Question

According to a Pew Research Center study, 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. Assume the significance level is ? = 0.10.

Which of the following is an appropriate conclusion for the hypothesis test?

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

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

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

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

Does secondhand smoke increase the risk of a low weight birth? A baby is "low birth weight" if it weighs less than 5.5 pounds at birth. According to the National Center of Health Statistics, about 7.8% of all babies born in the U.S. are categorized as low birth weight. Researchers randomly select 1200 babies whose mothers had extensive exposure to secondhand smoke during pregnancy. 10.4% of the sample are categorized as low birth weight.

Answer the following:

Which of the following are the appropriate null and alternative hypotheses for this research question.

H0: p = 0.078; Ha: p ? 0.078

H0: p = 0.078; Ha: p > 0.078

H0: p = 0.104; Ha: p ? 0.104

H0: ? = 0.104; Ha: ? > 0.104

Answer 1

Answer:

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

$p_v =P(z>1.82)=0.034$  

So the p value obtained was a very low value and using the significance level given $\alpha=0.1$ we have $p_v$ so we can conclude that we have enough evidence to reject the null hypothesis

And the best conclusion would be:

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

And for the second case the correct system of hypothesis is:

H0: p = 0.078; Ha: p > 0.078

Step-by-step explanation:

Data given and notation

n=300 represent the random sample taken

$\hat p=\frac{120}{300}=0.4$ estimated proportion of college students that have a smart phone

$p_o=0.35$ 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 proportion is >0.35.:  

Null hypothesis:$p\leq 0.35$  

Alternative hypothesis:$p > 0.35$  

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:  

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

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:  

$p_v =P(z>1.82)=0.034$  

So the p value obtained was a very low value and using the significance level given $\alpha=0.1$ we have $p_v$ so we can conclude that we have enough evidence to reject the null hypothesis

And the best conclusion would be:

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

And for the second case the correct system of hypothesis is:

H0: p = 0.078; Ha: p > 0.078