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- Let X be a random variable with a uniform distribution over the interval [-3,6]. Determine the mean of X. [The answer should be a number rounded to five decimal places, don't use symbols such as %]50. Let X₁, X2, ..., X50 be i.i.d. with E(X) = 25 and V(X₂) = 10. Consider the random variable defined by 50 X₁ the sample average X = Σ Apply the Central Limit Theorem to answer the following: n i=1 (a) What is the approximate distribution of X (include parameter(s))? (b) Approximate P(X> 25.5). (c) Approximate P(|X-25 <0.8). (d) Approximate the 0.975 quantile (97.5th%tile) of X.Let Zı, Z2, ... Zn represent a random sample of size n from a standard normal distribution. f n= 5; X= Z + Z+ Z and Y=Zi+Zs. What is the distribution of 2X/3Y? (Note: you must either prove or justify your argument using known theorems)
- Let X be a random variable with a uniform distribution over the interval [-6,5]. Compute P(X > -0.4) (The answer should be a number rounded to five decimal places, don't use symbols such as % Compute Fx(-2.2). (The answer should be a number rounded to five decimal places, don't use symbols such asSuppose that X~N(5000,400). You're considering taking a simple random sample and calculating the average of your sample. The Central Limit Theorem tells us that the distribution of all possible sample averages is Normally distributed with a mean of 5000. If you took a sample of size n = 10000, what is the standard deviation of that sample?3) Suppose test scores are measured by the Gaussian Normal Distribution N(X,75,10) calculate the following. a) Pr(80 < X < 90) b) Pr(50 < X < 70) c) The 95 th percentile
- The average wait time to get seated at a popular restaurant in the city on a Friday night is 12 minutes. Is the mean wait time greater for men who wear a tie? Wait times for 13 randomly selected men who were wearing a tie are shown below. Assume that the distribution of the population is normal. 13, 13, 11, 13, 13, 13, 13, 10, 12, 13, 13, 10, 13 What can be concluded at the the αα = 0.05 level of significance level of significance? The null and alternative hypotheses would be: H0 = (p,u) (<,>,=,not equal) to _____ H1 = (p,u) (<,>,=,not equal) to _____ The test statistic (t,z) = ___ The p-value = _____ Thus, the final conclusion is that ... The data suggest that the population mean wait time for men who wear a tie is not significantly more than 12 at αα = 0.05, so there is statistically insignificant evidence to conclude that the population mean wait time for men who wear a tie is more than 12. The data suggest the population mean is not significantly more than 12…Suppose that X~N(500,100). You're considering taking a simple random sample and calculating the average of your sample. The Central Limit Theorem tells us that the distribution of all possible sample averages is Normally distributed with a mean of 500. What is its standard deviation, if you took a sample of size n = 10000?The random variable x has a normal distribution with mean 50 and variance 9. Find the value of x, call it x0, such that: a) P(x ≤ xo) = 0.8413 b) P(x > xo) = 0.025 c) P(x > xo) = 0.95 d) P(41 ≤ x ≤ xo) = 0.8630
- Q4: Let i denote the mean of a random sample of size 25 from a gamma type distribution with a = 4 and p > 0. Then 1. 1. E(x) = 48 and Var(x) = 48² O True O False 2. I~N (4B, 4B²) 25 2. O True O FalseLet x be a random variable that represents red blood cell count (RBC) in millions of cells per cubic millimeter of whole blood. Then x has a distribution that is approximately normal, For the population of healthy female adults, suppose the mean of the x distribution is about 4.64, Suppose that a female patient has taken six laboratory blod tests over the past several months and that the RBC count data sent to the patient's doctor are as follows. 4.9 4.2 4.5 4.1 4.4 4.3 () Use a calculator with sample mean and standard deviation keys to find x and s. (Round your answers to two decimal places.) C) Do the given data indicate that the pooulation mean RBC count for this patient is lower than 4.647 Use a- 0.05. (a) What is the level of significance? State the nul and alternate hypotheses. O Hn: 4- 4.64; H: 4.64 O Ho: H> 4.64; H:u- 4.64 O Hn: H- 4.64; H: H 4.64 (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. O The standard normal,…8. Let X and Y be the means of two independent random samples, each of size n, from the respective distributions N(μ₁,0²) and N(μ₂,02), where the common variance is known. What is the value of n rounded to the closest integer so that P(X - Y -0/4 < µ₁ − µ₂ < Ñ -Ỹ + σ/4) = 0.90 ? a) 85 b) 87 c) 89 d) 173 e) 175 f) 53 g) 55
