Why is n=7 and not 8 in this $\chi^2$ goodness of fit test for Binomial Distribution?

Question

In this problem I clearly see 8 observations. In the solution, in Step 3, why is $n=7$ instead of $8$? To find $p$ why is the mean divided by $7$? Shouldn't it be divided by $8$?

Answer 1

Your use of the word “categories” might indicate that you’s confusing this with some other scenario.

Here the question is whether the data can be modelled by a binomial distribution. Since the students can use public transport at most $7$ days in a week, it’s implicit in the question that the parameter $n$ of the binomial distribution $\mathsf B(n,p)$, the number of trials, is $7$. A binomial distribution always takes $n+1$ possible values, from $0$ (all trials failed) to $n$ (all trials succeeded). So it’s not surprising but necessary that there are $n+1$ entries in the table. The mean of a binomial distribution is $np$, so the empirical mean is divided by the known parameter $n$ to estimate the unknown parameter $p$.