Describe the Distribution of the Sample Mean-Samples from a Population That Is Not Normal
-A farmer was interested 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 or 60 or more flights on average in the next 40 rows down which he drives his tractor?
A) $0.0612$
B) $0.4959$
C) $0.9388$
D) $0.0530$

Question

Quizplus · Accepted Answer

The Central Limit Theorem (CLT) applies here, stating that the distribution of the sample mean will approach a normal distribution as the sample size increases, regardless of the population's initial distribution. Given a sample size of 40, which is sufficiently large, we can use the normal distribution to calculate probabilities. The probability of observing a mean outside the range of 52 to 60 is found by calculating the Z-scores for these values and using the standard normal distribution. The correct probability is $0.0612$, indicating a relatively low chance of observing such extreme average counts in the next 40 rows.

Describe the Distribution of the Sample Mean-Samples from a Population 0.06120.06120.0612

Describe the Distribution of the Sample Mean-Samples from a Population $0.0612$