Suppose that Y₁, Y₂, ..., Y₁ is a random sample from a normal distribution with parameter mean 0 and variance 8, say N(0,0). If Mr Masela and Mr Sepuru desire to use 5% level of significance to test the null hypothesis Ho: 0 = 1 against alternative hypothesis H₁:0 > 1. i) Show that the uniformly most powerful test of their hypothesis rejects H, if Σ₁² ≥c, where c is a constant that solves the probability equation 0.05 = P1Y² ≥c|0 = 1). ii) What will be the value of c in part a) if n = 15.
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- The farmers' association of the Amatole region in the Eastern Cape wants to report on the average milk production in this region. It has been shown that the milk production in this region is normally distributed with mean u and variance o?. Suppose a random sample of farms,Y1,Y2, Y3, Y4 is taken from the region. Consider the following estimators of 0: Ô, = Ÿ and Ôô, = 2Y2+2Y4 _ 1. Identify the parameter of interest. 2. You have been presented with two estimators of the parameter of interest. (a) Find the expected value of each of the estimators. (b) Find the variance of each of the estimators. (c) Based on your calculations in (a) and (b), which estimator would you choose and why?Suppose that a simple random paired sample is taken from two populations to compare their means. Further suppose that the distribution of the paired-difference variable has a symmetric distribution. Under what conditions would the paired t test be preferable to the paired Wilcoxon signed-rank test? Explain your answer.A research tea believes that the weight of Grizzly adule male bears are normally distributed. A simple random saple of 23 Grizzly male bears yielded a mean weight of 277kg and a variance of 121. Let μ denote the mean weight for Grizzly adult male bears. Suppose the research teams wants to test, at the 5 % level of significance, whether the mean weight for Grizzly adult male bears is greater than 271kg. Suppose the assumptions to this hypothesis test are satisifed. d) what is the observed test statistic for hypothesis test? e) What is the correct conclusion for this hypothesis test? Since the observed test statistic is greater than the critical value we reject do not H0, at the 5% level of significance Since the observed test statistic is greater than the critical value we reject H0 at the 5% level of significance Since the observed test statistic is less than the critical value we do not reject H0 at the 5% level of significance Since the observed test statistic is less than the…
- One sample with n =8 has a S2 = 6 and a second sample n = 8 scores has a S2 = 10. If the pooled variance is computed for these two samples, then the value obtained will beA random sample of size n = 19 observations was drawn from a normal population with variance o? = 3.1. The sample mean was = 22.5. The researcher's interest was to defend that the population mean u > 25. (i) Write the null and alternative hypotheses for the test. (ii) Write the test statistic and any assumptions if you are making. (ii) Write the distribution of the test statistic under the null hypothesis.Let x1, X2, ..., Xn be a random sample from a normal population with unknown mean u and an unknown variance o?. At a given significance level a, derive the generalized likelihood ratio test for Ho : o 5. versus Completely specify the test when a = population: 0.05 and n = 26. Does the power function of the test depend on the mean of the The following `answers" have been proposed. %3B 26 (a) For a = 0.05 and n = 26 the test rejects Ho when E(X; – X„)² > 941.25. The power function of the test indirectly depends on the population mean u via = 7. (b) For a = 0.05 and n = 26 the test rejects Ho when E(X; – Xn)² > 941.25. The power function of the test does not depend on the population mean µ. (c) For a = 26 ,26 0.05 and n = 26 the test rejects Ho when E(X; – Xn)² > 188.25. The power function of the test does not depend on the population mean µ. (d) For a = 0.05 and n = 26 the test rejects Ho when E(X; – X„)² > 188.25. The power function of the test indirectly depends on the population mean…
- A research tea believes that the weight of Grizzly adule male bears are normally distributed. A simple random saple of 23 Grizzly male bears yielded a mean weight of 277kg and a variance of 121. Let μ denote the mean weight for Grizzly adult male bears. Suppose the research teams wants to test, at the 5 % level of significance, whether the mean weight for Grizzly adult male bears is greater than 271kg. Suppose the assumptions to this hypothesis test are satisifed. a) What are the null and alternative hypothesis, and level of significance, for this hypothesis test? b) What is the distribution of the test statistic for this hypothesis test? c)What is the decision rule for this hypthesis test?Suppose that the variable under consideration is normally distributed on each of two populations and that the population standard deviations are equal. Further suppose that you want to perform a hypothesis test to decide whether the populations have different means, that is, whether μ1 = μ2. If independent simple random samples are used, identify two hypothesis-testing procedures that you can use to carry out the hypothesis test.A random sample selected from a normal population with mean variance o gave the values 25, 31, 23, 33, 28, 36, 22. 26. Give the point estimators for u and o2 and find their point estimates. and