Question

According to a Pew Research Center, in May 2011, 35% of all American adults had a smart phone (one which the user can use to read email and surf the Internet). A communications professor at a university believes this percentage is higher among community college students.

She selects 300 community college students at random and finds that 120 of them have a smart phone. In testing the hypotheses: H0: p = 0.35 versus Ha: p > 0.35, she calculates the test statistic as Z = 1.82.

Use the Normal Table to help answer the p-value part of this question.

Click here to access the normal table.

1. There is enough evidence to show that more than 35% of community college students own a smart phone (P-value = 0.034).

2. There is enough evidence to show that more than 35% of community college students own a smart phone (P-value = 0.068).

3. There is not enough evidence to show that more than 35% of community college students own a smart phone (P-value = 0.966).

4. There is not enough evidence to show that more than 35% of community college students own a smart phone (P-value = 0.034).

Answer 1

Answer:

$p_v =P(z>1.82)=0.034$  

Assuming a standard significance level of $\alpha=0.05$ the best conclusion for this case would be:

4. There is not enough evidence to show that more than 35% of community college students own a smart phone (P-value = 0.034).

Because $p_v <\alpha$

If we select a significance level lower than 0.034 then the conclusion would change.

Step-by-step explanation:

Data given

n=300 represent the random sample taken

X=120 represent the people who have a smart phone

$\hat p=\frac{120}{300}=0.4$ estimated proportion of people who have a smart phone

$p_o=0.35$ is the value that we want to test

z would represent the statistic (variable of interest)

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

System of hypothesis

We need to conduct a hypothesis in order to test the claim that the true proportion of people who have a smart phone is higher than 0.35, the system of hypothesis are.:  

Null hypothesis:$p\leq 0.35$  

Alternative hypothesis:$p > 0.35$  

The statistic is given by:

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

Calculate the statistic  

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

$z=\frac{0.4 -0.35}{\sqrt{\frac{0.35(1-0.35)}{300}}}=1.82$  

Statistical decision  

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

$p_v =P(z>1.82)=0.034$  

Assuming a standard significance level of $\alpha=0.05$ the best conclusion for this case would be:

4. There is not enough evidence to show that more than 35% of community college students own a smart phone (P-value = 0.034).

Because $p_v <\alpha$

If we select a significance level lower than 0.034 then the conclusion would change.