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) 0.4959
B) 0.9388
C) 0.0612
D) 0.0530

Question

Quizplus · Accepted Answer

To find the probability of counting 52 or fewer flights or 60 or more flights on average in the next 40 rows, we use the normal distribution since the sample size is large enough (n=40) for the Central Limit Theorem to apply, even if the original distribution is not normal. The mean ($\mu$) is 57, and the standard deviation ($\sigma$) is 12. However, we're interested in the distribution of the sample mean for 40 rows, so the standard deviation of the sample mean ($\sigma_{\bar{x}}$) is $\sigma / \sqrt{n} = 12 / \sqrt{40}$.First, we calculate the Z-scores for 52 and 60:$$Z_{52} = \frac{52 - 57}{12 / \sqrt{40}}$$$$Z_{60} = \frac{60 - 57}{12 / \sqrt{40}}$$Using the Z-table or a calculator, we find the probabilities corresponding to these Z-scores. The probability of observing a mean of 52 or less is the area to the left of $Z_{52}$, and the probability of observing a mean of 60 or more is one minus the area to the left of $Z_{60}$. The total probability is the sum of these two areas.Given the options, the correct calculation leads to a probability that matches option C, indicating a relatively low probability of observing an average number of flights outside the range of 52 to 60 in the next 40 rows. This calculation involves using the standard normal distribution to find the areas (probabilities) corresponding to the calculated Z-scores and summing them for the final answer.

A Farmer Was Interested in Determining How Many Grasshoppers Were