Conduct the stated hypothesis test for μ 1− μ 2. μ 1− μ 2. Assume that the samples are independent and randomly selected from normal populations with equal population variances ( σ 12= σ 22)( σ 12= σ 22). H0 : μ 1− μ 2=0 H1 : μ 1− μ 2 ≠ 0 α =0.1 n1=16 x̄ 1=3,545 s1=305 n2=16 x̄ 2=3,769 s2=286 a. Calculate the test statistic.
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| H0 : μ 1− μ 2=0 | H1 : μ 1− μ 2 ≠ 0 | α =0.1 |
| n1=16 | x̄ 1=3,545 | s1=305 |
| n2=16 | x̄ 2=3,769 | s2=286 |
=
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- In a Nicd battery, a fully charged cell is composed of nickelic hydroxide. Nickel is an element that has multiple oxidation states. Assume the following proportions of the states: Nickel Proportions Charge Found 0.15 +2 0.35 +3 0.33 +4 0.17 Determine the mean and variance of the nickel charge. a. μ= Ε(Χ) = 2.37 o2 = V (X) = 1.47 Ο b. μ= Ε(Χ) 2.29 o = 1.53 |c. None among the choices O d. u = E(X) = 2.29 o² = V (X) = 1.53Conduct the stated hypothesis test for μ 1− μ 2. μ 1− μ 2. Assume that the samples are independent and randomly selected from normal populations. H0 : μ 1− μ 2=0H0 : μ 1− μ 2=0 H1 : μ 1− μ 2 ≠ 0H1 : μ 1− μ 2 ≠ 0 α =0.05 α =0.05 n1=36n1=36 x̄ 1=2,263 x̄ 1=2,263 σ 1=159.3 σ 1=159.3 n2=32n2=32 x̄ 2=2,315 x̄ 2=2,315 σ 2=171.5 σ 2=171.5 Standard Normal Distribution Table a. Calculate the test statistic. z=z= Round to two decimal places if necessary b. Determine the critical value(s) for the hypothesis test. + Round to two decimal places if necessary c. Conclude whether to reject the null hypothesis or not based on the test statistic. Reject Fail to RejectConduct the stated hypothesis test for μ 1− μ 2. μ 1− μ 2. Assume that the samples are independent and randomly selected from normal populations with equal population variances ( σ 12= σ 22)( σ 12= σ 22). H0 : μ 1− μ 2=0�0 : μ 1− μ 2=0 H1 : μ 1− μ 2 < 0�1 : μ 1− μ 2 < 0 α =0.05 α =0.05 n1=30�1=30 x̄ 1=8.61 x̄ 1=8.61 s1=1.39�1=1.39 n2=22�2=22 x̄ 2=9.37 x̄ 2=9.37 s2=1.23�2=1.23 T-Distribution Table a. Calculate the test statistic. t= Round to three decimal places if necessary b. Determine the critical value(s) for the hypothesis test. = Round to three decimal places if necessary c. Conclude whether to reject the null hypothesis or not based on the test statistic. Reject Fail to Reject
- 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…Let x p X2 ............ X be a random sample drawn from a given population with mean u and variance o2 show that sample mean x is an unbiased estimator of population mean µ, E ( x ) = µ.Test whether mu 1 less than mu 2μ1<μ2 at the alphaα=0.01 level of significance for the sample data shown in the accompanying table. Assume that the populations are normally distributed. Determine the null and alternative hypothesis for this test. A. H0:mu 1<mu 2μ1<μ2 H1:mu 1= mu 2μ1=μ2 B. Upper H 0 :H0:mu 1 equals mu 2μ1=μ2 Upper H 1 :H1:mu 1 not equals mu 2μ1≠μ2 C. Upper H 0 :H0:mu 1 not equals mu 2μ1≠μ2 Upper H 1 :H1:mu 1 less than mu 2μ1<μ2 D. Upper H 0 :H0:mu 1 equals mu 2μ1=μ2 Upper H 1 :H1:mu 1 less than mu 2μ1<μ2 Your answer is correct. Detemine the P-value for this hypothesis test. Pequals=nothing (Round to three decimal places as needed.)
- 8. Let y be a normal random variable with a constant mean E(y) = 4, constant variance var(y.) = o², and a covariance cov(y, y;) = 0 for t# j. Consider the sample mean j = E %3D A. Show that the variance of the sample mean y is .Suppose a random sample of size 10 is taken from a normally distributed population, and the sample mean and variance are calculated to be x¯=46.1x¯=46.1 and s2=4.5s2=4.5respectively. Use this information to test the null hypothesis H0:μ=50H0:μ=50versus the alternative hypothesis HA:μ≠50HA:μ≠50, at the 5% level of significance. a) What is the value of the test statistic tt, for testing the null hypothesis that the population mean is equal to 50?Suppose Y₁, Y2,…, Yn is an iid sample from a beta (a, ß) distribution with a = ß = 0, 0 > 0 a. Use the Factorization Theorem to show that n T = []Y;(1 — Y;) i=1 is a sufficient statistic. b. The sample variance n 1 = ΣΥ; - Y)2 n 1 i=1 is an unbiased estimator of the population variance 1 4(20 + 1)* Is S² the MVUE of o2? Explain.
- 10. Let Y₁, Y₂,..., Yn be a random sample of size n from a gamma distribution with parameters and a both being 1. Is Y a pivotal quantity? Show why and why not is a pivotal quantity. 11. Consider a random sample Y₁, Y₂, ..., Yn from a normal population Y~N(u,0²) where the population variance and mean are unknown. We want to construct a Σ(X₁-X)² Show 0² 100(1a)% confidence interval for the population variance if: whether or not is a pivotal quantity and construct a 100(1-a) confidence interval.1.2. Let X and Y be independent standard normal random variables. Determine the pdf of W = x² + y². Find the mean and the variance of U = W.3-4.3 Arandom variable X has a variance of 9 and a statistically independent random variable Y has a variance of 25. Their sum is another random variable Z = X + Y. Without assuming that either random variable has zero mean, find a) the correlation coefficient for X and Y b) the correlation coefficient for Y and Z c) the variance of Z.