Mathematics, 19.12.2019 04:31, kkcheers16

According to a pew research center study, in may 2011, 38% 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 366 community college students at random and finds that 160 of them have a smart phone. then in testing the hypotheses: h 0 : p = 0.38 versus h a : p > 0.38, what is the test statistic? z = . ( round your answer to two decimal places.)

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answered: tsigler6425

$z=\frac{0.437 -0.38}{\sqrt{\frac{0.38(1-0.38)}{366}}}=2.25$

p_v =P(Z2.25)=0.012

Step-by-step explanation:

1) Data given and notation

n=366 represent the random sample taken

X=160 represent the people with smartphones

$\hat p=\frac{160}{366}=0.437$ estimated proportion of people with smartphones

p_o=0.38 is the value that we want to test

$\alpha$ represent the significance level

z would represent the statistic (variable of interest)

p_v represent the p value (variable of interest)

2) Concepts and formulas to use

We need to conduct a hypothesis in order to test the claim that the true proportion of people with smatrphones is higher than 0.38:

Null hypothesis: $p \leq 0.38$

Alternative hypothesis: p 0.38

When we conduct a proportion test we need to use the z statisitc, 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 .

3) Calculate the statistic

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

$z=\frac{0.437 -0.38}{\sqrt{\frac{0.38(1-0.38)}{366}}}=2.25$

4) 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 assumed is $\alpha=0.05$ . 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(Z2.25)=0.012

So the p value obtained was a very low value and using the significance level assumed $\alpha=0.05$ we have p_v so we can conclude that we have enough evidence to reject the null hypothesis, and we can said that at 5% of significance the true proportion of people with smatphones is higher than 0.38.