 # Confidence interval Q

An investment analyst takes a random sample of 100 aggressive equity funds and calculates the average beta as 1.7. The sample betas have a standard deviation of 0.4. Using a 95% confidence interval and a z-statistic, which of the following statements about the confidence interval and its interpretation is most accurate? The analyst can be confident at the 95% level that the interval: A. 1.580 to 1.820 includes the mean of the sample betas. B. 1.622 to 1.803 includes the mean of the sample betas. C. 1.622 to 1.778 includes the mean of the population beta. D. 1.634 to 1.766 includes the mean of the population beta. why it doesn’t work when i apply the formula here (mean +/- 1,65 sd) ?? 1,7 +/- 1,65x0,4 give none of this anwser …

1.7 +/- (1.96)(0.4/root 100) should give C I think

Answer should be C. You need to divide std by 10.

1.96 for a 95% confidence interval…answer is C. At first i was looking for a population estimation of the mean, but it doesnt say it anywhere in this. If it asked you for a 95% confidence interval for the population mean you would have used the standard error instead of the standard deviation…just a reminder

1.65 is for 90% 1.96 is for 95%

it did ask for the standard error, no? Isn’t std/Sqrt(n) standard error?

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disregard what i just said pls

This is an awful question and if it confuses you, that’s probably good. Those betas just aren’t samples from the same population so just about everything after that is pretty messed.

1.7+/- [(1.96)x(.4/10)] = 1.622 to 1.778

Please explain. Aren’t they all aggressive equity funds? We are looking at their volatility around the market index…

I’m confused… why do we have to do sd/root 100???

Do we think E(beta fund 1) = E(beta fund 2)? In fact, I think we believe that E(beta fund 1) = population beta fund 1 which might be an important criterion for deciding which fund to invest in.

If I’m not mistaken, it’s because this is a sample…

But all of the betas are in relation to the market, and each fund has one beta. Why is that different from expected mean of fund1, expected mean of fund 2, etc, as compared to market mean?

So all the stuff you learned about C.I.'s says you calculate a statistic and determine its sampling distribution. If the sampling distribution is normal, then do the statistic ± z*s.e. thing. That doesn’t look like what we are doing here and yet we are trying to calculate a C.I. according to the question. That’s a serious pedagogical issue. So to step back - suppose that all these funds were constructed using some random procedure of selecting from the same group of stocks. The each of the betas would be a random sample from the same distribution and the average of the betas would be normally distributed because of the CLT. You could then do this x-bar ± z*s.e. to get an approximate C.I. It’s approximate because you are using z and the CLT. On the other hand there is serious question about whether that would be the best way to estimate the population mean (it wouldn’t be because you are double-using the CLT). Without telling you that the mutual funds all have the same underlying beta or are somehow constructed randomly, you don’t have any basis for any of this.