Question

Radioactive fallout from testing atomic bombs drifted across a region. There were 220 people in the region at the time and 36 of them eventually died of cancer. Cancer experts estimate that one would expect only about 33 cancer deaths in a group this size. Assume the sample is a typical group of people. ​a) Is the death rate observed in the group unusually​ high? ​b) Does this prove that exposure to radiation increases the risk of​ cancer?

Answer 1

Answer:

(a) Yes, the death rate observed in the group unusually​ high.

(b) Yes, this prove that exposure to radiation increases the risk of​ cancer.

Step-by-step explanation:

In a region where radioactive fallout from testing atomic bombs drifted across, 36 of the 220 people died of cancer.

The sample proportion of the number of people dying of cancer in this region is:

$\hat p =\frac{36}{220} =0.164$

It is estimated by the cancer expert that there will be 33 cancer deaths in a group of this size.

The population proportion is: $p=\frac{33}{220} =0.15$

A hypothesis test for single proportion can be performed to test if the proportion of cancer deaths was high or not and whether it was due to the increase in radiation.

The hypothesis can be defined as:

H₀: The proportion of death due to cancer in this region is 0.15, i.e. p = 0.15.

Hₐ: The proportion of death due to cancer in this region is more than 0.15, i.e. p > 0.15.

Assuming the population is Normally distributed since the sample size is large.

Assume the level of significance is α = 0.05.

The test statistic is:

$z=\frac{\hat p-p}{\sqrt{\frac{p(1-p)}{n}}}\\= \frac{0.164-0.15}{\sqrt{\frac{0.15(1-0.15)}{220}}} \\=0.581$

The test statistic value is 0.581.

Decision rule:

If the p-value of the test is less than the significance level then the null hypothesis is rejected and vice versa.

The p-value of the test is:

$P(Z>0.581)=1-P(Z$

**Use a z-table for the p-value.

The p-value = 0.281 > α = 0.05.

The null hypothesis was failed to be rejected.

(a)

As the null hypothesis was not rejected at 5% level of significance it can be concluded that the death rate due to cancer observed in this group of people is unusually high.

Yes, the death rate observed in the group unusually​ high.

(b)

It was previously mentioned that a radioactive fallout from testing atomic bombs drifted across this region. Due to this 36 people died of cancer in this region.

As the death rate observed in the group unusually​ high, it can be said that the increase in the death rate due to cancer was due to the excessive exposure to radiation.

Yes, this prove that exposure to radiation increases the risk of​ cancer.