B2. Consider the testing problem where we have observed a random sample x₁,...,n fromthe model X₁,..., Xn ~ N(µ, o²), where the X, are independent, and we want to testthe hypothesis Ho: o² = o against the alternative H₁: 0² o. For the test we usethe test statisticny = Σi=1(x₁ - x)²0²where is the sample mean of the ₁.(a) What are the type I errors and type II errors in this testing problem?(b) If Ho is true, what is the distribution of the random variable Y corresponding to y?(c) Assume that n = 20 and we reject Ho if and only if y> 32.852. What is theprobability of a type I error in this case?(d)now that H₁ is true for the test in part (c). Draw a sketch of the type IIerror as a function of o2. (There is no need to calculate explicit values.)-(e) Explain why the test described in part (c) is not a good test for testing the hypoth-esis Ho: 0² o against the alternative H₁: o² o2. How would you improvethe test?

As per our guidelines, we are supposed to answer only 3 sub-parts.In this problem we are performing…

Answered: B2. Consider the testing problem where…

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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1. A study compared the number of tree species in unlogged forest plots to similar plots logged 8 years earlier. Independent random samples of both logged and unlogged plots were analyzed and the number of tree species was recorded. Tables 1 and 2 present the Shapiro-Wilk normality test for both types of forest plots and the Independent Samples t-test results. Use the information presented in these tables to answer the questions. Table 1: Tests of Normality Tests of Normality Kolmogorov-Smirnov Shapiro-Wilk Forest Plot Statistic df Sig. Statistic df Sig. Tree_Species Logged .181 12 .200 .936 12 .444 Unlogged .110 14 .200 .945 14 .480 *. This is a lower bound of the true significance. a. Lilliefors Significance Correction Table 2: Independent Samples Test Independent Samples Test Levene's Test for Equality of Variances t-test for Equality of Means 95% Confidence Interval of the Difference Mean Std. Error F Sig. t df Sig. (2-tailed) Difference Difference Lower Upper Equal variances…
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Test the claim about the difference between two population means mu 1μ1 and mu 2μ2 at the level of significance alphaα. Assume the samples are random and independent, and the populations are normally distributed. Claim: mu 1μ1equals=mu 2μ2; alphaαequals=0.01 Population statistics: sigma 1σ1equals=3.3, sigma 2σ2equals=1.5 Sample statistics: x overbar 1x1equals=17, n1equals=29, x overbar 2x2equals=19, n2equals=28 Determine the alternative hypothesis. Upper H Subscript aHa: mu 1μ1 ▼ not equals≠ greater than or equals≥ less than or equals≤ greater than> less than< mu 2μ2 Determine the standardized test statistic. zequals=nothing (Round to two decimal places as needed.) Determine the P-value. P-valueequals=nothing (Round to three decimal places as needed.) What is the proper decision? A. Reject Upper H 0H0. There is enough evidence at the 1% level of significance to reject the claim. B. Fail to reject Upper H 0H0. There is not…
How do I prove the left equals the right? Using a one-way ANOVA model
Suppose you have a normally distributed population such that o? = 7. Using the sampling distribution of S?, we find that from a random sample of size n = 11, s² = 16.053. Does our population variance seem || reasonable? Note: We will consider our population variance reasonable if our x value falls within the interval x0.975 (df), x'o.025 (df)]. 1) Find the values of the interval [xo.975(df), x²o.025(df)]. x°0.975(df) = (Round to 3 decimals.) x'0.025(df) - (Round to 3 decimals.) 2) Find the value of x that comes from the data you collected in the problem statement. (Round to 3 decimals.) 3) Is the population variance reasonable? Yes O No
My questions pertain to part (a), (b) and (c). Thank you.
= A19 Assume we have observed independent samples ₁,..., 5 and y₁,... 13. The sample averages are x = 12.8 and y 11.2, and the population variances are known to be o² = 0.83 and ² = 0.23. Let μ and uy be the population means. We want to perform a z-test for the hypothesis Ho: x=y, with alternative Ho: My. What is the value of the test statistic for this test?
We test the null hypothesis Ho:μ = 10 against the alternative Ha:μ<10 for a normal population with σ = 4. A random sample of 16 observations is drawn from the population, and we find the sample mean of these observations to be x̅ = 12. The P-value is?
2. Review on estimators. Let Y = (Y1, 2, 3, 4) be a random sample from the normal distribution N(μ, 2²), where the expected value of the normal distribution μER is unknown, and the variance parameter has four (4) as a value. We examine the following parameter μ estimator 1 ₪ (Y) = ± ± (Y₁ + Y₂ + Y3 + 2Y₁) 4 Calculate the bias of the estimator (Y). Is the estimator unbiased? Calculate the variance and the mean squared error (MSE) of the estimator. The problems deal with the confidence intervals (Chapter 5 topics) and especially testing theory (Chapter 6).
4.54 Figure 4.16 shows a randomization distribution for testing Ho 1=2 versus Ha statistic used for each sample is D = x, – x2. In each case, use the distribution to decide which value is closer to the p-value for the observed difference in : H1 # µz. The sample means. (a) The p-value for D = x - I = -2.9 is closest to: %3D 0.01 or 0.25? (b) The p-value for D = x, – x, = 1.2 is closest to: %3D 0.30 or 0.60? 90 80 70 60 50 40 30 20 10 무 -3 -2 -1 1 2 3 Differences Figure 4.16 Randomization distribution for Exercises 4.54 to 4.58 Frequency

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