Question

25.5 Visitors to the Monterey Bay Aquarium, continued. The two-way table in Exercise 25.1 displays data on the relationship between age and ethnicity for a random sample of visi- tors to the Monterey Bay aquarium. The null hypothesis is that there is no relationship be- tween ethnicity (Caucasian vs. Hispanic) and age category. AGEDIST (a) If this hypothesis is true, what are the ex- pected counts for the four age categories for Hispanic visitors? This is one column of the two-way table of expected counts. Find the column total and verify that it agrees with the column total for the ob- served counts. (b) Hispanic visitors tend to be younger than Caucasian visitors to the aquarium. How is this reflected when comparing the ob- served and the expected accounts for His- panic visitors?
please show all work and calculations! Thank you in advance!

Answer 1

First we get the column and row totals for the table as given below

Age Category	Caucasian	Hispanic	Total
18 to 24	785	586	1371
25 to 34	1413	938	2351
35 to 44	1027	594	1621
45 to 54	859	180	1039
55 and over	1255	160	1415
Total	5339	2458	7797

Now let (considering the first row under the Hispanic column)

A= the event that a visitor belongs to the age category 18-24

B=The event that a visitor is Hispanic

If null hypothesis were true (that is if there is no relationship between the ethnicity and age category), then we can say that the above events A and B are independent.

The probability that a randomly selected visitor is in age category 18-24 and Hispanic is

$\begin{align*} P(A\cap B)&=P(A)\times P(B)\quad\text{because A and B are independent}\\ &=\frac{1371}{7797}\times \frac{2458}{7797} \end{align*}$

The expected count for this cell then is

$\begin{align*} \text{Expected count}=P(A\cap B)\times N&= &=\frac{1371}{7797}\times \frac{2458}{7797}\times 7797=\frac{1371\times 2458}{7797} \end{align*}$

We can generalize the above and say that if there is no relationship between the ethnicity and age category then the expected frequency of any cell can be calculated as

$\begin{align*} f_e=\frac{RT\times CT}{N} \end{align*}$

where RT= the row total for the row corresponding to the cell

CT= the row total for the row corresponding to the cell

N=Total number of observations

Now we calculate the expected frequency for Hispanic visitors

$\begin{align*} f_e&=\frac{1371\times2458}{7797}=432.21\quad\text{: Age Category:18 to 24}\\ f_e&=\frac{2351\times2458}{7797}=741.15\quad\text{: Age Category:25 to 34}\\ f_e&=\frac{1621\times2458}{7797}=511.02\quad\text{: Age Category:35 to 44}\\ f_e&=\frac{1039\times2458}{7797}=327.54\quad\text{: Age Category:45 to 54}\\ f_e&=\frac{1415\times2458}{7797}=446.08\quad\text{: Age Category:55 and over}\\ \end{align*}$

The Following table of expected count can be made

Age Category	Caucasian	Hispanic	Total
18 to 24	938.79	432.21	1371
25 to 34	1609.85	741.15	2351
35 to 44	1109.98	511.02	1621
45 to 54	711.46	327.54	1039
55 and over	968.92	446.08	1415
Total	5339	2458	7797

We can verify that the column count for Hispanic visitors agrees with the count 2458 of observed counts

b) The observed and expected counts for Hispanic visitors are

Age Category	Observed	Expected
18 to 24	586	432.21
25 to 34	938	741.15
35 to 44	594	511.02
45 to 54	180	327.54
55 and over	160	446.08

If we look at the younger age category, the expected counts are lower than the observed counts for age groups 18 to 24 and 25 to 34

In the older age groups of 45 and above, we can see that the expected count is larger than the observed.

Hence we can see that although the observed counts show us that the Hispanic visitors are more likely to be younger, the expected count tries to spread this out across the age groups more in line with the overall count of visitors when we ignore the ethnicity. That is because we calculated the expected count assuming that there is no relationship between the ethnicity and the age groups.