I am lost with this question. I spent a few hours trying to recheck my calculations, but something is wrong.
The spinning time of a newly developed wheel = 75rpm under specified test conditions. The spinning time is known to be normally distributed with mean value 75rpm and standard deviation 9 rpm. Engineers have proposed new brakes to decrease average spinning time. It is believed that the spinning time with new brakes will remain normally distributed with standard deviation = 9. Because of the expense associated with new brakes, evidence should strongly suggest a reduction in average spinning time before such a conclusion is adopted. Only if the null hypothesis can be rejected will the new brake be declared successful and then be used. If mu denotes the true average spinning time when the new brakes are used, compute the type 2 error when mu = 72. What will happen to Beta if the sample size increases from n=25 initially to n=100? {Cutoff value for Type-1 error = 1% is 70.8 rpm (lower-tailed). }
a) Decreases
b) Increases
c) Remains constant
OA - Decreases.
However, as per my calculation, Beta is increasing. H0: mu = 75 H(alternative): mu < 75. alpha = 1%, cutoff = 70.8
Beta = type 2 error when mu =72, H0 is accepted when it is false. Hence, I need to focus on the region P(X>70.8, where 70.8 was the cutoff for lower tailed Type1 error)
X = random variable that denotes the spinning time from a normal distribution wiht mean value mu and standard deviation = 9.
Case 1: n =25 Sigma/sqrt(n) = 9/5 = 1.8 Beta with n=25 = P(type 2 error, when mu =72) =P(X > 70.8 when mu =72) =P(Z > (70.8 - 72)/ 1.8)) =0.7486 Now, Case 2: n =100 Sigma/sqrt(n) = 9/10 = 0.9 Beta with n=100 = P(type 2 error, when mu =72) =P(X > 70.8 when mu =72)
=P(Z > (70.8 - 72)/ 0.9)) =1 - phi((70.8 - 72)/ 0.9) =Phi(1.33) =0.9082 Shouldn’t Beta decrease because of increasing n and constant type-1 error? Can someone please help me? Thanks in advance.
I noticed that in your question you say the sample increases from 25 to 1000 but you use 25 and 100 in your calculations…
However, I don’t think you need to use calculations for this. Does your answer key use them?
For one, I figured since the only change was the increase in sample size that beta decreased. To me they are trying to see if you understand the relationship of beta with sample size, alpha, etc, rather than if you can calculate something.
I looked at your question, and my first thought was, “Gross!”
I haven’t analyzed the specific calculations, and doubt that I will. You needn’t do anything remotely this elaborate on the CFA exams (any level). What you need to understand is that:
If the probability of a Type I error increases (with a given sample size), the probability of a Type II error decreases, and vice versa
To reduce the probability of both Type I and Type II errors, you need more data
From these, you can infer that if the amount of data increases and the probability of a Type I error (α) remains constant, the probability of a Type II error will decrease.
You won’t ever have to calculate the probability of a Type II error on a CFA exam.
Thank you S2000magician. I believe I am following your advice of understanding the concepts rather than memorizing a bunch of plug-and-chug formulas. So, I thought of solving this question. (I do have a bit of a background in Stochastic Processes so I thought of verifying bullet#2 in your post, which is also there in the curriculum.) I took this question, and thought of mathematically proving that when Type 1 error is constant and when the sample data increases, Type 2 error goes down. Unfortunately, and very sadly, I couldn’t arrive at an answer. Do you mind explaining what is my mistake? This will help me to internalize these concepts very well.
Moreover, I am sorry if you are pissed off with me.
I’m not angry at all. What I meant was, “Gross! There’s no way a question on the Level I exam (or the other two, for that matter) is going to require this much work to solve.”
I infer that beta is the probability of a Type II error, but I have no idea what the phi function is. (I presume that it’s not the Euler phi function, that gives the number of divisors of a positive integer.) Unfortunately, I’m in the midst of preparing for a three-day Level II review in Los Angeles, so I won’t have much time to look at it until next week.
S2000Magician is da bomb. Thank you for your answer which confirms the answer I arrived at in my head for the same reasons. Glad to hear this kind of calculation is reserved for the Level 4 exam.
