Let X1 and X2 be independent and identical normal random variables with common mean 3 and standard deviation 1. Consider the sample mean U = Xi+X2 a. What is the moment-generating function my (t) of U? [Hint: Refer to Proposition 3.1, mu (t) = mx,(G)mx, (t) - ] b. What is the distribution of U? c. Find the probability that the observed sample mean is between 2.8 and 3.2.
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- Suppose that X has a Weibull distribution with β = 2 and δ = 2400. Determine the following. a. P(X > 5000) = b. For an exponential random variable with the same mean as the Weibull distribution P(X > 5000) =The random variable X is described by using the gamma distribution with α = 5 and λ = 1. Find the standard deviation of X.Let 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 4.78. Suppose that a female patient has taken six laboratory blood 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 (i) Use a calculator with sample mean and standard deviation keys to find x and s. (Round your answers to two decimal places.) x = s = (ii) Do the given data indicate that the population mean RBC count for this patient is lower than 4.78? Use ? = 0.05. (a) What is the level of significance?State the null and alternate hypotheses. H0: ? < 4.78; H1: ? = 4.78H0: ? = 4.78; H1: ? ≠ 4.78 H0: ? = 4.78; H1: ? < 4.78H0: ? = 4.78; H1: ? > 4.78H0: ? > 4.78; H1: ? = 4.78 (b) What sampling…
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- I need help with parts D and C only. Let x be a random variable representing dividend yield of bank stocks. We may assume that x has a normal distribution with σ = 2.9%. A random sample of 10 bank stocks gave the following yields (in percents). 5.7 4.8 6.0 4.9 4.0 3.4 6.5 7.1 5.3 6.1 The sample mean is = 5.38%. Suppose that for the entire stock market, the mean dividend yield is μ = 4.1%. Do these data indicate that the dividend yield of all bank stocks is higher than 4.1%? Use α = 0.01. (a) What is the level of significance? (Enter a number.)State the null and alternate hypotheses. Will you use a left-tailed, right-tailed, or two-tailed test? H0: μ = 4.1%; H1: μ > 4.1%; right-tailedH0: μ = 4.1%; H1: μ < 4.1%; left-tailed H0: μ > 4.1%; H1: μ = 4.1%; right-tailedH0: μ = 4.1%; H1: μ ≠ 4.1%; two-tailed (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. The Student's t, since n is large with…Let 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 blood 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 (1) Use a calculator with sample mean and standard deviation keys to find x and s. (Round your answers to two decimal places.) x=| S= (ii) Do the given data indicate that the population mean RBC count for this patient is lower than 4.64? Use a = 0.05. (a) What is the level of significance? State the null and alternate hypotheses. Ο Hg: μ= 4.64; H1: μ 4.64 Ο Ηρ: μ> 4.64; H1: μ = 4.64 Ho: u = 4.64; H1: u * 4.64 (b) What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. O…The times to pop a regular bag of microwave popcorn without burning it are Normally distributed with a mean time of 140 seconds and a standard deviation of 20 seconds. The times to pop a mini bag of microwave popcorn without burning it are Normally distributed with a mean time of 90 seconds and a standard deviation of 15 seconds. Suppose two independent random samples, 25 of each, are taken and the mean popping times are calculated. Let R = the popping time of a randomly selected regular-sized bag and M = the popping time of a mini-sized bag. Which of the following best describes the shape of the sampling distribution of ? Normal because both population distributions are Normal uniform because the both sample sizes are less than 30 skewed right because the difference in times cannot be negative skewed left because the sample sizes are less than 30 and the sampling variability is unknown
- Let 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,…There are two independent random variables X and Y. X has mean 4 and standard deviation , and Y also has mean 4 and standard deviation . A random variable W is the difference between X and Y. That is, W = X - Y. Calculate the mean of W.Let the random variable T follow a Student's t distribution with 9 degrees of freedom. State the distribution of the random variable T. Group of answer choices T∼Student′st(9) T∼Student′st(8) T∼Normal(0,1) T∼Student′st(0,1) Compute the probability that T is between −0.1 and 0.6. True or False: The larger the degrees of freedom in a Student's t distribution, the closer to a standard normal distribution the Student's t distribution becomes.