3. (Sample size determination for hypothesis testing) Suppose that X₁,..., Xnknown. We would like to testi.i.d.~N(μ,02) and σ² isWe consider the test statisticHo Ho H₁p>μ-and the critical region at significance level aT=X{ }C =; To+Za(a) We write (') to be the probability of type II error given the true value of μ='. Prove that▷ (ZaB(µ') = za +where (z) = P(Z) is the cdf of Z~ N(0, 1).μο - μ'(b) We would like to determine the sample size n so that the probability of type I error is a and theprobability of type II error is ẞ. Using part (a), prove that'σ (za +23)2n =- Orf(c) Let μ denote the true average tread life of a certain type of tire. Consider testing Hoμ= 25,000versus H₁ >25,000 from a normal population distribution with σ = 1300. Assuming the truevalue of is 27,000. What is the sample size n so that the probability of type I error isa = 0.05 and the probability of type II error is B = 0.025?

Answered: 3. (Sample size determination for…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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A population of values has a normal distribution with μ=150μ=150 and σ=18.8σ=18.8. You intend to draw a random sample of size n=207n=207.Find the probability that a sample of size n=207n=207 is randomly selected with a mean less than 149.3.P(¯xx¯ < 149.3) = Enter your answers as numbers accurate to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.
A population of values has a normal distribution with μ=128.3μ=128.3 and σ=87σ=87. You intend to draw a random sample of size n=242n=242.Find the probability that a sample of size n=242n=242 is randomly selected with a mean between 121.6 and 127.2.P(121.6 < ¯xx¯ < 127.2) = Enter your answers as numbers accurate to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.
is being approxin Let X1,..., Xn be an iid with mean, µ = E(X), and variance, o2 = Var(X). True or False, and briefly explain: The sample average, X, can be expected to be closer and closer to µ as n gets bigger and pigger. Confidence Intervals
A population of values has a normal distribution with μ=244.3μ=244.3 and σ=12.9σ=12.9. You intend to draw a random sample of size n=32n=32.Find the probability that a sample of size n=32n=32 is randomly selected with a mean greater than 245.7.P(M > 245.7) = Enter your answers as numbers accurate to 4 decimal places.
Parts (a) and (b) are both related to normal distribution but are not directly connected to each other. Please complete both parts. 12. (a) Find zo.8729- Assume that X is a normal random variable, with u= -1 (b) and o = 2. Find x such that P(X > x)= 0.06.
11
You wish to test Ho: μ₁ μ2 versus Ha:μ₁ μ₂ at a = = 0.10. You obtain a sample of size n₁ 1 = 58.2 and a standard deviation of $₁ = 20.9 from the first population. You obtain a sample 40.2 and a standard deviation of $2 = 5.6 from the second 9 with a mean of 2 a mean of of size n2 population. = = What is the test statistic? = = 6 with Assume that the data come from normal populations with equal variances. Round your answers to three decimal places, and round any interim calculations to four decimal places.
Solve the problem and draw the normal curve with the shaded region representing the given scenario in the problem. The accountants of a known auditing firm have a mean daily salary Php 850 with a standard deviation of Php 82. Find the median and the mode, and compare it with the given mean value. Determine the intervals within 1, 2 and 3 standard deviations from the mean, that is, the µ ± 1σ, µ ± 2σ, µ ± 3σ. Verify that the empirical rule holds that is, approximately, 68%, 95%, and 99% of the data lie within ± 1σ, ± 2σ, ± 3σ units of the mean, respectively. If an accountant from the firm is chosen at random, what is the probability that he /she earning Php 950 to Php 1,000? What percentage of the accountants will fall in between the salary of Php 300 and Php 500?
A population of values has a normal distribution with μ=184.5μ=184.5 and σ=15.5σ=15.5. You intend to draw a random sample of size n=68n=68.Find P40, which is the mean separating the bottom 40% means from the top 60% means.P40 (for sample means) = Enter your answers as numbers accurate to 1 decimal place. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.
1. In the past, a chemical company produced 880 pounds of a certain type of plastic per day. Now, using a newly developed and less expensive process, the mean daily yield of plastic for the first 50 days of production is 871 pounds; the standard deviation is 21 pounds. Do the data provide sufficient evidence to indicate that the mean daily yield for the new process is less than that of the old procedure? (Use α=0.05) (d) Conclusion of the test above is Reject the null hypothesis and the mean daily yield for the new process is less than that of the old procedure. Reject the null hypothesis and the mean daily yield for the new process is not less than that of the old procedure. Do not reject the null hypothesis and the mean daily yield for the new process is less than that of the old procedure. Do not reject the null hypothesis and the mean daily yield for the new process is not less than that of the old procedure.
Vehicle speeds at a certain highway location are believed to have approximately a normal distribution with mean µ = 50 mph and standard deviation σ = 5 mph. The speeds for a randomly selected sample of n = 25 vehicles will be recorded. (a) Give numerical values for the mean and standard deviation of the sampling distribution of possible sample means for randomly selected samples of n = 25 from the population of vehicle speeds. Mean = s.d.(x) = (b) Use the Empirical Rule to find values that fill in the blanks in the following sentence. For a random sample of n = 25 vehicles, there is about a 95% chance that the mean vehicle speed in the sample will be between and mph. (c) Sample speeds for a random sample of 25 vehicles are measured at this location, and the sample mean is 59 mph. Given the answer to part (b), explain whether this result is consistent with the belief that the mean speed at this location is μ = 50 mph. A sample mean of 59 mph (when n = 25) --Select--- be consistent with…

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