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

Answer:

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

Step-by-step 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:  

$z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}$ (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.