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.

  • $\begingroup$ Hi @user516076 - I just tried to give some direction to your question. If you need more assistance, please let me know! $\endgroup$ – 324 Dec 11 '19 at 16:02

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.

  • $\begingroup$ Ummm.. yes thanks anyway. I think i made a wrong question. What i'm asking about is, how $x$ can be determined? I 've read about Continuity Correction Factor, and that was my main question about. But i can't really relate that with my thought. For example, if the discrete $x=6$ then, the continuous is $5.5< x < 6.5$. And if the discrete $x>6$, then the continuous is $x>6.5$. And for the last example, if the discrete $x\leq 6$, then the continuous is $x<6.5$. But, where do they come from?????? $\endgroup$ – user516076 Dec 11 '19 at 22:58

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