Question

Test the hypothesis using the​ P-value approach. Be sure to verify the requirements of the test. Upper H 0H0​: pequals=0.40.4 versus Upper H 1H1​: pgreater than>0.40.4 nequals=125125​; xequals=5555​, alphaαequals=0.01

Answer 1

Answer:

$z=\frac{0.44 -0.4}{\sqrt{\frac{0.4(1-0.4)}{125}}}=0.913$  

$p_v =P(z>0.913)=0.181$  

So the p value obtained was a very high value and using the significance level given $\alpha=0.01$ 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 1% of significance the population proportion is not higher than 0.4.  

Step-by-step explanation:

Data given and notation

n=125 represent the random sample taken

$\hat p=\frac{55}{125}=0.44$ estimated proportion

$p_o=0.4$ is the value that we want to test

$\alpha=0.01$ represent the significance level

Confidence=99% or 0.99

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 is greater than 0.4:  

Null hypothesis:$p\leq 0.4$  

Alternative hypothesis:$p >0.4$  

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  

Since we have all the info requires we can replace in formula (1) like this:  

$z=\frac{0.44 -0.4}{\sqrt{\frac{0.4(1-0.4)}{125}}}=0.913$  

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 $\alpha=0.01$. The next step would be calculate the p value for this test.  

Since is a bilateral test the p value would be:  

$p_v =P(z>0.913)=0.181$  

So the p value obtained was a very high value and using the significance level given $\alpha=0.01$ 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 1% of significance the population proportion is not higher than 0.4.