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Basic Question 5 of 13
A manufacturer of light bulbs claims that the distribution of the light bulb life span has a mean of 60 hours and a standard deviation of 4 hours. The competition decides to check this claim by purchasing 30 light bulbs and testing them to determine the life span of the light bulbs. If the competition finds that the mean of the sample population is 55 hours, what can be said about the manufacturer's claim that the light bulbs' mean life span is 60 hours?
B. The competition could not state that the manufacturer's claim is probably true.
C. The competition could state that the manufacturer's claim is probably untrue.
D. The competition could state that the manufacturer's claim is probably true.
A. The competition could not state that the manufacturer's claim is untrue.
B. The competition could not state that the manufacturer's claim is probably true.
C. The competition could state that the manufacturer's claim is probably untrue.
D. The competition could state that the manufacturer's claim is probably true.
User Contributed Comments 25
| User | Comment |
|---|---|
| quincy | why is A, not C? |
| cfaman | the null hypothesis is "the mean is less than 60 hours". The competition fails to reject its null hypothesis so D is correct. |
| Aimy | cfaman, fail to reject null hypothesis means going along with "the mean is less than 60hours". Looks like C is correct. |
| tony1973 | the null hypothesis is what the manufacturer claims: the average life is 60 hours (or more). Even if the competition fails to reject the null hypothesis, it cannot conclude the claim is true: it can only conclude that it has not strong evidence to that it's untrue. So A is correct. |
| vincenthuang | agree with tony1973, the test mean is 55 which is less than 60, it seems that the manufacturer's claim is untrue, but we cannot conclude it. |
| gruszewski | why do you think we failed to reject Ho? |
| Janey | This is saying that we dont have enough proving evidence that the manufacturer is wrong, A is correct. |
| surob | Comletely confused |
| Khadria | We can't say if the null hypothesis is less than or more than 60 hrs. Hence H(0) MU = 60 hrs. The claim can be less than equal to OR more than equal to (WE DONT KNOW). Since the competitotr found MU < 60 hrs, H(0) can be rejected DEFINITELY (not PROBABLY). |
| mchu | Lifespan means maximum life. so in this case,it is less than 60(not more than 60). since the sample mean is 55. it is also less than 60. so the competitor could not state that the manufacturer's claim is untrue. |
| tom1980 | here is my reasoning, correct me if i'm wrong: it may depend on the null hypothesis whether the life span has 60 hours If 1)yes, then the computed statistic -6.846 seems large enough to reject the null hypothesis and the statement is untrue 2)no, then the statement is true. so A may be correct? |
| bobert | OK I think I made sense out of it. This is the way I understand null/alternate hypothesis, which seems to work in every case I've tried. Not the way the book/notes really explain it though. Please read on though it may help tremendously. Notes Def: Null: The statement to be tested, and will be either rejected or fail to be rejected. Alternate: The statement accepted if the sample data provides sufficient data that the null hypothesis is false. This is the way I go about it which for explanation's sake I feel is much easier. I'll use this question as the example. 1) See what is trying to be determined. So for this one, it appears that we want to see if their claim is true that the light bulbs last 60hours on average (at least from the competition's point of view). 2) Find the null. From a knee jerk reaction I said, "hmm, well if x-bar is 55, then we probably want to see that 55 < 60 right" This sounds like what the problem wants you to think. You know you start with null, you may think (as I briefly did) null is wrong (which it is) but again, from the competition's point of view, x-bar < 60 is what is trying to be proved, and therefore the H(A). This makes the null x-bar => 60 3) My third step is generally finding the alternate, which we already did. From here things should be clearer to you. Conclusion to my "lesson": H0: x-bar > 60 HA: x-bar <= 60 X-bar = 55 Therefore, we do not reject the null. Because the null is rejected, we know there cannot be any conclusions drawn. Reasons for NOT the other choices: B. The competition could not state that the manufacturer's claim is probably true. --> The first part of B is true, they really cannot state anything as no conclusion is drawn. The second half of the answer, which is worded very craftily and almost got me, is just saying that their claim is true. However from the sample mean, it is not true. C. The competition could state that the manufacturer's claim is probably untrue. --> For both C and D, the immediate indicator they are wrong is that they both say they "state the manufacturer's claim" meaning conclude, and no conclusions can be derived from a fail to reject null. D. The competition could state that the manufacturer's claim is probably true. --> See explanation to (C). Therefore: (finally!) A. The competition could not state that the manufacturer's claim is untrue. IS CORRECT Therefore (A) IS correct. Keep working at it, and it will all make sense eventually. |
| StanleyMo | hello all, this is my thinking. After calculating, we got Z<-6.xx and we notice that the biggest probability for Z table is 4 or -4!, how do you calculate the probability out of the Z table. i think this might be some good clue. |
| chamad | This is how I solved the question: H0: mean = 60h. std dev = 4. H0 true means mean is between 56(60 - 4) and 64 (60 + 4). since 55 is outside this range H0 is false. A CORRECT answer |
| SuperKnight | I am a bit confused, it almost seems like everyone is just trying to fit something to make this question correct, I don't understand why A is the correct answer. Perhaps its in the wording? The way I looked at it was like this, the competition is trying to prove that the mean amount of hours is below 60, so H(0) >= 60 (claim of the manufacturer), H(A) < 60 .. ok.. so the test statistic = (sample statistic - parameter value under H(0))/standard error of the statistic.... test statistic = (55 - 60)/(4/sqrt(3)) = -6.85... now if you look at the z distribution chart you will see its off the grid.. this is very significant evidence to reject H(0) .. which would be answer C. So why is the correct answer A? is it in the wording? or is the power of the test too weak? |
| alallstar | It might be something to do with the fact that we have the population SD instead of sample SD. |
| erinelize | Answer A just doesn't make logical sense to me. If the competition is trying to prove that the lightbulbs last less than 60 hours and they come up with a mean of 55 hours and the standard deviation is only 4 AND the test statistic is -6.85 how is it that they can't state that the claim of 60 hours is untrue? What was the point of testing the bulbs in the first place? At what mean would the competition be able to say that the claim was probably untrue? Will someone please explain? |
| dcfa | this is what I think - the competetion is testing manufacturers claim so the test is Ha:mu >= 60 Ho:mu < 60 n=30 so z-statistic can be used decision rule: reject Ho if test-statistic < Z-alpha test-statistic=x-bar - mu / sd/sqrt(n)= - 6.8465 -6 < -4 sd(any one sided Z-alpha) so reject Ho in favor of Ha i.e., cannot say that the manufacturers claim is untrue |
| ialink | do not divide the SD by sqrt(N) |
| Vicc | Its all semantics. You can never draw conclusions, i.e. "state," "prove," or "disprove" H0. However, you can draw conclusions about HA. To clear things up, the right answers should look like this: 1. The competition can state that the average life span is probably less than 60. (Accepting HA) 2. The competition can state that the data is inconsistent with the manufature's claim. (Reject H0). |
| johntan1979 | You all think too much and missed the MOST IMPORTANT point in hypothesis testing... if the significance level is not defined, how on earth can you conclude the test result, regardless of how large or how small that value may be? |
| andyfink | johntan, I believe that when significance level is not stated, the general rule is to use 95% level of confidence/5% significance. That said, I tend to agree with Vicc. |
| jrojasut09 | I agree wuth johntan - the wording of the answer implies that without the significance level, we can not prove the claim is untrue |
| sshetty2 | I see johntan/AN's logic but with a z score that's so low I think you should be able to say that's it's probable that the null would be rejected as stated above |
| teodomi | you cannot accept Ha, you can just reject/fail to reject H0. Second, H0 is always stated at the point of equality(<=, =>,=). |
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Martin Rockenfeldt
Learning Outcome Statements
explain hypothesis testing and its components, including statistical significance, Type I and Type II errors, and the power of a test
CFA® 2024 Level I Curriculum, Volume 1, Module 8.