a. Show that another expression for the chi-squared statistic is Why is it more...

a. Show that another expression for the chi-squared statistic is

Why is it more efficient to compute χ² using this formula?

b. When the null hypothesis is (H₀: p₁ = p₂ =………….= (i.e., for all i), how does the formula of part (a) simplify? Use the simplified expression to calculate χ² 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 360°). 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.