Answer:
The population does not need to be normally distributed for the sampling distribution of [tex]\bar{X}[/tex] to be approximately normally distributed. Because of the central limit theorem. The sampling distribution of [tex]\bar{X}[/tex] is approximately normal.
Step-by-step explanation:
We have a random sample of size [tex]n = 57[/tex] from a population with [tex]\mu = 69[/tex] and [tex]\sigma = 2[/tex]. Because n is large enough (i.e., n > 30) and [tex]\mu[/tex] and [tex]\sigma[/tex] are both finite, we can apply the central limit theorem that tell us that the sampling distribution of [tex]\bar{X}[/tex] is approximatelly normally distributed, this independently of the distribution of the random sample. [tex]\bar{X}[/tex] is asymptotically normally distributed is another way to state this.
