Using the Central Limit Theorem, it is found that since the sample size is greater than 30, a normal approximation can be used, hence the test can be made.
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the sampling distribution is also approximately normal, as long as n is at least 30.
In this problem, the distribution of lengths is skewed, however, since the sample size is of 100 greater than 30, a normal approximation can be used, hence the test can be made.
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