Question

t2.13 a study recorded the amount of oil recovered from the 64 wells in an oil field, in thousands of barrels. here are descriptive statistics for that set of data from statistical software. descriptive statistics: oilprod variable n mean median stdev min max q1 q3 oilprod 64 48.25 37.80 40.24 2.00 204.90 21.40 60.75 based on the summary statistics, is the distribution of amount of oil recovered from the wells in this field approximately normal? justify your answer.

Explanation:

Step1: Recall normal - distribution properties

For a normal distribution, the mean and median are approximately equal, and the inter - quartile range ($IQR = Q_3 - Q_1$) is related to the standard deviation ($\sigma$) by $IQR\approx1.35\sigma$.

Step2: Compare mean and median

The mean of the oil production data is $\bar{x}=48.25$ and the median is $M = 37.80$. The difference between the mean and median is $48.25−37.80 = 10.45$, which is relatively large.

Step3: Calculate the inter - quartile range and compare with standard deviation

First, calculate the $IQR$: $IQR=Q_3 - Q_1=60.75 - 21.40=39.35$. The standard deviation is $s = 40.24$. If the data were normally distributed, we would expect $IQR\approx1.35\sigma$. Here, $1.35\times40.24 = 54.324$. The ratio $\frac{IQR}{1.35\sigma}=\frac{39.35}{54.324}\approx0.72$. The discrepancy between the expected and actual relationship between $IQR$ and $\sigma$ is significant.

Answer:

No, the distribution of the amount of oil recovered from the wells in this field is not approximately normal. The mean and median are quite different, and the relationship between the inter - quartile range and the standard deviation does not follow the pattern expected for a normal distribution.