# Testing Hypotheses about Means and Proportions

1. Oct 19, 2014

### Sharp005

1. The problem statement, all variables and given/known data
Hello Everyone!

I am having trouble with one of my questions it goes:

It is claimed that fewer than 30% of Thunder Bay households have home delivery of the newspaper. To test this claim, a random sample of 390 Thunder Bay households reveals that only 95 get the paper delivered. Using an alpha value of 0.01, what conclusion can be reached about the claim? (Test Statistic z* or t*)

2. Relevant equations
t* or z*= x-Mu/S/sqrt(n)

3. The attempt at a solution
I think:
H0: P=0.3
H1: P not equal to 0.3

I am pretty sure I am doing t* because the sample size is greater than 30, but the formula we are to us is x-Mu/S/sqrt(n)
x=95
Mu=0.3
S=?
n=390 (I'm not sure if those are right either)
I don't know how to find the standard deviation (s) I tried using S= sqrt[(np)(1-p)] where n=390 and p=30% (0.3) but my final answer is 9.05, and when I plug it into the first formula I get a large number 206.7 and I don't think that is right.

I just don't know how to find standard deviation for this equation and what to do after I solve the equation
Thanks!

2. Oct 19, 2014

### Staff: Mentor

You need parentheses. The right side you wrote isn't what you meant, and would be interpreted as
$$x - \frac{\frac{\mu}{S}}{\sqrt{n}}$$
No. From your problem description, the claim is that "fewer than 30% of Thunder Bay households" get the newspaper at their home.
How should you write H0 and H1 to account for that?
Getting the null and alternate hypotheses right will also help you determine whether to use a one-tailed test or two-tailed test.

3. Oct 19, 2014

### Ray Vickson

The test statistic is

$$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0\,(1-p_0)}{n} }}$$
(See, eg., https://onlinecourses.science.psu.edu/stat200/node/53 ) Here, $p_0$ is the $p$ value in the null hypothesis, $\hat{p}$ is the observed $p$ and $n$ is the sample size: $p_0 = 0.3, n = 390$ in your case.

Note: you are trying to test against the alternative that $p > 0.3$, so the null hypotheses is that $p = 0.3$. If the observed $p$ is $\leq 0.3$ you do not want to reject the null hyposthesis, and if the observed $p$ is only slightly greater than 0.3, you still want to accept the null hypothesis (since random fluctuations could take an true $p$ slightly less than 0.3 into an observed $p$ slightly more than 0.3---no surpises there). However, if the observed $p$ is substantially greater than 0.3 you would be inclined to say the true $p$ is also, probably, greater than 0.3. So, you devise a critical threshold $p_c$ and reject the null hypothesis if the observed $p$ is $\hat{p} > p_c$. You want to keep the probability of incorrectly rejecting the null hypothesis low.

4. Oct 19, 2014

### Sharp005

Thanks, I looked on the site you provided, and I understand everything more! I am still unsure how to find the p^ for the equation, is it 95?
equation

5. Oct 19, 2014

### Ray Vickson

You tell me. I purposely did not tell you what it is in your case; just remember: it is a proportion.

6. Oct 19, 2014

### Sharp005

Oh!! ok I think I got it, I divided 95 by 390 to find my p^ and got ~0.244 and my final answer was -2.14 and on the Z table it is 0.4920

7. Oct 19, 2014

### Sharp005

Now would I accept the null or is there another step I am missing?

8. Oct 19, 2014

### Staff: Mentor

What do you have for your null and alternate hypotheses? They were wrong in your first post. Also what is your calculation for z?

9. Oct 19, 2014

### Sharp005

Ho: P=>0.3
H1: P<0.3

10. Oct 19, 2014

### Staff: Mentor

You have them backwards. "Fewer than" means strictly less than. The first sentence I quoted is the null hypothesis.

11. Oct 19, 2014

### Sharp005

Ok so
Ho: P<0.3 and H1: P=>0.3

12. Oct 19, 2014

### Ray Vickson

It is not usual in statistics to deal with a "composite" null hypothesis, such as H0: p <= 0.3. The problem is that you have no idea what number <= 0.3 you are supposed to use in the test quantity z (or whatever). That is why a null hypothesis would almost always be stated as a single value, such as H0: p = 0.3. Of course, the alternative H1: p > 0.3 is certainly composite in this case.