Describe the Distribution of the Sample Mean-Samples from a Population That Is Not Normal
-A farmer was interest in determining how many grasshoppers were in his field. He knows that the distribution of grasshoppers may not be normally distributed in his field due to growing conditions. As he drives his tractor down each row he counts how many grasshoppers he sees flying away. After several rows he figures the mean number of flights to be 57 with a standard deviation of 12 . What is the probability of the farmer will count 52 or fewer flights on average in the next 40 rows down which he drives his tractor?
A) $0.0041$
B) $0.4959$
C) $0.9959$
D) $0.0410$

Question

Quizplus · Accepted Answer

The Central Limit Theorem states that the distribution of the sample mean will be approximately normal if the sample size is large enough (usually n > 30 is considered sufficient). Here, the sample size is 40, which is large enough. The standard error of the mean is $ \frac{12}{\sqrt{40}} \approx 1.897 $. The z-score for 52 is $ \frac{52 - 57}{1.897} \approx -2.637 $. Using the standard normal distribution table, the probability of a z-score being less than -2.637 is approximately 0.0041.

Describe the Distribution of the Sample Mean-Samples from a Population 0.00410.00410.0041

Describe the Distribution of the Sample Mean-Samples from a Population $0.0041$