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#5. Test of hypothesis: Book problems, section 9.1: 1, 3, 6, 8, 11, 13, 14 1. Consider the hypothesis H0: God exists. State the appropriate alternative hypothesis, (b) What is a type II error, (b) what is a type II error? 2. Which of the following statements about hypothesis testing are true? a. A type I error occurs if H0 is rejected when it is true b. A type II error occurs if H0 is rejected when it is true c. Type I errors are always worse than type II errors. A only, b only, c only, a and c only b and c only 3. You are given the following about a random sample: Y=X1+X2+…+Xn where the sample size n=25, and the random variables are i.i.d. Each Xi has the Poisson distribution with parameter λ. H0: λ=0.1, H1: λ<0.1. The critical region is C={Y≤3}. Calculate the significance level of the test. 4. You are given a random sample from a normal loss distribution: the sample mean is 42,000, the sample standard deviation is 8,000, there are 25 loss observations in the sample. Using a two sided test with H0: µ=45,000 and H1: µ≠45,000, at what level of α would you reject the null hypothesis. A) α=.01, b) does not reject at α=.01, but reject at α=.02, c) does not reject at α=.02, but reject at α=.05, d) does not reject at α=.05, but reject at α=.1, e) do not reject at α=.1. 5. A random sample of 21 observations from a normal distribution yields the following result: sample mean=3.5, sample variance=0.6156, n=21. You are testing the hypothesis H0: µ=3 vs H1: µ≠3. Calculate the p-value. 6. A survey claims that the average cost of a hotel room in Tulsa Oklahoma is less than $49.21. In order to test this claim, a researcher selects a sample of 26 hotel rooms and finds an average cost equal to 47.37, with a sample standard deviation of 6.42. Assume the population random variable to be normal. At α=.05 comment on the claim. 7. Drivers are classified as “good” or “bad”. Results of the classification are assumed to be binomially distributed with probability of being “good” equal to p. A sample consists of 100 drivers. Determine the critical value for testing the hypothesis p<0.5 with significance level α of at most 0.05 using normal approximation. 8. Aaron was shooting his Koosh basketball into the hoop above his door. He claims that he is an 80% shooter. In a sample Aaron made 25 of 36 shoots. Comment on his claim at the 5% level. 9. X1,X2,…,X11 is a random sample of size 11 from a normal distribution with mean µ1 and variance σ12, with a sample mean=2, and sample variance=4. Y1,Y2,…,Y13 is a random sample of size 13 from a normal distribution with mean µ2 and variance σ22, with a sample mean=1.5, and sample variance=5. H0: µ1= µ2, vs H1: µ1> µ2. The variance is unknown but it is assumed to be the same between the samples. Calculate the test statistic to evaluate the null hypothesis and compute the p-value. 10. X1,X2,…,X11 is a random sample of size 11 from a normal distribution with mean µ1 and variance σ12, with a sample mean=10, and sample variance=4. Y1,Y2,…,Y11 is a random sample of size 11 from a normal distribution with mean µ2 and variance σ22, with a sample mean=9, and sample variance=12. H0: µ1= µ2, vs H1: µ2> µ1. The underlying variance is assumed to be the same for both distributions. Let t be the sample t-statistic used to test the difference of the means from the two normally distributed random samples. Let T be the critical value at α=.05 . Calculate |t-T|. 11. X1,X2,…,X8 is a random sample where X is normally distributed, the sample mean =4, and sample variance=16. Y1,Y2,…,Y9 is a random sample, where Y is normally distributed, with a sample mean=3, and sample variance=9. H0: σ12= σ22, vs H1: σ12> σ22. Calculate the test statistic f, and the critical value F, then find |f-F|. 12. The independent random variables X,Y are from separate normal distributions. Random samples from each distribution are shown below: X Y 1 5 9 10 17 36 1 4 5 5 15 Using the sample data and a 5% significance level, which of the following are true statements regarding the hypothesis testing? a) Reject H0: σX2=50 in favor of Ha: σX2>50 b) Reject H0: σy2=50 in favor of Ha: σY2<50 c) Reject H0: σX2= σy2 in favor of Ha: σX2> σy2 I. a) only II. b) only III. a) and b) IV. a) and c) V. b) and c)