Normal distribution question I encountered a question here:
Sodium content of hot dogs are normally distributed with mean = 140 mg and standard deviation = 8 mg. If we randomly select 49 hot dogs, what is the probability that their mean sodium content will be greater than 145 mg?
I know this question can be vague, but I assumed the sodium content from 49 hot dogs follows normal distribution, so I found z score that corresponds to 145 given N(140,64) and found the probability to be 0.266.
 A: We're talking about the mean of iid random variables, so we need to use the central limit theorem.
Briefly, the CLT says that the sample mean's sampling distribution is normal with mean $\mu$ and variance $\sigma^2/n$.  Here, $\mu$ and $\sigma^2$ are population level parameters.
So what is the probability that 49 hot dogs have  a mean sodium content larger than 145 given the information in the question?  The z statistic is
$$ z = \dfrac{145 - 140}{8/7} = 4.375 $$ 
and using r
1 - pnorm(4.375)
>>> 6.07e-06

So it is incredibly improbable. We can verify this by just drawing from a normal with mean 140 and sd 8.  Looks like we get one observation in almost 166,000 observations, so let's just sample a million points and see what our probability is empirically.
s = replicate(1000000,{
  x = rnorm(49, 140, 8)

  mean(x)
})


mean(s>145)
>>>4e-6

That's on the right order of magnitude, so I feel pretty comfortable with my answer.
A: Comment:  Here is a figure to go with the Answer of @DemitriPananos (+1):
The blue curve is for the population density $X\sim\mathsf{Norm}(\mu=140,\sigma=8).$ The maroon curve is for the distribution of
$\bar X,$ the average for $n = 49$ hot dogs:
$$\bar X \sim \mathsf{Norm}(\mu_{\bar X} = 140, \sigma_{\bar X} = 8/7).$$
This curve is 1/7th as 'wide" as the population curve, so in order to
enclose area $1,$ it must be 7 times as tall.
The area you seek is the (tiny) area under the maroon curve to the right
of the vertical black line: $P(\bar X > 45) = 1 - P(\bar X \le 45) \approx 0,$ computed in R statistical software where pnorm is a normal CDF. 
1 - pnorm(145, 140, 8/7)
[1] 6.071624e-06


