Nate is a competitive eater specializing in eating hot dogs. From past experience we know that it takes him on average $15$ seconds to consume one hot dog, with a standard deviation of $4$ seconds. In this year's hot dog eating contest he hopes to consume $64$ hot dogs in just $15$ minutes. Use the CLT to approximate the probability that he achieves this feat of skill.

I am having trouble setting up this problem.

Outline: Let $S_n:=X_1+X_2+\dots X_{64}$ where each $X_i$ is the $i^{th}$ hot dog consumed. We have $E[S_n]=64*15=960$, $Var(S_n)=64*4=256.$ Using CLT, I get $0.6293$ but the answer key says its $.0304$. I'm guessing my setup is wrong? I'm not sure whether I should work with in terms of seconds or minutes.

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    $\begingroup$ Assuming i.i.d. $\{X_i\}$ we get $Var(S_n) = \sum_{i=1}^{64}Var(X_i) = \sum_{i=1}^{64}4^2 = 1024$ seconds-squared. Recall that variance $\sigma^2$ is the square of standard deviation $\sigma$. (Here $\sigma=4$ seconds.) It seems easiest to work everything in units of seconds. $\endgroup$ – Michael May 16 at 20:14
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    $\begingroup$ Wow, I forgot about that. Thanks. $\endgroup$ – Tomás Palamás May 16 at 20:18

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