Restaurateur Denny Valentine is evaluating two sites, Raymondville and Rosenberg, for his next restaurant. He wants to prove that Raymondville residents (population 1) dine out more often than Rosenberg residents (population 2). Denny plans to test this hypothesis using a random sample of 81 families from each suburb. His null hypothesis is .

represent the sample for the sample 1

represent the sample for the sample 2

represent the deviation for the sample 1

represent the deviation for the sample 2  

represent the sample size for the group 1

represent the sample size for the group 2

And on this case if we create the system of hypothesis we have this:

Null hypothesis :

Alternative hypothesis:

Step-by-step explanation:

Previous concepts

A hypothesis is defined as "a speculation or theory based on insufficient evidence that lends itself to further testing and experimentation. With further testing, a hypothesis can usually be proven true or false".  

The null hypothesis is defined as "a hypothesis that says there is no statistical significance between the two variables in the hypothesis. It is the hypothesis that the researcher is trying to disprove".  

The alternative hypothesis is "just the inverse, or opposite, of the null hypothesis. It is the hypothesis that researcher is trying to prove".  

On this caseHe wants to prove that Raymondville residents (population 1) dine out more often than Rosenberg residents (population 2). So this statement need to be on the alternative hypothesis. And the complement of the alternative hypothesis would be on the null hypothesis.  

Solution to the problem

We assume that we have the following statistics:

represent the sample for the sample 1

represent the sample for the sample 2

represent the deviation for the sample 1

represent the deviation for the sample 2  

represent the sample size for the group 1

represent the sample size for the group 2

And on this case if we create the system of hypothesis we have this:

Null hypothesis :

Alternative hypothesis: