Question

A newsletter publisher believes that under 48H% of their readers own a personal computer. Is there sufficient evidence at the 0.050.05 level to substantiate the publisher's claim? State the null and alternative hypotheses for the scenario.

Answer 1

Answer:

We need to conduct a hypothesis in order to test the claim that true proportion os readers with a personal computer is less than 0.48 or 48%, the system of hypothesis would be.:  

Null hypothesis:$p\geq 0.48$  

Alternative hypothesis:$p < 0.48$  

Step-by-step explanation:

Data given and notation  

n represent the random sample taken

$\hat p$ estimated proportion of readers own a personal computer

$p_o=0.48$ is the value that we want to test

$\alpha=0.05$ represent the significance level

Confidence=95% or 0.95

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 true proportion os readers with a personal computer is less than 0.48 or 48%, the system of hypothesis would be.:  

Null hypothesis:$p\geq 0.48$  

Alternative hypothesis:$p < 0.48$  

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  

We just need to replace the into formula (1) and find the statistic $z_{calc}$.

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 $\alpha=0.05$. The next step would be calculate the p value for this test.  

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

$p_v =P(z<z_{calc})$  

If the p value is lower than 0.05 we reject the null hypothesis. And in the other case if the p value is higher than 0.05 we FAIL to reject the null hypothesis.