Arandom sample of 378 hotel guests was taken one year ago, and it was found that 178 requested nonsmoking rooms. recently, a random sample of 516 hotel guests showed that 320 requested nonsmoking rooms. do these data indicate that the proportion of hotel guests requesting nonsmoking rooms has increased?

The p value is a very low value and using any significance level for example always so we can conclude that we have enough evidence to reject the null hypothesis, and we can say the the proportion of proportion of hotel guests requesting nonsmoking rooms has increased.  

Step-by-step explanation:

1) Data given and notation  

represent the number of requested nonsmoking rooms before

represent the number of requested nonsmoking rooms after

sample of number of requested nonsmoking rooms before

sample of number of requested nonsmoking rooms after

represent the proportion of requested nonsmoking rooms before

represent the proportion of requested nonsmoking rooms after

z would represent the statistic (variable of interest)  

represent the value for the test (variable of interest)

2) Concepts and formulas to use  

We need to conduct a hypothesis in order to check if the proportion of requested nonsmoking rooms after is higher then the proportion before , the system of hypothesis would be:  

Null hypothesis:  

Alternative hypothesis:  

We need to apply a z test to compare proportions, and the statistic is given by:  

  (1)

Where  

3) Calculate the statistic

Replacing in formula (1) the values obtained we got this:  

4) Statistical decision

For this case we don't have a significance level provided , but we can calculate the p value for this test.  

Since is a one side test the p value would be:  

So the p value is a very low value and using any significance level for example always so we can conclude that we have enough evidence to reject the null hypothesis, and we can say the the proportion of proportion of hotel guests requesting nonsmoking rooms has increased.  

step-by-step explanation: