a. Answer TRUE/FALSE: 1. If the p-value for a test is 0.036, the null hypothesis can be rejected at the ? = 0.05 level of significance. 2. In a formal test of hypothesis, ? is the probability that the null hypothesis is incorrect. 3. If the p-value is very small for a test to compare two population means, the difference between the means must be large. 4. Power is always computed by assuming that the null hypothesis is true. 5. If 0.01 < p-value < 0.025, the null hypothesis can always be rejected at the ? = 0.02 level of significance. 6. When developing a likelihood ratio test, it is possible that L(??) > L(??0). 7. The noncentrality parameter for the noncentral t distribution can be negative. 8. The unbiased estimator of ?² in simple regression with an intercept follows ?((n?2)/2, 2?²/(n?2)). Assume the assumptions hold and normally distributed error terms. b. For each of the following questions please answer: could happen, impossible, or certainly. 1. Based on a sample of n = 131 yes-or-no values, the 95% confidence interval for the population proportion p was found to be 0.711 ± 0.078. 2. Also, with the same data the null hypothesis H0: p = 0.785 (tested against the alternative Ha: p ? 0.785) was rejected at the 5% level of significance. 3. The hypothesis H0: ? = 1.8 was tested against the alternative Ha: ? ? 1.8 at the 0.05 level of significance, using a sample size of n = 85. 4. Unknown to the statistical analyst, the true value of ? was 1.74 and yet H0 was accepted. 5. A hypothesis test was done at the 5% level of significance, and H0 was not rejected. With the same data using the 1% level of significance, H0 was not rejected. 6. A hypothesis test was done at the 5% level of significance, and H0 was rejected. With the same data, but using the 1% level of significance, H0 was not rejected. 7. A hypothesis test was done at the 5% level of significance, and H0 was not rejected. With the same data, but using the 1% level of significance, H0 was rejected. 8. A sample of size n = 36 was taken from a population that has a known population mean ? = 3.5 and standard deviation ? = 1.71 yet the 95% confidence interval that was constructed failed to cover 3.5.
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036, we compare it to the significance level α = 0.05. If the p-value is less than α, we reject the null hypothesis. Therefore, the statement is TRUE. Show more…
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PROBLEM #1: For each of the following statements, determine whether it is true or false. Label "T" if it is true, otherwise label "F". a. The null hypothesis is the claim that is initially assumed to be true while the alternative hypothesis is the assertion that is contradictory to the null hypothesis. They are two competing hypotheses. b. Depending on the form of the alternative hypothesis, we have three different forms of hypothesis tests, namely, upper-tailed, lower-tailed, and two-tailed tests. c. A test statistic and a rejection region are two basic ingredients of a hypothesis test. d. For a hypothesis test, we have two results: rejecting H0 and failing to reject H0. e. P-value is the probability of obtaining results as extreme as the observed results of a statistical hypothesis test, assuming that the null hypothesis is correct. f. For a hypothesis test, if the calculated P-value is less than or equal to the specified significance level, we should reject the null hypothesis. g. In a hypothesis test, Type I error is the error made when the null hypothesis is rejected when in fact the null hypothesis is true; Type II error is the error made when the null hypothesis is not rejected when it is false. h. For a hypothesis test regarding the population mean, we should use t-test if the underlying distribution of the observations is normal, the sample size is not large enough, and the population variance is unknown. However, if we know the population variance, we should use z-test. i. For a z-test regarding the population mean, if Ha : μ > μ0 is the alternative hypothesis, then the rejection region should be z ≥ zα where zα is the z critical value associated with the significance level α. j. For a t-test regarding the population mean, if Ha : μ > μ0 is the alternative hypothesis, then the rejection region should be t > tα,n−1 where tα,n−1 is the t critical value associated with the significance level α (suppose that the given sample is of size n).
Madhur L.
1.) A critical region describes low probability outcomes under a false null hypothesis. True False 2.) To maximize the likelihood of rejecting a null hypothesis, the test statistic should be composed of a large numerator and a small denominator. True False 3.) If the null hypothesis is true, then a sampling distribution composed of random samples drawn from a population will: A. have the same variability as the population. B. be more dispersed than the population. C. always have a mean value that is different from the population mean. D. be centered at the value provided by the null hypothesis. 4.) Given: Ho ≤ 10; standard error = 3.125 If the value of the test statistic equals: z = -2.00, what conclusion should you make regarding the null hypothesis? A. Fail to reject B. Reject at alpha equals 5%, but fail to reject at alpha equals 1% C. Reject at alpha equals 5% D. Reject at alpha equals 2.5% 5.) Assuming that the null hypothesis is true, and holding alpha level constant (e.g., 5%), a one-tailed test will always result in a greater likelihood of rejecting the null hypothesis, relative to a two-tailed test. True False 6.) Two samples are drawn from a population. Both are used in separate, two-tailed hypothesis tests with alpha levels of 1%. The first sample generates a test statistic of z = +3. The second sample generates a test statistic of z = -2.9. What is the correct conclusion regarding the status of the null hypothesis? A. Only the first sample results in rejection of the null hypothesis. B. The first sample results in rejection of the null hypothesis, whereas the second sample only results in null hypothesis rejection if the hypothesis test is converted to a one-tailed test. C. The samples cancel each other out; thus, the null hypothesis is preserved. D. Both samples result in rejection of the null hypothesis.
Chai S.
(i) What is the level of significance? State the null and alternate hypotheses. (ii) Check Requirements What sampling distribution will you use? What assumptions are you making? What is the value of the sample test statistic? Compute the corresponding $z$ or $t$ value as appropriate. (iii) Find (or estimate) the $P$ -value. Sketch the sampling distribution and show the area corresponding to the $P$ -value. (iv) Based on your answers in parts (i)-(iii), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level a? (v) Interpret your conclusion in the context of the application.Note: For degrees of freedom $d . f$. not in the Student's $t$ table, use the closest $d . f$. that is smaller. In some situations, this choice of $d . f .$ may increase the $P$ -value a small amount, and therefore produce a slightly more "conservative" answer.Answers may vary due to rounding. wanagement: Intimidators and Stressors This problem is based on information regarding productivity in leading Silicon Valley companies (see reference in Problem 21 ). In large corporations, an "intimidator" is an employee who tries to stop communication, sometimes sabotages others, and, above all, likes to listen to him-or herself talk. Let $x_{1}$ be a random variable representing productive hours per week lost by peer employees of an intimidator.$$\begin{array}{llllllll}x_{1}: & 8 & 3 & 6 & 2 & 2 & 5 & 2\end{array}$$. A "stressor" is an employee with a hot temper that leads to unproductive tantrums in corporate society. Let $x_{2}$ be a random variable representing productive hours per week lost by peer employees of a stressor.Use a calculator with mean and standard deviation keys to verify that $\bar{x}_{1}=4.00, s_{1} \approx 2.38$ $\bar{x}_{1}=5.5,$ and $s_{2} \approx 2.78$ (a) Assuming that the variables $x_{1}$ and $x_{2}$ are independent, do the data indicate that the population mean time lost due to stressors is greater than the population mean time lost due to intimidators? Use a $5 \%$ level of significance. (Assume that the population distributions of time lost due to intimidators and time lost due to stressors are each mound-shaped and symmetrical.) (b) Find a $90 \%$ confidence interval for $\mu_{1}-\mu_{2}$. Explain the meaning of the confidence interval in the context of the problem.
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