If there is no seasonal effect on human births, we would expect equal numbers of children to be born in each season (winter, spring, summer, and fall). A student takes a census of her statistics class and finds that of the 120 students in the class, 25 were born in winter, 35 in spring, 32 in summer, and 28 in fall. She wonders if the excess in the spring is an indication that births are not uniform throughout the year. a) What is the expected number of births in each season if there is no "seasonal effect" on births? b) Compute the x2 statistic. c) How many degrees of freedom does the x2 statistic have? d) Find the a = 0.05 critical value for the x2 distribution with the appropriate number of df. e) Using the critical value, what do you conclude about the null hypothesis at a = 0.05?

The objective is to test if there is no seasonal effect on human births. If this is true, then you would expect the proportion of births to be the same through all seasons. This means that, theoretically, you could expect that P(W)=P(Sp)=P(Su)=P(F)=1/4

To test this out a student made a census in her class and obtained the following observations:

n= 120 students surveyed

25 were born in Winter

35 were born in Spring

32 were born in Summer

28 were born in Fall

The appropriate statistic analysis to test the student's presumption is a Goodness to Fit Chi-Square test, with hypotheses:

[tex]H_{0}: P_{W}=P_{Sp}=P_{Su}=P_{F}=1/4[/tex]

H₁: At least one of the proportions is different from the others.

a) To calculate the expected frequency for each category you have to use the following formula Ei= n*Pi

Where Pi represents the expected proportion for the i-category.

In this particular case, the expected proportion for all categories is the same so the expected frequency will also be the same:

[tex]E_{W}= n*P_{W}= 120*1/4= 30\\E_{Sp}= n*P_{Sp}= 120*1/4=30\\E_{Su}= n*P_{Su}= 120*1/4=30\\E_{F}= n*P_{F}= 120*1/4=30[/tex]

b)

X²=∑[tex]\frac{(Oi-Ei)^2}{Ei}[/tex]~[tex]X^2_{k-1}[/tex]

[tex]X^2_{H_0}= \frac{(25-30)^2}{30} + \frac{(35-30)^2}{30} + \frac{(32-30)^2}{30} + \frac{(28-30)^2}{30} = 1.93[/tex]

c)

The statistic for this test has k-1 degrees of freedom, where k represents the number of categories of the variable. In this case, the statistic has 4-1=3 degrees of freedom.

d)

The rejection region for the Goodnes to Fit test is always one-tailed to the right, wich means that you'll reject the null hypothesis to big values of X²:

[tex]X^2_{k-1;1-\alpha }= X^2_{3;0.95}= 7.815[/tex]

The decision rule is:

If [tex]X^2_{H_0}[/tex] ≥ 7.815, reject the null hypothesis

If [tex]X^2_{H_0}[/tex] < 7.815, do not reject the null hypothesis

e)

[tex]X^2_{H_0}[/tex]= 1.93 is less than the critical value 7.815, so the decision is to not reject the null hypothesis.

Using a 5% level of significance, there is no significant evidence to reject the null hypothesis. Then you can conclude that there is no seasonal effect on human birth, i.e. the expected birth proportions are the same through all seasons.

I hope this helps!