Final answer:
To find the approximate value of pr{4<=Y DASH <=5}, standardize the values using the formula z = (x - mean) / standard deviation and use a standard normal distribution table or calculator. The approach used in part (a) is approximately correct due to the Central Limit Theorem, which states that as the sample size increases, the sample mean approaches a normal distribution. However, the same approach would not be valid for a sample size of 2.
Step-by-step explanation:
To find the approximate value of pr{4<=Y DASH <=5}, we can use the standard normal distribution since the sample size is large. First, we need to standardize the values of 4 and 5 using the formula z = (x - mean) / standard deviation. Substitute the values for mean and standard deviation given in the question to get the standardized values. Then, use a standard normal distribution table or calculator to find the probability between these two standardized values.
For part (b), the approach used in part (a) is approximately correct even though the population distribution is not normal because of the Central Limit Theorem. The theorem states that as the sample size increases, the distribution of the sample mean approaches a normal distribution regardless of the shape of the population distribution. This means that for a large enough sample size, the same approach can be used regardless of the population distribution. However, for a sample size of 2, the Central Limit Theorem does not apply, and the same approach would not be valid.