Re: Central limit theorem

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**Tweety** 25 children in a class each roll a fair die 30 times and record the number of sixes they obtain. Find an estimate of the probability that **the mean number of sixes** recorded for the class is less than 4.5.

"Mean **number** of sixes" implies a count -- it is a 6 or it is not a 6. This problem does not require any calculations with respect to the mean value of a collection of die roll values.

Think binomial.

EDIT: See my post below, please.

Re: Central limit theorem

Let $\displaystyle X$ be a RV indicating the number of sixes a child obtains from 30 (independent) single rolls of a fair die.

The probability that (one) roll of a (fair) die is 6 is given by $\displaystyle \hat{p}=\tfrac{1}{6}$.

We use the sample size $\displaystyle n=30$ in our calculations. Why not use $\displaystyle n=25$? Because 25 is the size of the sampling distribution of sample means. The CLT's requirement on the sampling distribution (of sample means) is that it is normal, which is going to approximately be the case when the sampling distribution consists of 25 sample means. For instance, $\displaystyle \{\bar{X}_1, \bar{X}_2, \ldots, \bar{X}_{25}\}$ is approximately normally distributed. Same deal if there were, say, 30, or 100 sample means. All you need to glean from the number 25 is that it is a sufficient quantity of sample means from which we may infer that the sampling distribution is approximately normal.

$\displaystyle \mu_{X} = E[X] = n\hat{p} = (30)(\tfrac{1}{6}) = 5 $

$\displaystyle \sigma_{X}=\sqrt{V(x)} = \sqrt{n\hat{p}(1-\hat{p})} = \sqrt{(30)(\tfrac{1}{6})(\tfrac{5}{6})} = \sqrt{4.1\overline{666}} \approx 2.04124 $

$\displaystyle \mu_{\bar{X}}=\mu_{X}=5 $

$\displaystyle \sigma_{\bar{X}}=\frac{\sigma_{X}}{\sqrt{n}} \approx \frac{2.04124}{\sqrt{30}} \approx 0.372678 $

$\displaystyle P\left(\bar{X}<4.5\right) = P\left(\frac{\bar{X}-\mu_{\bar{x}}}{\sigma_{\bar{X}}}<\frac{4.5-\mu_{\bar{x}}}{\sigma_{\bar{X}}}\right) \approx P\left(Z<\frac{4.5-5}{0.372678}\right) $$\displaystyle \approx P(Z<-1.34) \approx 0.0901$

Re: Central limit theorem

Thank you, but my book says the correct answer is awrt 0.11??