Question

If there is no seasonal effect on human births, one 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, 26 were born in winter, 37 in spring, 31in summer, and 26 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?

Answer 1

(a) We have given the information that, if there is no seasonal effect on human births, then number of children born in each season expected to be equal.

The general way of finding the Expected frequency is:

But since, in question it is mention that we expect that equal number of children born in each season, so we the expected frequencies is as follows:-

	winter	spring	summer	fall	Total
Observed	26	37	31	26	120
Expected	30	30	30	30	120

(b) The chi-square test statistics is given by:

So, the - statistic is calculated as

(c) The degrees of freedom (df) for the -test is given by:

where, k is the number of categories, here in this example the categories are the different seasons, i.e., winter, summer, spring and fall.

Since there are only 4 seasons, so k=4

Hence the degrees of freedom for the -statistic is