Question

1. 100,000 Massachusetts adults were randomly sampled with two factors recorded: whether or not the individual

had diabetes, and whether or not the person ate Kale. The following gives a table of the results.

Diabetes No Diabetes

Kale: 796 9187

No Kale: 9900 80117

(a) We write p(Diabetes | Kale) for the probability that a Massachusetts adult who eats kale has diabetes.

Either give a value for p(Diabetes | Kale) or explain why it cannot be computed.

(b) We write ^p(Diabetes | Kale) for the proportion of Kale-eating members of our sample above that had

diabetes. Either give a value for ^p(Diabetes | Kale) or explain why it cannot be computed.

(c) Give 95% confidence intervals for the probability of having diabetes for both the kale-eating and non-kale-

eating members of our Massachusetts adults.

(d) Can you conclude that the kale-eaters are less likely to have diabetes? Explain your reasoning.

(e) Can you conclude that kale consumption causes a lower diabetes rate in this population? Explain your

reasoning.

(f) Come up with a possible theory that explains why kale-eaters have a lower rate of diabetes, but does not

assume that kale causes the lower rate.

Answer 1

given that

	diabetes	no diabetes	total
kale	796	9187	9983
no kale	9900	80117	90017
total	10696	89304	100000

a)

for we cant calculate P(diabetes |kale) because for that we need population probabilities that we don't have so we cant calculate ,we can only calculate sample proportions based on the given sample data

b)

now

c)

proportions of kale eating diabetes persons =P=796/9983=0.0797

so 95% confidence interval is given by

Hence interval is (0.0744,0.0850)

Proportions of no kale eating diabetes persons =9900/90017=0.11

now 95% confidence interval is given by

Hence required interval is (.108,0.112)

d)

if we look at the intervals we have calculated above in part (c) ,interval for probability of diabetes in kale eaters contains values from 0.0744 to 0.0850 while for no kale eaters this value is more than 0.108 hence yes we can say that kale eaters are less likely to have diabetes,