Weisberg (1975) reports the following data on the number of boys among the first seven children born to a collection of 1,334 Swedish ministers.

Number of Boys 0 1 2 3 4 5 6 7 Frequency 6 57 206 362 365 256 69 13 Assume that the number of boys has a Bin(7, .5) distribution. Compute the probabilities for each of the eight categories 0, 1,..., 7. From the sample of 1,334 families, what is the expected frequency for each category? What is the distribution of the number of families that fall into each category?

Summarize the fit of the assumed binomial model by computing X2 =

7 i=0

(Observationi − Expectedi)

2 Expectedi

.

The statistic X2 is known as Pearson’s chi-square statistic. For large samples such as this, if the Expected values are correct, X2 should be one observation from a χ2(7) distribution. (The 7 is one less than the number of categories.) Compute X2 and compare it to tabled values of the χ2(7)
distribution. Does X2 seem like it could reasonably come from a χ2(7)?
What does this imply about how well the binomial model fits the data?
Can you distinguish which assumptions made in the binomial model may be violated?