Question

The Pew Research Center reported that 73% of Americans who own a cell phone also use text messaging. In a recent local survey, 155 out of 200 cell phone owners used text messaging.
Since a Z test is appropriate, test whether the population proportion of Americans who use text messaging is different from 73%. Use level of significance α = 0.10.
Hint: Do you need to conduct a t-test or a z-test? Next, find the p-value, using p-value, and level of significance, you can see if the decision (Reject or Do Not reject H0.) You can also find the critical value(s) to finalize your decision.

Answer 1

Answer:

$z=\frac{0.775 -0.73}{\sqrt{\frac{0.73(1-0.73)}{200}}}=1.433$  

Now we can find the p value. Since we have a bilateral test the p value would be:  

$p_v =2*P(z>1.433)=0.152$  

Since the p value is higher than the significance level of 0.1 we have enough evidence to FAIL to reject the null hypothesis and the best conclusion for this case would be:

Do Not reject H0

Step-by-step explanation:

Information provided

n=200 represent the sample size slected

X=155 represent the cell phone owners used text messaging

$\hat p=\frac{155}{200}=0.775$ estimated proportion of cell phone owners used text messaging

$p_o=0.73$ is the value to verify

$\alpha=0.1$ represent the significance level

We need to conduct a z test for a proportion

z would represent the statistic

$p_v$ represent the p value

System of hypothesis

We want to verify if the true proportion of cell phone owners used text messaging is different from 0.73 so then the system of hypothesis are:

Null hypothesis:$p=0.73$  

Alternative hypothesis:$p \neq 0.73$  

The statistic to check this hypothesis is given by:

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

Replacing the data given we got:

$z=\frac{0.775 -0.73}{\sqrt{\frac{0.73(1-0.73)}{200}}}=1.433$  

Now we can find the p value. Since we have a bilateral test the p value would be:  

$p_v =2*P(z>1.433)=0.152$  

Since the p value is higher than the significance level of 0.1 we have enough evidence to FAIL to reject the null hypothesis and the best conclusion for this case would be:

Do Not reject H0