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- 1a. A random variable Z is normally distributed with mean 0 and standard deviation 1. It is known that P(Z 2.4) = b. This is shown in the following diagram. a -1.6 2.4 Find P(-1.6The random variable X is related to e2. Find P(X>0.4).X is a discrete random variable with the following PMF: P(X = -9.2) = 0.25 P(X = -2.5) = 0.25 P(X = 7.7) = 0.5 Find the standard deviation of X.Assume the random variable X is normally distributed, with mean μ=47 and standard deviation σ=8. Find the 14th percentile.Assume the random variable X is normally distributed with mean μ=50 and standard deviation σ=7. Find the 77th percentile. The 77thpercentile is:Agricultural scientists are testing a new chicken feed to see whether it increases the number of eggs laid. The scientists divided a flock in half and gave half of the chickens the new feed, while the other half were given a regular feed. The eggs laid in a one-year period for a random sample of 10 chickens using the new feed and 10 chickens using the regular feed are recorded. Assume that the population standard deviation of number of eggs laid is 12 eggs per year for both groups and that the number of eggs laid per year is normally distributed. Let the chickens given the new feed be the first sample, and let the chickens given the regular feed be the second sample. The scientists conduct a two-mean hypothesis test at the 0.05 level of significance to test if there is evidence that the new feed increases the number of eggs laid. (a) Which answer choice shows the correct null and alternative hypotheses for this test? Select the correct answer below: H0:μ1=μ2;…Suppose that you randomly draw one card from a standard deck of 52 cards. After writing down which card was drawn, you replace the card, and draw another card. You repeat this process until you have drawn 16 cards in all. What is the probability of drawing at least 7 clubs? For the experiment above, let X denote the number of clubs that are drawn. For this random variable, find its expected value and standard deviation. E(X) = σ =In each of Exercises, we have provided a scenario for a hypothesis test for a population mean. Decide whether the z-test is an appropriate method for conducting the hypothesis test. Assume that the population standard deviation is known in each case. A normal probability plot of the sample data shows no outliers and is quite linear. The sample size is 12.The random variables X is the best described by a normal distribution with = 30 and a = 6. Find the z-score that corresponds to the following X values.:1. X = 212 X = 26.5The answer choices for expected value of x is E(X): 0.561 0.531 0.550 0.697 The answer choices for the standard deviation of the random variable X, denoted σ,: 0.80.60.70.9 and which player is more likely to help the team score runsThe mean is μ = 15 and the standard deviation is σ = 0.9.Find the probability that X is between 14.3 and 16.1.A random variable X is normally distributed. It has a mean of 270 and a standard deviation of 31. For a sample of size 21, the mean is 270 and the standard deviation is 6.76. 1. Find the probability of getting a sample mean greater than 277.85 when taking a sample of size 21 from this population.