Question

When 3010 adults were surveyed in a​ poll, 27​% said that they use the Internet. Is it okay for a newspaper reporter to write that ​"1 divided by 4 of all adults use the​ Internet"? Why or why​ not? Identify the null​ hypothesis, alternative​ hypothesis, test​ statistic, P-value, conclusion about the null​ hypothesis, and final conclusion that addresses the original claim. Use the​ P-value method. Use the normal distribution as an approximation of the binomial distribution.

The test statistic is z = ?. (Round to two decimal places as needed.)The P-value is ?. (Round to four decimal places as needed.)Identify the conclusion about the null hypothesis and the final conclusion that addresses the original claim. (Assume a 0.05 significance level.)

Answer 1

Null hypothesis:
`p=0.25`

Alternative hypothesis:
`p \\eq 0.25`

z=2.53

pv=0.0114

So based on the p value obtained and using the significance given
`\alpha=0.05` we have
`p_v<\alpha` so we can conclude that we reject the null hypothesis, and we can said that at 5% of significance the proportion of people who says that they use the Internet differs from 0.25 or 25% .

Explanation:

1) Data given and notation

n=3010 represent the random sample taken

X represent the people who says that said that they use the Internet.

`\hat p=(X)/(106)=0.27` estimated proportion of people who says that said that they use the Internet.

`p_o=0.25` 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 50% of people who says that they would watch one of the television shows.:

Null hypothesis:
`p=0.25`

Alternative hypothesis:
`p \\eq 0.25`

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{(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{0.27 -0.25}{\sqrt{(0.25(1-0.25))/(3010)}}=2.53`

4) Statistical decision

P value method or p value approach . "This method consists on 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.

We have the significance level provided
`\alpha=0.05`. The next step would be calculate the p value for this test.

Since is a bilateral test the p value would be:

`p_v =2*P(z>2.53)=2*(0.0057)=0.0114`

So based on the p value obtained and using the significance given
`\alpha=0.05` we have
`p_v<\alpha` so we can conclude that we reject the null hypothesis, and we can said that at 5% of significance the proportion of people who says that they use the Internet differs from 0.25 or 25% .