Question

We feel that more than 10% of Americans lost someone due to inability to pay for healthcare from 2014-2019. From 655 Americans we find 88 that did lose someone due to inability to pay for healthcare. Test the claim using a = 0.10

1. What is the claim? What are the null and alternate hypotheses? Will it be a one or two tailed test?

2. Do we use z-scores or nothing? Why?

3. Define the rejection region. Put it into words also.

4. Summarize the sample information and then find z*. Also find the p-value.

5. What is your conclusion?

6. Looking back at number 1, state the two types of error possible.

Answer 1

Here the claim is about whether someone from us died because of inability to pay for healthcare .

Null hypothesis would probablity that a person is not able to pay for health Care is 0.1 against the alternative that it is not 0.1.

Alternative hypothesis is two tailed.

So here the test statistic is X the number of people who cannot afford to pay for healthcare in a sample of n=655 people. X follows Binomial distribution with parameter n=655 and under the null hypothesis p=0.1. so here the sample size is large and so we use Central limit theorem. That is we use z score. Level of significance is 10 percent.

E(X)=np ,V(X)=npq

Rejection region is the set of values of (X1,x2,....,xn) for which if it falls under this region we reject the null hypothesis with probability 0.1 when the null hypothesis is true.Here the rejection region is if |z|>1.96 then we reject the null hypothesis with probability 0.1 when the null hypothesis is true.

Here |z|=2.93 and p value is P(|Z|>2.93)=2(1-P(Z>2.93))=2(0.00169)=0.00338

Which is less than the level of significance 0.10. hence we conclude that we reject the null hypothesis at 0.1 level of significance.

P(Type 1 error)=0.1

P(type 2 error)=1-P(|Z|>1.96| alternative is true)

That is when p not equal to 0.1.