STAT 330: 98-1

Assignment 6 Solutions

  1. Chapter 9 Q 54: Let tex2html_wrap_inline109 be the population standard deviation for control rats and tex2html_wrap_inline111 be the treatment population standard deviation. Then to test tex2html_wrap_inline113 against the one sided alternative tex2html_wrap_inline115 we compute tex2html_wrap_inline117 and get upper tail P-values from F tables with 19 and 22 degrees of freedom. We get F = 2.85 and P just a bit bigger than 0.01 so that we conclude that the treatment population is more variable. (While the question asks a one-tailed question it is not entirely clear to me that the tail was not chosen after seeing the data; if so you should double the P-value. The conclusion is not really changed.)
  2. Chapter 9 Q 56: The quantity tex2html_wrap_inline129 has an F distribution so that


    The right hand inequality holds if and only if


    Similarly, the first inequality can be rearranged to be


    so that the range between these two limits is a level tex2html_wrap_inline139 confidence interval for tex2html_wrap_inline141 . For the data we have tex2html_wrap_inline143 and tex2html_wrap_inline145 . The critical point tex2html_wrap_inline147 while to find the lower tail critical point we use tex2html_wrap_inline149 . The interval is then 0.00549/(0.0258*9.28) to (9.28)(0.00549)/0.0258. Note that some people may have put tex2html_wrap_inline155 on top.

  3. Chapter 10 Q 10:

    1. tex2html_wrap_inline157 . Use tex2html_wrap_inline159 and the fact that summing over j just multiplies by J to get tex2html_wrap_inline165 .
    2. tex2html_wrap_inline167 . The variance of any average of J independent quantities each with variance tex2html_wrap_inline171 is just tex2html_wrap_inline173 so we get tex2html_wrap_inline175 .
    3. tex2html_wrap_inline177 which, using (a) and the rule in (b) is tex2html_wrap_inline179 .
    4. We have tex2html_wrap_inline181 . Expand out the square and use the fact that tex2html_wrap_inline183 to see that


      Take expected values and put in the results of (b) and (c) to get


    5. Under tex2html_wrap_inline187 the second of these terms is 0 so tex2html_wrap_inline189 while under the alternative the second term is positive so that tex2html_wrap_inline191 .

  4. Chapter 10 Q 42: For tex2html_wrap_inline193 we have tex2html_wrap_inline195 , tex2html_wrap_inline197 , and all the other tex2html_wrap_inline199 so that the confidence interval for tex2html_wrap_inline193 is tex2html_wrap_inline203 (where we use the level tex2html_wrap_inline205 for a 95% confidence interval). For the other intervals it is the values of the tex2html_wrap_inline207 which change. They are (1,0,-1), (0,1,-1) and (0.5,0.5,-1) respectively. Only the contrast tex2html_wrap_inline215 is judged significantly different from 0. (Note the use of tex2html_wrap_inline217 not tex2html_wrap_inline219 ; these are 95% confidence intervals.)
  5. Chapter 10 Q 44: We have tex2html_wrap_inline221 and tex2html_wrap_inline223 . Subtracting we get tex2html_wrap_inline225 so the the SSE for the y's is tex2html_wrap_inline231 times the SSE for the x's. Similarly we have tex2html_wrap_inline237 so that the new SSTr is tex2html_wrap_inline231 times the old SSTr. The factors tex2html_wrap_inline231 then cancel out in the formula for the F statistic so that the new F statistic is exactly equal to the old F statistic.
  6. Chapter 10 Q46:

    1. The two samples are now tex2html_wrap_inline253 for tex2html_wrap_inline255 and tex2html_wrap_inline257 for tex2html_wrap_inline259 . The two sample variances are tex2html_wrap_inline261 and tex2html_wrap_inline263 . Then the pooled estimate of tex2html_wrap_inline171 is


      which is just


      which is in turn just the MSE.

    2. The pooled statistic squared is just


      Now write tex2html_wrap_inline273 as tex2html_wrap_inline275 to see that


      Now examine the Treatment Sum of Squares. First note that tex2html_wrap_inline279 . Use this to see that the Treatment Sum of Squares is given by


      which simplifies to


      This last is the numerator of tex2html_wrap_inline283 .

Richard Lockhart
Fri Mar 6 10:16:01 PST 1998