Question

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 Xshould 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 0.10 to see whether the data supports the hypothesis.

Direction	0?<45°	45?<90°	90?<135°	135?<180°
Frequency	10	17	17	15

Direction	180?<225°	225?<270°	270?<315°	315?<360°
Frequency	13	20	17	11

State the rejection region and compute the test statistic value. (Round your answers to three decimal places.)

rejection region	?2	?
test statistic	?2	=

Answer 1

The null and alternative hypothesis are:

Rejection region

To find the rejection region, first, find chi-square critical value.

To find critical value, we need alpha(level of significance) and degrees of freedom

alpha = 0.10 and degrees of freedom = number of categories - 1 = 8 - 1 = 7

The Chi-square test is a right-tailed test,

By using chi-square table the critical value at 0.10 significance level and DF = 7 is, 12.017

So the rejection region is,

Test statistics:

The formula of Chi-square test statistics is,

where O - observed frequencies which are given in 8 categories

E - Expected frequencies, have to find out it.

n = number of pigeons = 120 and p = 1/8. since there are 8 categories and which are uniformly distributed means equally likely.

E = 120 * (1/8) = 15, each category have same expected frequency.

O	E	(O-E)2 / E
10	15	1.666667
17	15	0.266667
17	15	0.266667
15	15	0
13	15	0.266667
20	15	1.666667
17	15	0.266667
11	15	1.266667
Sum		5.466667

The chi-square test statistics,