Question

Question regarding hypotheses testing (screenshot attached)

Answer 1

I believe that the rejection region is indicated when \(\mu \cancel{=} \mu_0\):
\[Z \ge Z_{\alpha /2} ~~~\text{and}~~~Z \le -Z_{\alpha /2}\]Right? Given that the test statistical value would be\[Z=\frac{ \bar{x}-\mu_0 }{ \sigma / \sqrt{n} }\]

Answer 2

yes that looks right

Answer 3

Okay, so is the SD given by the 40% Is this just 40% of the population? 0.4*159?

Answer 4

the test statistic formula is the wrong one though

Answer 5

although, I think I see what you mean

Answer 6

Yeah =/

Answer 7

test statistic for proportion

\[\Large z = \frac{\hat{p} - p}{\sqrt{\frac{p*(1-p)}{n}}}\]

Answer 8

p = hypothesized proportion = 0.40
p-hat = sample proportion = 83/159 = 0.52201257861636
n = sample size = 159

Answer 9

Ahh, that makes sense. X)
So for part b). should I assume that \(\alpha=0.01\)?

Answer 10

yeah `significance level of 0.01` ----> alpha = 0.01

0.01/2 = 0.005 is the area in each tail

Answer 11

Right, that makes sense, hehe X)
Did you find that\[Z \ge 2.575 ~~~ and ~~~ Z \le -2.575\]

Answer 12

I'm getting -2.576 and 2.576 but close enough

Answer 13

I used this

http://onlinestatbook.com/2/calculators/inverse_normal_dist.html

and typed in 0.99 as the area and clicked the "between" radio button

Answer 14

Ohh, that's a very handy website.. I think I'll save that one, hehe X)

Answer 15

Did you get 3.14 for the test statistic?

Answer 16

Yeah I'm getting z = 3.140490955

Answer 17

Perfect! Now what do I compare this value to in order to either reject or fail to reject \(\text{H}_0\)?

Answer 18

yes you compare that to the first part

Answer 19

We hardly ever go through examples so I probably seem really stupid right now X)

Answer 20

you're doing fine

Answer 21

Oh right! So since it's greater than 2.576 then that would mean that we should reject it

Answer 22

correct

Answer 23

z = 3.140490955 makes z >= 2.576 true, so this test statistic is in the rejection region

Answer 24

This makes sense now! Thank you! :)

Answer 25

no problem