### CFA Practice Question

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### CFA Practice Question

Indicate which assumptions are needed to use the sample mean and normal tables to test a hypothesis about a population mean, μ, and known variance, σ2. Which of the following assumptions are needed to use x-bar, the mean of the data, and normal tables to test a hypothesis about μ?

I. It is a random sample.
II. The population distribution is normal.
III. The sample size is large.

A. I, II, and III
B. I and either II or III
C. II and III

User Comment
surob Why B? Can someone explain please?
dhiru Because if the sample size is large you can apply the central limit theorm and even if the population is not normally distributed it doesn't matter. but if the population is distributed normally the simple size is less of an issue.
Hopefully my reasoning is correct.
studyprep dhiru + this: If you pick a small sample size (usually less than 30) and population is not normalized too, then the results would be biased and won't be the population parameters representative. Issue can be resolved as given in B.
apiccion To you use a Z-score table the underlying distribution needs to be normal.

However, recall that the central limit theorem states that the sampling distribution of the mean x-bar approaches a normal distribution as the sample size increases.

What this means is that suppose you have a population and you want to find infer it's mean value. You select a small random sample of the population (say 30) and calculate the mean value of that sample. Now imagine that you repeat that step over and over again say 100 times.

You then plot a graph of the frequency distribution of those sample means. The central limit theorem says that your graph will be approximately normal REGARDLESS of whether or not the population mean is normal.
karnick dhiru is correct!(i and ii are obvious). The concern is only about why i and iii?: (Refer to page 482 of book 1: "The central limit theorem allows us to make a quite precise probability statements about the population mean by using the sample mean, whatever the distribution of the population, because the sample mean follows an approximate normal distribution for large-size samples.")