Note : I'm not asking you for solving the entire of question, i just want to focus on how to determine the value of $X$ and i'm in hurry, so please help me. Gimme some hints and i'll do the rest.
Given Problem :
A fabric manufacturer believes that the proportion of orders for raw material arriving late is $p =0.6$. If a random sample of $50$ orders shows that $24$ or fewer arrived late, the hypothesis that $p =0.6$ should be rejected in favor of the alternative $p\lt 0.6$.
Use the normal distribution.
(a) Find the probability of committing a type I error if the true proportion is $p =0.6$.
(b) Find the probability of committing a type II error for the alternatives $p =0.3$, $p =0.4$, and $p =0.5$.
We know :
$\begin{align}
n=50\\
X \le 24\\
q=1-p=0,4\\
H_0: p=0,6\\
H_1: p \lt 0,6
\end{align}$
My attempt :
From the problem above, i conclude that if i wanna find a type I error ($\alpha$), it's mean i have to reject the $H_0$ when it's true and absolutely need to find the probability when $X\le 24$.
Then, to find a type II error ($\beta$), i have to receive the $H_0$ when it's false or it's equivalent with find the probability when $X\gt 24$
$\begin{aligned}
Z&=\dfrac{X-\mu}{\sigma}\\
&=\dfrac{X-np}{\sqrt{npq}}\\
&=\dfrac{\mathbb{23,5}-(50)(0,6)}{\sqrt{(50)(0,6)(0,4)}}\\
\end{aligned}$
Nah, that is what i'm confusing about.
How to determine the value of $X$?
Example on my book (Probability and Statistics by Walpole) said in first case that i have $X\leq 24$, i have to put $23,5$ on $Z$. Second case, when i have $X\geq 24$, i have to put $24,5$ on $Z$
I'm not sure with those, then i checked the solution manual on slader there are 2 different answers.
First answer:
We don't need to change the $X$, so if you have either $X\leq 24$, or $X\geq 24$, it doesn't matter and just put $X=24$ on $Z$
Second answer:
If we have $X\leq 24$, you have to put $X=23,5$, and $X=24,5$ for $X\geq 24$.
Another case, if we have $X\le 24$ it's mean $X\leq 23$, and definitely we have to put $X=22,5$. And $X\ge 24$ gives you $X\geq 25$ and $X=25,5$
With all of those different answers, which one is true?
I'm getting confused.
Actually is there a good site or good book or anything that i can learn about hypothesis test and it trusted?
Especially, how to determine the $X$ value on $Z$?
Sorry if my explanation is bad. But i believe this is easy to understand.
Nb: i use
$$Z=\frac{X-\mu}{\sigma}$$
instead of
$$ Z=\frac{X-\mu}{\sigma_x/\sqrt{n}}$$
Just because of i don't have enough information about the standard deviation.
Please help and comment or reply me if my question isn't clear. Thanks.
Best Answer
X is the total mean while $\mu$ is what you 'think' the mean will be (i.e., what you say $\mu$ is in your null hypothesis. Hopefully this gets you started in the right direction.