Two Sample Population Proportion Significance Tests:

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 341 deaths among 34282 African miners and 12 deaths
among 1635 European miners in the Gold Coast. Consider this year as a sample
from the prewar era in Africa.

a) State the null and alternative hypotheses.

b) Calculate a test statistic.

c) Give a P-value as exact as the tables in the text allow.

d) Is there good evidence that the proportion of African miners who died during a year
was higher than the proportion of European miners who died?

a) In this case, the null hypothesis (H0) would be that there is no difference in the death rates between African miners and European miners. The alternative hypothesis (Ha) would be that the death rate is higher among African miners compared to European miners.

b) To calculate the test statistic, we need to compute the test statistic Z. The formula for the test statistic is:

Z = (p̂1 - p̂2) / sqrt((p̂(1-p̂) / n1) + (p̂(1-p̂) / n2))

Where:
p̂1 and p̂2 are the sample proportions of deaths among African miners and European miners respectively.
n1 and n2 are the sample sizes of African miners and European miners respectively.

To determine p̂1, we divide the number of deaths among African miners (341) by the total number of African miners (34282) and calculate the proportion. Similarly, to determine p̂2, we divide the number of deaths among European miners (12) by the total number of European miners (1635) and calculate the proportion.

p̂1 = 341 / 34282 = 0.009953
p̂2 = 12 / 1635 = 0.007337

Now, we can substitute these values into the formula to calculate the test statistic Z.

Z = (0.009953 - 0.007337) / sqrt((0.009953 * (1-0.009953)) / 34282 + (0.007337 * (1-0.007337)) / 1635)

c) To find the p-value, we need to look up the test statistic Z in the standard normal distribution table (also known as Z-table). The p-value is the probability of obtaining a test statistic as extreme as the calculated Z (or more extreme) under the null hypothesis.

We locate the Z-value from step b) in the Z-table and find the corresponding p-value. Keep in mind that the table may not have a direct exact value, so we need to determine the closest approximate value.

d) If the p-value is less than the predetermined significance level (often 0.05 or 0.01), we can reject the null hypothesis and conclude that there is evidence to support the alternative hypothesis. If the p-value is greater than the significance level, we do not have enough evidence to reject the null hypothesis.

Based on the calculated p-value, we can make a determination regarding the evidence for the difference in death rates between African miners and European miners.