10. Let x be a binomial random variable with n = 150 and p = 0.4. Use the normal approximation to find the following. a) P(48 < X < 66) μ = np = 60 o = √np(1-p) = 6 P(65.5) - P(47.5) = $(0.917)-0(-2.08) = 0.8024 b) P(X> 69) P(X>69) = 1- P(X = 69.5) = 1−ø(- c) P(X ≤ 70) 70.5-60. 69.5–60. 6 P(X <=70) = P(X = 70.5) = (- -) = = 0.9599 -) = 0.0571
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- 2A normal random variable X is given with a mean = 22 and a standard deviation o = 17. Give an approximate value of the z such that: Pr(X ≤ 1) = 0.375 Give approximate values of an z and y such that: Pr(y ≤X ≤1) = 0.431. NOTE: For this problem, both z AND y must be FINITE so you can't just take y to -00 and solve the problem akin to what you did in (a). Your strategy here should be to find z and y such that Pr(X ≤1) - Pr(X ≤y) gets as close to 0.431 as you want. Read the Z table and play with the arithmetic!Suppose that average male weight in the US is 175 pounds with a standarddeviation of 25 pounds. Suppose you randomly select 1,000 male Americans and ask their weight, and average the 1,000 numbers to compute a sample mean Xn. A. What is the variance of the sample mean Xn? B. Use your answer to part (A), and Chebyshev’s inequality, to come up with a quantitative upper bound for the probability that sample mean Xn is more than a certain distance of 175
- Let m denote margin of error, n sample size and σ standard deviation. m= zα/2(σn) Solve the above equation for n.4. The standard deviation of X, denoted SD(X), is given by SD(X) = Var(X). Find SD(aX + b) if X has variance of o?.Suppose a random sample of size 55 is selected from a population with σ = 8. Find the value of the standard error of the mean in each of the following cases (use the finite population correction factor if appropriate). The population size is infinite (to 2 decimals). The population size is N = 50,000 (to 2 decimals). The population size is N = 5,000 (to 2 decimals). The population size is N = 500 (to 2 decimals).
- How were the answers of 1.0000 and 54.5982 attained?16. Let X, X2 X, be a random sample from a normal distribution, X, N(6, 25), and denote by X and S2 the sample mean and sample variance. Use tables from Appendix C to find each of the following: (a) P[3 < X < T. (b) P[L860< 3(X-6)/S]. (c) PIS <319375]Suppose in a local Kindergarten through 12th grade (K -12) school district, 49% of the population favor a charter school for grades K through 5. A simple random sample of 144 is surveyed. a. Find the mean and the standard deviation of X of B(144, 0.49). Round off to 4 decimal places. O = b. Now approximate X of B(144, 0.49) using the normal approximation with the random variable Y and the table. Round off to 4 decimal places. Y - N( c. Find the probability that at most 81 favor a charter school using the normal approximation and the table. (Round off to z-values up to 2 decimal places.) P(X 75) - P(Y > a (Z > e. Find the probability that exactly 81 favor a charter school using the normal approximation and the table. (Round off to z-values up to 2 decimal places.) P(X = 81) - P(A random sample of size n₁ = 14 is selected from a normal population with a mean of 76 and a standard deviation of 7. A second random sample of size n₂ = 9 is taken from another normal population with mean 71 and standard deviation 11. Let X₁ and X₂ be the two sample means. Find: (a) The probability that X₁ – X₂ exceeds 4. 1 2 (b) The probability that 4.3 ≤ X₁ – X2 ≤ 5.6. Round your answers to two decimal places (e.g. 98.76). (a) i (b) iSuppose in a local Kindergarten through 12th grade (K -12) school district, 49% of the population favor a charter school for grades K through 5. A simple random sample of 144 is surveyed. a. Find the mean and the standard deviation of X of B(144, 0.49). Round off to 4 decimal places. O = b. Now approximate X of B(144, 0.49) using the normal approximation with the random variable Y and the table. Round off to 4 decimal places. Y - N( c. Find the probability that at most 81 favor a charter school using the normal approximation and the table. (Round off to z-values up to 2 decimal places.) P(X 75) - P(Y > a (Z > e. Find the probability that exactly 81 favor a charter school using the normal approximation and the table. (Round off to z-values up to 2 decimal places.) P(X = 81) - P(8. The time (in minutes) that it takes a car mechanic to replace a car battery is a random variable X that follows an exponential distribution with mean 15 minutes. a) Find P(14 < X < 22). Round off your answer to two decimal places. x 22 1 P(14 < X <22) = √₂/1² 15 e 15dx = P(x < 22) — P(x < 14) x 15 122 = e 15 22 1 - e 14 or 0.8 = = 22 14 = −e 15 + e15 = 0.16254≈ 0.16 - (1 – b) Find the 80th percentile. Round off to the nearest whole number. 1 X S 15 14 e 15 t e 15 dt X 0.8 = 1 e 15 X e 15 = = 0.2 14 22 = e 15 e 15 = 0.16 x — = ln (0.2). This gives x = 24.14156 15SEE MORE QUESTIONSRecommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON