Mistake in book. This is an example from an indtroduction to the central limit theorem written by a professor at my university. Is this a typo?

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    $\begingroup$ The first sentence in the exercise is grammatically incomplete (the adjective "independent" isn't referred to any noun), therefore yes, something is missing. $\endgroup$ – user239203 Jun 18 at 23:58
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    $\begingroup$ @Gae.S. Shouldn't the math be $P\{T_{144}<150\}=P\{\frac{T_{144}-144}{12}<\frac{150-144}{12}\}=P\{Z_{144}<.5\}\approx .69$? $\endgroup$ – ernesto Jun 19 at 0:01
  • $\begingroup$ @ernesto Yes, your computation is correct. The book seems to have made several typos. $\endgroup$ – angryavian Jun 19 at 0:21
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    $\begingroup$ This question is extremely vague about what the "typo" is referring to. In fact one might reasonably assume that the typo is referring to the handwritten correction, but your comment suggests otherwise. @ernesto Please edit the question to clarify. $\endgroup$ – Erick Wong Jun 19 at 0:37
  • $\begingroup$ @ernesto If the text doesn't describe the distribution of $T_{144}$, then addressing the computations is pointless: if I evaluate them according to the correction you suggested, then I can't know whose mistake it is. $\endgroup$ – user239203 Jun 19 at 7:16

You hit the jackpot on this one: Three errors instead of one.

(a) As @Gae.S. says, and you have suggested in your handwritten note, there needs to be mention of the exponential distribution with parameter $\lambda=1.$

(b) As you have commented, the z score should be $(150-144)/12,$ not $(144-140)/12$ so the normal approximation is $0.6915 \approx 0.69.$

[1] 0.6914625

(c) In addition, $n = 144$ is not quite large enough for a good normal approximation. The correct value is $P(T_{144} < 150) = 0.6988 \approx 0.70,$ as computed in R because $T_{144}\sim \mathsf{Gamma}(\mathrm{shape}=144,\mathrm{rate}=1),$ as can be shown by moment generating functions.

pgamma(150, 144, 1)
[1] 0.6987842

If the purpose is to show how fast or slowly the CLT converges, there may be some point to using a normal approximation. But if one wants the correct answer to even two places it is necessary to use the exact distribution theory for $n = 144.$ Because of its marked skewness the exponential distribution "converges slowly to normal'.

A better exercise might be to look at $P(T_{625}< 630) = 0.58,$ to two place accuracy by either method.

pgamma(630, 625, 1)
[1] 0.5842562
[1] 0.5792597

Note: Any textbook author has to face the fact that his or her book will contain errors. I am always appreciative when errors are brought to my attention, hoping in vain for an error-free second (or $n+1$st) edition. I hope you will send the professor a brief 'possible erratum' email about (a) and (b), and perhaps noting the numerical discrepancy in (c).


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