91影视

(a)The cell totals, in the standard order, are

The other cell totals are

An efficient method for hand computation due to Yates is: write total of 8 cells totals in the standard order. Obtain next 8 cells with just adding d to each combination to obtain total of 16 cells. Add four more columns to the table of 16 cells. The entries in the columns represent sum of particular entries of the previous columns - first eight are sum of 1 and 2 ; 3 and 4 ; 5 and 6 ; 7 and 8 ; etc; and the last eight are differences between 2 and 1 ; 4 and 3 ; 6 and 5 ; 7 and 8 ; etc. The table becomes:

To understand the table above, see the following details:

First (second) column consist of corresponding sums of particular factors at corresponding levels ,

eg.\({x_{1111}}\)is the sum when factor\(A\)is held at level 1 , factor \(B\)is held at level 1 , factor\(C\)is held at level 1 , and factor\(D\)is held at level 1 . Third column is constructed as follows:

first row is sum of \({x_{1111.}}\)and\({x_{2111.}}\)

\({x_{1111.}}\)+\({x_{2111.}}\)=7+11=18;

Second rowis sum of \({x_{1211}}\)and\({x_{2211}}\)

\({x_{1211}}\)+\({x_{2211}}\)=7+12=19;

Third rowis sum of \({x_{1121.}}\)and\({x_{2121}}\)

\({x_{1121.}}\)+\({x_{2121}}\)=21+41=62;

Fourth row \({x_{1221}}\)and\({x_{2221}}\)

\({x_{1221}}\)+\({x_{2221}}\)=27+48=75;

Continue until eight row.

Ninth rowis um of \({x_{2111.}}\)and\({x_{1111}}\)

\({x_{2111.}}\)+\({x_{1111}}\)=11-7=4;

Tenth row is um of \({x_{2211}}\)and\({x_{1211.}}\)

\({x_{2211}}\)+\({x_{1211.}}\)=12-7=5;

Eleventh rowis sum of \({x_{2121.}}\)and \({x_{1211}}\)

\({x_{2121.}}\)+ \({x_{1211}}\)=41-21=20;

Twelve row is sum of \({x_{2221}}\)and\({x_{1221}}\)

\({x_{2221}}\)+ \({x_{1221}}\)=48-27=21;

Continue until the last row.

In a similar manner obtain fourth, fifth, and sixth column. The formula to obtain the seventh column is given in the table.

The estimate are computed using the formula

\({\rm{\;estimate\;}} = \frac{{{\rm{\;effect contrast\;}}}}{{{2^p}}}\)

Where \(p = 4\)in this case this is the estimate because \(n = 1\). The estimate of main effects are

\({\hat \alpha _1} = \frac{{144}}{{{2^4}}} = 9\)

\({\hat \beta _1} = \frac{{36}}{{{2^4}}} = 2.25\)

\({\hat \delta _1} = \frac{{272}}{{{2^4}}} = 17\)

\({\hat \gamma _1} = \frac{{336}}{{{2^4}}} = 21\)

Estimates of two-factor interaction effects for this experiment are

\({(\widehat {\alpha \beta })_{11}} = \frac{0}{{{2^4}}} = 0\)

\({(\widehat {\alpha \delta })_{11}} = \frac{{32}}{{{2^4}}} = 2\)

\({(\hat \alpha \gamma )_{11}} = \frac{4}{{{2^4}}} = 2.75\)

\({(\widehat {\beta \delta \delta })_{11}} = \frac{{12}}{{{2^4}}} = 0.75\)

\({(\hat \beta \gamma )_{11}} = \frac{8}{{{2^4}}} = 0.5\)

\({(\widehat {\delta \gamma })_{11}} = \frac{{72}}{{{2^4}}} = 4.5\)

(b)

Order the effects

-32,-12,-4,-4,-4

0,8,12,32,36,44,72,144,272,336,684

Then find the corresponding \(z\)values and create the normal probability plot The plot suggest the interactions \(A,C,{\rm{\;and\;}}D\) are significant.

Thus the result is \({\rm{\;estimate\;}} = \frac{{{\rm{\;effect\;}}}}{{{2^p}}}{\rm{; b}}{\rm{. Interactions\;}}A,C,{\rm{\;and\;}}D{\rm{\;are significant\;}}\).