Question

Corn. The average yield of corn (bushels per acre) for Iowa counties during 2018 can be described by a Normal distribution with a mean of 192.6 bushels per acre and a standard deviation of 20.3 bushels per acre. Use the 68-95-99.7 Rule (Empirical Rule) to answer the following questions.


(a) Create a well labeled normal curve for the average corn yield (bushels per acre) for Iowa counties during 2018. On this graph numerically label the mean (iv), the center 68% (iii) and (v), the center 95% (ii) and (vi) and the center 99.7% (i) and (vii). Report answers (i) through (vii.) in Canvas. You should also use this picture to help you answer the next few questions.
(b) The middle 95% of counties have a corn yield between what two values?
c) What is the value of the 84th percentile of corn yield for counties in Iowa?
(d) What proportion of counties have a corn yield between 152.0 and 212.9 bushels per acre?
(e) 0.15% of counties have a corn yield more than or equal to what value?
(f) What proportion of counties have a corn yield of at most 233.2 bushels per acre?

Answer 1

Using the empirical rule for standard deviation the graph for 68-95-99.7 is plotted as:

b) So, from the graph above the middle 95 % is within 2 standard deviation that is 152 to 233.2.

c) The 84th percentile value is the value that lies at +1 standard deviation above the mean because middle 68% is within 1 standard deviation and also the graph is symmetric so, left to -1 standard deviation is 16% so, 84th percentile is at +1 standard deviation. Thus the value will be 212.9

d)From the graph above it is clear that 152 lie at -2 standard deviation and 212.9 lies ta +1 standard deviation so, the percentage will be

=84-2.5

=81.5 % or 0.815.

e) 0.15% area from the left in the graph shown above is at -3 standard deviation below the mean and 0.15% are from the right tail is the value at which the value lies to which corn yield is equal or more to the value. and this value is at +3 standard deviation above the mean so, the value is 253.5

f) Since 233.3 lies at +2 standard deviation above the mean so, 0.95+0.025 =0.975 is the proportion.