Prove the 2-tailed case of the One-Sample t-test, i.e., if X₁, X2,..., Xn is a random sample from a normal distribution with mean and unknown variance o2, then define 71 T= χ-μο S/√n with S2 (X-X). Hence for the test of hypothesis Ho: μ = po n-1 i=1 versus H₁ μμo, at the level of significance o, reject Ho iff either t ≤-ta/2(n-1) or t≥ ta/2(n-1).
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- You wish to test the following claim (HaHa) at a significance level of α=0.002. Ho:p1=p2 Ha:p1≠p2 You obtain 383 successes in a sample of size n1=498 from the first population. You obtain 318 successes in a sample of size n2=443 from the second population. For this test, you should NOT use the continuity correction, and you should use the normal distribution as an approximation for the binomial distribution.What is the test statistic for this sample? (Report answer accurate to three decimal places.)test statistic = What is the p-value for this sample? (Report answer accurate to four decimal places.)p-value = The p-value is... less than (or equal to) αα greater than αα This test statistic leads to a decision to... reject the null accept the null fail to reject the null As such, the final conclusion is that... There is sufficient evidence to warrant rejection of the claim that the first population proportion is not equal to the second population proprtion. There is…You wish to test the following claim (HaHa) at a significance level of α=0.001α=0.001. Ho:p=0.8Ho:p=0.8 Ha:p<0.8Ha:p<0.8You obtain a sample of size n=232n=232 in which there are 177 successful observations. For this test, you should NOT use the continuity correction, and you should use the normal distribution as an approximation for the binomial distribution.What is the test statistic for this sample? (Report answer accurate to three decimal places.)test statistic = What is the p-value for this sample? (Report answer accurate to four decimal places.)p-value =You wish to test the following claim (HaHa) at a significance level of α=0.002. Ho:p1=p2 Ha:p1>p2You obtain 256 successes in a sample of size n1=326 from the first population. You obtain 469 successes in a sample of size n2=686 from the second population. For this test, you should NOT use the continuity correction, and you should use the normal distribution as an approximation for the binomial distribution. What is the p-value for this sample? (Report answer accurate to four decimal places.)p-value =
- A population of values has a normal distribution with μ=244.3μ=244.3 and σ=12.9σ=12.9. You intend to draw a random sample of size n=32n=32.Find the probability that a sample of size n=32n=32 is randomly selected with a mean greater than 245.7.P(M > 245.7) = Enter your answers as numbers accurate to 4 decimal places.Let a = 0.05. (a) Suppose that we sample from a bivariate normal distribution with p = 0.5. Assuming that we observe r = 0.5, how large a sample will be needed to reject Ho: p= 0 in favor of Ho: p=0? (b) Suppose that we sample from a bivariate normal distribution with p = 0.1. Assuming that we observe r = 0.1, how large a sample will be needed to reject Ho: p= 0 in favor of Ho: p0?You wish to test the following claim (Ha) at a significance level of α=0.10 Ho:p1=p2 Ha:p1≠p2You obtain 60.5% successes in a sample of size n1=650 from the first population. You obtain 57% successes in a sample of size n2=258 from the second population. For this test, you should NOT use the continuity correction, and you should use the normal distribution as an approximation for the binomial distribution.What is the test statistic for this sample? (Report answer accurate to three decimal places.)test statistic = What is the p-value for this sample? (Report answer accurate to four decimal places.)p-value = The p-value is... less than (or equal to) αα greater than αα This test statistic leads to a decision to... reject the null hypothesis accept the null hypothesis fail to reject the null hypothesis As such, the final conclusion is that... There is sufficient evidence to warrant rejection of the claim that the first population proportion is not equal to the second…
- You wish to test Ho: μ₁ μ2 versus Ha:μ₁ μ₂ at a = = 0.10. You obtain a sample of size n₁ 1 = 58.2 and a standard deviation of $₁ = 20.9 from the first population. You obtain a sample 40.2 and a standard deviation of $2 = 5.6 from the second 9 with a mean of 2 a mean of of size n2 population. = = What is the test statistic? = = 6 with Assume that the data come from normal populations with equal variances. Round your answers to three decimal places, and round any interim calculations to four decimal places.You wish to test the following claim (Ha) at a significance level of α=0.01. Ho:p1=p2 Ha:p1>p2You obtain 54.3% successes in a sample of size n1=529 from the first population. You obtain 47.9% successes in a sample of size n2=555 from the second population. For this test, you should NOT use the continuity correction, and you should use the normal distribution as an approximation for the binomial distribution.What is the test statistic for this sample? (Report answer accurate to three decimal places.) test statistic = What is the p-value for this sample? (Report answer accurate to four decimal places.) p-value = The p-value is... less than (or equal to) α greater than αYou wish to test the following claim (HaHa) at a significance level of α=0.001 Ho:p1=p2 Ha:p1<p2You obtain 4.7% successes in a sample of size n1=299 from the first population. You obtain 7.8% successes in a sample of size n2=602 from the second population. For this test, you should NOT use the continuity correction, and you should use the normal distribution as an approximation for the binomial distribution.What is the test statistic for this sample? (Report answer accurate to three decimal places.)test statistic = What is the p-value for this sample? (Report answer accurate to four decimal places.)p-value = The p-value is... less than (or equal to) αα greater than αα This test statistic leads to a decision to... reject the null hypothesis accept the null hypothesis fail to reject the null hypothesis As such, the final conclusion is that... There is sufficient evidence to warrant rejection of the claim that the first population proportion is less than the second…
- A simple random sample of size n=40 is drawn from a population. The sample mean is found to be 107.6, and the sample standard deviation is found to be 20.2. Is the population mean greater than 100 at the α=0.025 level of significance? Determine the null and alternative hypotheses. H0: mu equals 100μ=100 mu less than 100μ<100 mu equals 100μ=100 mu greater than 100μ>100 mu less than 107.6μ<107.6 mu equals 107.6μ=107.6 mu greater than 107.6μ>107.6 H1: mu greater than 100μ>100 mu less than 100μ<100 mu equals 100μ=100 mu greater than 100μ>100 mu less than 107.6μ<107.6 mu equals 107.6μ=107.6 mu greater than 107.6μ>107.6 Compute the test statistic. t 0t0 t 0t0 z 0z0 = (Round to two decimal places as needed.) Determine the P-value. The P-value is . (Round to three decimal places as needed.) What is the result of the hypothesis test? the null hypothesis because the P-value is the level of significance.…You wish to test the following claim (Ha) at a significance level of α=0.005 Ho:p1=p2 Ha:p1>p2You obtain 88.3% successes in a sample of size n1=642 from the first population. You obtain 82% successes in a sample of size n2=748 from the second population. For this test, you should NOT use the continuity correction, and you should use the normal distribution as an approximation for the binomial distribution.What is the test statistic for this sample? (Report answer accurate to two decimal places.)test statistic = What is the p-value for this sample? (Report answer accurate to four decimal places.)p-value = The p-value is... less than (or equal to) α greater than α This test statistic leads to a decision to... reject the null accept the null fail to reject the null As such, the final conclusion is that... There is sufficient evidence to warrant rejection of the claim that the first population proportion is greater than the second population proportion. There is not…Consider a paint-drying situation in which drying time for a test specimen is normally distributed with o = 6. The hypotheses H,: µ = 74 and H: µ < 74 are to be tested using a random sample of n = 25 observations. (a) How many standard deviations (of X) below the null value is x = 72.3? (Round your answer to two decimal places.) 1.42 standard deviations (b) If x = 72.3, what is the conclusion using a = 0.004? Calculate the test statistic and determine the P-value. (Round your test statistic to two decimal places and your P-value to four decimal places.) z = p-value = State the conclusion in the problem context. O Do not reject the null hypothesis. There is not sufficient evidence to conclude that the mean drying time is less than 74. O Do not reject the null hypothesis. There is sufficient evidence to conclude that the mean drying time is less than 74. O Reject the null hypothesis. There is sufficient evidence to conclude that the mean drying time is less than 74. O Reject the null…