Question

The Gold Coast. A historian examining British colonial records for the Gold Coast in Africa suspects that the death rate was higher among African miners than among European miners. In the year 1936, there were 223 deaths among 33,809 African miners and 7 deaths among 1541 European miners on the Gold Coast. (The Gold Coast became the independent nation of Ghana in 1957.) Consider this year as a random sample from the colonial era in West Africa. Is there good evidence that the proportion of African miners who died was higher than the proportion of European miners who died?

Answer 1

There is no enough statistical evidence that the proportion of African miners deaths is higher than Europeans.

Explanation:

We want to perform a hypothesis test on the difference of proportions.

The null and alternative hypothesis are:

`H_0: \pi_1=\pi_2\\\\ H_1: \pi_1\\eq\pi_2`

We assume a level of significance of 0.05.

The proportion of European miners that died in 1936 is:

`p_1=7/1541=0.0045`

The proportion of African miners that died in 1936 is:

`p_1=223/33809=0.0066`

The difference in proportions is

`\Delta p =p_2-p_1=0.0066-0.0045=0.0021`

The weighted average of the proportion to estimate the standard deviation is:

`p=(n_1*p_1+n_2*p_2)/(n_1+n_2)=(1541*0.0045+33809*0.0065)/(1541+33809)=(7+223)/(1541+33809)=(230)/(35350)=0.0065`

The estimated standard deviation is:

`s=\sqrt{(p(1-p))/(n_1)+(p(1-p))/(n_2)}=\sqrt{(0.0065*0.9935)/(1541)+(0.0065*0.9935)/(33809)}=0.0021`

The z-value can now be calculated

`z=(p_2-p_1)/s=(0.0066-0.0045)/0.0021=1`

The P-value for z=1 is

`P(z>1)=0.32`

The P-value (0.32) is greater than the significance level (0.05), so we failed to reject the null hypothesis.

There is no evidence the proportion of African miners deaths is higher than Europeans.