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\fa. Show that another expression for the chi-squared statistic is N 2 Y = np jo Why is it more efficient to compute + using this formula? b. When the null hypothesis is (Hop, = P, = . . . = P: = 1/k (i.e., Pa = 1/k for all /), how does the formula of part (a) simplify? Use the simplified expression to cal- culate 12 for the pigeon/direction data in Exercise 4. Why is it more efficient to compute y" using this formula? b. When the null hypothesis is (Ho: p, = P2 =.... PA - 1/k (i.e., po 1/k for all ?). (i.e., for all i). how does the formula of part (3) simplify? Use the simplified expression to calculate y"for the pigeon/direction data in Exercise 4. Reference exercise 4 It is hypothesized that when homing pigeons are disoriented in a certain manner, they will exhibit no preference for any direction of flight after takeoff (so that the direction X should be uniformly distributed on the interval from 0' to 380*). To test this. 120 pigeons are disoriented, let loose, and the direction of flight of each is recorded; the resulting data follows. Use the chi-squared test at level .10 to see whether the data supports the hypothesis. Direction 0-