The table below shows historical data for the distribution of the number of customers, in half-hour time periods, who visit the electronics department of a retail store. For example, in 25 percent of the time periods for which data were collected, no customers were observed in the electronics department of the store.

$$ \begin{array}{|l|c|c|c|c|} \hline \text{Number of Customers} \hskip{2em} & 0 & 1 & 2 & 3 \text{ or more} \\ \hline \text{Proportion of time periods} & 0.25 & 0.20 & 0.30 & 0.25 \\ \hline \end{array} $$

To investigate if the distribution has changed, the number of customers who visited the electronics department of the store was recorded for each of 50 randomly selected time periods. The results are shown in the table below.

$$ \begin{array}{|l|c|c|c|c|} \hline \text{Number of Customers} \hskip{2em} & 0 & 1 & 2 & 3 \text{ or more} \\ \hline \text{Number of time periods} & 4 & 13 & 14 & 19 \\ \hline \end{array} $$

A chi-square goodness-of-fit test was conducted to determine whether the data provide convincing evidence that the distribution has changed. The test statistic was 10.13 with a p-value of 0.0175. Which of the following statements is true?

Summary Submit Skip Question

Since the \(p\)-value is less than \(\alpha\), we can reject the null hypothesis that the distributions are the same. There is evidence the distribution changed.

The observed frequency can be less than 5, since the expected frequency is 5 or greater. We can find the expected frequency by multiplying 50 and the proportion of time periods.

There are only 3 degrees of freedom (4-1).