# Quick question about : how to determine the $X$ value for hypothesis test (Z-Test)

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.

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.
• 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?????? – user516076 Dec 11 '19 at 22:58