A 20-minute consumer survey mailed to 400 adults aged 25-34 included a $5 Starbucks gift certificate. The same survey was mailed to 400 adults aged 25-34 without the gift certificate. There were 57 responses from the first group and 31 from the second group.
(a) Perform a two tailed test comparing the response rate (proportions) at a = .05
(b) Form a confidence interval for the difference of proportions, without pooling the sample. Does it includes zero?

a)  

If we compare the p value and the significance level given we see that so we can conclude that we have enough evidence to reject the null hypothesis, and we can say the the two proportions are different at 5% of significance. .  

b)

And the 95% confidence interval would be given (0.02187;0.10813).  

And as we can see the confidence interval not contains the 0.

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".  

The margin of error is the range of values below and above the sample statistic in a confidence interval.  

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

represent the real population proportion for the gift

represent the estimated proportion for the gift

is the sample size required for the gift

represent the real population proportion for no gift

represent the estimated proportion for no gift

is the sample size required for no gift

represent the critical value for the margin of error  

The population proportion have the following distribution  

Part a

We need to conduct a hypothesis in order to check if the proportions are equal , 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

Calculate the statistic  

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

Statistical decision  

We have a significance level provided , and now we can calculate the p value for this test.  

Since is a two tailed test the p value would be:  

If we compare the p value and the significance level given we see that so we can conclude that we have enough evidence to reject the null hypothesis, and we can say the the two proportions are different at 5% of significance. .  

Part b

The confidence interval for the difference of two proportions would be given by this formula  

For the 95% confidence interval the value of  and , with that value we can find the quantile required for the interval in the normal standard distribution.  

And replacing into the confidence interval formula we got:  And the 95% confidence interval would be given (0.02187;0.10813).  

We are confident at 95% that the difference between the two proportions is between

And as we can see the interval not contains the 0.

starting with:    

5 + x - 12 = 2x - 7   

5 + x - 12 - x = 2x - 7 - x (subtract x from both sides)   

5 - 12 = x - 7 (combine and cancel x's from both sides)   

5 - 12 + 7 = x - 7 + 7 (add 7 to both sides)   

0 = x (add numbers together on both sides)       

so in order for the equation to be true, x must be 0.   

trying x = -0.5   

5 + x - 12 = 2x - 7 (original equation)   

5 + -0.5 - 12 = 2(-0.5) - 7 (plug in value of x)   

5 + -0.5 - 12 = -1 - 7 (did multiplication)   

-7.5 = -8 (did addition on both sides)       

since -7.5 is obviously not equal to -8, the value of -0.5 for x is wrong.   

trying x = 0   

5 + x - 12 = 2x - 7 (original equation)   

5 + 0 - 12 = 2(0) - 7 (plug in value of x)   

5 + 0 - 12 = 0 - 7 (did multiplication)   

-7 = -7 (did addition on both sides)       

since -7 equals -7, that demonstrates that the value of 0 for x is correct.   

trying x = 1   

5 + x - 12 = 2x - 7 (original equation)   

5 + 1 - 12 = 2(1) - 7 (plug in value of x)   

5 + 1 - 12 = 2 - 7 (did multiplication)   

-6 = -5 (did addition on both sides)       

since -6 is obviously not equal to -5, the value of 1 for x is wrong.

27 +27 or 27times 2=54

the diffrence is 0