Question

In: Statistics and Probability

Consider the following competing hypotheses and accompanying sample data drawn independently from normally distributed populations. Use...

Consider the following competing hypotheses and accompanying sample data drawn independently from normally distributed populations. Use Table 2. H0: μ1 − μ2 ≥ 0 HA: μ1 − μ2 < 0 x−1 = 256 x−2 = 269 s1 = 37 s2 = 15 n1 = 9 n2 = 9 a-1. Calculate the value of the test statistic under the assumption that the population variances are unknown but equal. (Negative values should be indicated by a minus sign. Round all intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.) Test statistic a-2. Calculate the critical value at the 10% level of significance. (Negative value should be indicated by a minus sign. Round your answer to 3 decimal places.) Critical value a-3. Do you reject the null hypothesis at the 10% level? Yes, since the value of the test statistic is not less than the critical value. No, since the value of the test statistic is less than the critical value. No, since the value of the test statistic is not less than the critical value. Yes, since the value of the test statistic is less than the critical value. b-1. Calculate the value of the test statistic under the assumption that the population variances are unknown and are not equal. (Negative values should be indicated by a minus sign. Round intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.) Test statistic b-2. Calculate the critical value at the 10% level of significance. (Negative value should be indicated by a minus sign. Round your answer to 3 decimal places.) Critical value b-3. Do you reject the null hypothesis at the 10% level? Yes, since the value of the test statistic is not less than the critical value. No, since the value of the test statistic is not less than the critical value. Yes, since the value of the test statistic is less than the critical value. No, since the value of the test statistic is less than the critical value.

Solutions

Expert Solution

a-1) The pooled variance(sp2) = ((n1 - 1)s21^2 + (n2 - 1)s2^2)/(n1 + n2 - 2)

                                                = (8 * (37)^2 + 8 * (15)^2)/(9 + 9 - 2)

                                                = 797

The test statistic t = ()/sqrt(sp2/n1 + sp2/n2)

                             = (256 - 269)/sqrt(797/9 + 797/9)

                             = -0.98

a-2) df = 9 + 9 - 2 = 16

At 10% significance level, the critical value is t* = -1.337

b-3) Since the test statistic value is not less than the critical value(-0.98 > -1.337), so we should not reject the null hypothesis.

No, Since the value of the test statistic is not less than the critical value.

b-1) The test statistic t = ()/sqrt(s1^2/n1 + s2^2/n2)

                                     = (256 - 269)/sqrt((37)^2/9 + (15)^2/9)

                                     = -0.98

df = (s1^2/n1 + s2^2/n2)^2/((s1^2/n1)^2/(n1 - 1) + (s2^2/n2)^2/(n2 - 1))

    = ((37)^2/9 + (15)^2/9)^2/(((37)^2/9)^2/8 + ((15)^2/9)^2/8)

    = 11

b-2) At 10% significance level, the critical value is t* = -1.363

b-3) Since the test statistic value is not less than the critical value(-0.98 > -1.363), so we should not reject the null hypothesis.

No, Since the value of the test statistic is not less than the critical value.


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