Question

A hypothesis test is conducted with a significance level of 10%. The alternative hypothesis states that more than 15% of a population has at least one sibling. The p-value for the test is calculated to be 0.27. Which statement is correct?

A. We can conclude that more than 27% of the population has at least one sibling.

B. We can conclude that exactly 27% of the population has at least one sibling.

C. We can conclude that more than 15% of the population has at least one sibling.

D. We cannot conclude that more than 15% of the population has at least one sibling.

E. There is not enough information given to make a conclusion.

Answer 1

D. We cannot conclude that more than 15% of the population has at least one sibling.

Explanation:

Data given and notation

n represent the random sample taken

X represent the number of people who has at least one sibling

`\hat p` estimated proportion of people who has at least one sibling

`p_o=0.15` is the value that we want to test

`\alpha=0.1` represent the significance level

Confidence=90% or 0.90

z would represent the statistic (variable of interest)

`p_v` represent the p value (variable of interest)

Concepts and formulas to use

We need to conduct a hypothesis in order to test the claim that the true proportion of people who has at least one sibling is more than 0.15:

Null hypothesis:
`p\leq 0.15`

Alternative hypothesis:
`p > 0.15`

When we conduct a proportion test we need to use the z statistic, and the is given by:

`z=\frac{\hat p -p_o}{\sqrt{(p_o (1-p_o))/(n)}}` (1)

The One-Sample Proportion Test is used to assess whether a population proportion
`\hat p` is significantly different from a hypothesized value
`p_o`.

Calculate the statistic

Let's assume that the calculated value for this case is
`z_(calc)`

Statistical decision

It's important to refresh the p value method or p value approach . "This method is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.

The significance level provided
`\alpha=0.1`. The next step would be calculate the p value for this test.

Since is a right tailed test the p value would be:

`p_v =P(z> z_(calc))=0.27`

So the p value obtained was a very high value and using the significance level given
`\alpha=0.1` we have
`p_v>\alpha` so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and we can said that at 10% of significance the proportion of people who has at least one sibling is not significanlty higher than 0.15.

So then the best answer for this case would be:

D. We cannot conclude that more than 15% of the population has at least one sibling.