Answer:
Step-by-step explanation:
Hello!
Given the variable of interest:
X: Number of births in each season of the year, categorized: W: Winter, Sp: Spring, Su: Summer and F: Fall
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:
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:
b)
X²=∑~
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 <u>always</u> one-tailed to the right, wich means that you'll reject the null hypothesis to big values of X²:
The decision rule is:
If ≥ 7.815, reject the null hypothesis
If < 7.815, do not reject the null hypothesis
e)
= 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!