Getting P-value While Using VarianceSuppose we observe a random sample of five measurements: 10, 13, 15, 15, 17, from a normal distribution with unknown mean μ 1 and unknown variance σ 1 2 , A second random sample from another normal population with unknown mean μ 2 and unknown variance σ 2 2 yields the measurements: 13, 7, 9, 11.a) Test for evidence that σ 1 &gt; 1.0. Complete the P-value for this test as accurately as possible. Draw a conclusion at α = 0.05.Here's what I've done so far:Step 1: Calculate σ 1 σ 1 = ( 10 − 14 ) 2 + ( 13 − 14 ) 2 + ( 15 − 14 ) 2 + ( 15 − 14 ) 2 + ( 17 − 14 ) 2 5 = 28 5 = 2.366Step 2: Set up Hypothesis Test H 0 : σ 1 = 2.366 H a : σ 1 &gt; 1.0How do I proceed from here? Thanks.EDIT:Also have this question, and would appreciate some insight.b) Use the pivotal method(and a pivotal statistic with F distribution) to derive a 95% confidence interval for σ 2 σ 1 . Work it out for these data. And test the null hypothesis that σ 2 = σ 1 at the 5% level of significance.

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Getting P-value While Using VarianceSuppose we observe a random sample of five measurements: 10, 13, 15, 15, 17, from a normal distribution with unknown mean       μ    1   and unknown variance       σ    1    2  , A second random sample from another normal population with unknown mean       μ    2   and unknown variance       σ    2    2   yields the measurements: 13, 7, 9, 11.a) Test for evidence that       σ    1    &amp;gt;  1.0. Complete the P-value for this test as accurately as possible. Draw a conclusion at   α  =  0.05.Here&#039;s what I&#039;ve done so far:Step 1: Calculate       σ    1        σ    1    =                    (        10        −        14                  )          2                +        (        13        −        14                  )          2                +        (        15        −        14                  )          2                +        (        15        −        14                  )          2                +        (        17        −        14                  )          2                    5        =            28      5        =  2.366Step 2: Set up Hypothesis Test      H    0    :      σ    1    =  2.366      H    a    :      σ    1    &amp;gt;  1.0How do I proceed from here? Thanks.EDIT:Also have this question, and would appreciate some insight.b) Use the pivotal method(and a pivotal statistic with F distribution) to derive a 95% confidence interval for             σ      2              σ      1      . Work it out for these data. And test the null hypothesis that       σ    2    =      σ    1   at the 5% level of significance.

retalibry9 · Accepted Answer

Step 1for part a), you&#039;ve made a wrong null hypothesis test. You want to test for evidence that       σ    1    &amp;gt;      1    →      σ    1    2    &amp;gt;      1  . Thus, your hypotheses should be:      H    0    :      σ    1    2    =  1      H    1    :      σ    1    2    &amp;gt;  1We need       n    1   which is 5. We also need our sample variance, which you have the square root of. Thus,       S    2    =  5.6 and now we calculate our chi-square statistic, which is      χ    2    =            (      n      −      1      )              s        2                    σ      0      2        =            (      5      −      1      )      (      5.6      )        1    =  22.4We know our   α  =  .05, so we need to find       χ          α        2    (  n  −  1  )  =      χ          .05        2    (  4  )  =  9.488.. Since   22.4  ≥  9.488, we reject       H    0    . By p-value, we can see that at   α  =  .01  ,  P  (  W  ≥      13.28    )  =  .01, so our p-value is less than .01. Also,   P  (  W  ≥      14.86    )  =  .005, so it would appear that our p-value for this problem is VERY small indeed!Step 2for part b), we know that       s    1    2    =  5.6 and       s    2    2    =            (      13      −      10              )        2            +      (      7      −      10              )        2            +      (      9      −      10              )        2            +      (      11      −      10              )        2              4    =  5.. ALso, let m be the number of measurements from the first random sample and let n be the number of measurements from the second random sample. Considering the scope of deriving the confidence, I give it to you freely here:  .95  C  I      (                  σ        2        2                    σ        1        2              )    =      [          1                        F                                    α              2                                      (        n        −        1        )        (        m        −        1        )                            s        2        2                    s        1        2              ,          F                        α          2                      (    m    −    1    )    (    n    −    1    )                  s        2        2                    s        1        2              ]    =      [          1                        F                      .025                          (        4        )        (        3        )                    5.6      5        ,          F              .025              (    3    )    (    4    )          5.6      5        ]  Since I gave you the freebee, you should try and finish the hypothesis test for the ratio using what you know of the F-statistic and the information provided...

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