# Eating competition central limit problem

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.

• 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. – Michael May 16 at 20:14
• Wow, I forgot about that. Thanks. – Tomás Palamás May 16 at 20:18