b) Use your MGF from part a) to find the mean and variance of X.
1. Give examples of sample statistic and population parameter. 2. Give some properties of any normal distribution. 3. What is the total area under a normal distribution curve? 4. What is the mean and standard deviation of standard normal distribution? 5. what percentage of the area under the normal curves lies a) To the right of µ b) Between µ - 3σ and µ + 3σ (within three standard deviation of the mean)
We can now use the Standard Normal Distribution Table to find the probability P(-0.25 sz s 1). 0.05 0.06 0.07 0.08 0.09 -0.2 0.4013 0.3974 0.3936 0.3897 0.3859 0.00 0.01 0.02 0.03 0.04 Using these 1.0 0.8413 0.8438 0.8461 0.8485 0.8531 The table entry for z = -0.25 is 0.00 and the table entry for z = 1 is values to calculate the probability gives the following result. PC-0.25 sz s 1) P(Z < 1) - P(Z 5 -0.25) 10....