Question 1191810
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We're going to use a Two Proportion Z-Test.


Hypotheses:
Ho: p1 = p2
Ha: p1 < p2
The claim is in the alternative hypothesis and the claim is p1 < p2
The inequality sign in the alternative hypothesis says we have a left-tailed test.


Sample sizes
n1 = 766
n2 = 649


Sample proportions
phat1 = 0.864
phat2 = 0.911


phat = x/n
x = n*phat


x1 = n1*phat1
x1 = 766*0.864
x1 = 661.824
x1 = 662


x2 = n2*phat2
x2 = 649*0.911
x2 = 591.239
x2 = 591


pbar = pooled or average sample proportion
pbar = (x1+x2)/(n1+n2)
pbar = (662+591)/(766+649)
pbar = 0.885512
Think of pbar in the same light as something like xbar
It's the letter p with a horizontal bar over top.


SE = standard error
SE = sqrt( pbar*(1-pbar)*(1/n1 + 1/n2) )
SE = sqrt( 0.885512*(1-0.885512)*(1/766 + 1/649) )
SE = 0.016987


z = test statistic
z = (phat1 - phat2)/SE
z = (0.864 - 0.911)/0.016987
z = -2.767


Since the test statistic is recorded to three decimal places, we can't use a standard Z table unfortunately.
Instead, we can only use a p-value calculator.
There are countless many such free calculators out there, so feel free to pick your favorite.
One such option is this really neat online calculator
<a href = "https://davidmlane.com/normal.html">https://davidmlane.com/normal.html</a>
Make sure the mean and standard deviation are 0 and 1 respectively.


The p-value you should get is roughly 0.0028
Keep in mind this is a left-tailed test.


Comparing this to the level of significance alpha = 0.005, we see that the p-value is smaller. 


Whenever the p-value is smaller than alpha, we reject the null. 
A useful saying is "If the p-value is low, then the null must go".


Since we rejected the null, we accept the claim that p1 < p2.


Translating back to the original problem, we have the conclusion that 
(c)The sample data support the claim that the first population proportion is less than the second population proportion.
Once again, the claim is in the alternative hypothesis.


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<font color=red>Answers:</font>
test statistic = -2.767
p-value = 0.0028
The p-value is <u>less</u> than alpha = 0.005
It means we <u>reject the null</u>


Conclusion: 
(c)The sample data support the claim that the first population proportion is less than the second population proportion.

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