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The number of participants in a clinical trial involving lung cancer are tabulated under different ethnic groups.

The expected population distribution under each ethnic group is also provided.

Assume subjects are randomly selected.

Let O denote the observed frequencies of people of different races.

The following values are obtained:

\(\begin{aligned}{l}{O_1} = 3855\\{O_2} = 60\\{O_3} = 316\\{O_4} = 54\\{O_5} = 12\end{aligned}\)

The sum of all observed frequencies is computed below:

\[\begin{aligned}{c}n = 3855 + 60 + ... + 12\\ = 4297\end{aligned}\]

Let E denote the expected frequencies.

It is expected that the frequencies should fit well with the population distribution.

Therefore, the population distribution of each race is given as follows:

\(\begin{aligned}{c}{p_1} = \frac{{75.6}}{{100}}\\ = 0.756\\{p_2} = \frac{{9.1}}{{100}}\\ = 0.091\end{aligned}\)

\(\begin{aligned}{c}{p_3} = \frac{{10.8}}{{100}}\\ = 0.108\\{p_4} = \frac{{3.8}}{{100}}\\ = 0.038\end{aligned}\)

\(\begin{aligned}{c}{p_5} = \frac{{0.7}}{{100}}\\ = 0.007\end{aligned}\)

Now, the expected frequencies are computed below:

\(\begin{aligned}{c}{E_1} = n{p_1}\\ = 4297\left( {0.756} \right)\\ = 3248.532\end{aligned}\)

\(\begin{aligned}{c}{E_2} = n{p_2}\\ = 4297\left( {0.091} \right)\\ = 391.027\end{aligned}\)

\(\begin{aligned}{c}{E_3} = n{p_3}\\ = 4297\left( {0.108} \right)\\ = 464.076\end{aligned}\)

\(\begin{aligned}{c}{E_4} = n{p_4}\\ = 4297\left( {0.038} \right)\\ = 163.286\end{aligned}\)

\(\begin{aligned}{c}{E_5} = n{p_5}\\ = 4297\left( {0.007} \right)\\ = 30.079\end{aligned}\)

Also,the expected values are greater than 5.

Thus, the requirements of the test are satisfied.

The hypotheses is stated as follows:

\({H_o}:\)The distribution of observations fits the distribution of population

\({H_a}:\)The distribution of observations does not fit the distribution of the population.

The test is right-tailed.

The table below shows the necessary calculations:

There is enough evidence to conclude that the participants are not distributed according to the population distribution.

Since there are a few participants that belong to the American Indian/Alaskan Native ethnic group as well as the Asian/Pacific Islander ethnic group, it can be said that these two races are underrepresented.