Question

Corn. The average yield of corn (bushels per acre) for Iowa counties during 2019 can be described by a Normal distribution with a mean of 180.5 bushels per acre and a standard deviation of 16.8 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)

(b) The middle 95% of counties have a corn yield between what two values?

(c) What is the value of the 16th percentile of corn yield for counties in Iowa?

(d) What proportion of counties have a corn yield between 146.9 and 197.3 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 214.1 bushels per acre?

Answer 1

Given:

= 180.5, = 16.8

a)

b) The middle 95% of counties have a corn yield between 146.9 & 214.1

C)

Find: 16th Percentile

P(X < Xo) = 16% = 0.16

P(X < -0.9945) = 0.16 ....................................From Normal distribution table,

Or by using Invnorm function, Invnorm(0.16) = - 0.9945

Therefore, The required value of X is,

Z = (X - ) /

-0.9945 = (X - 180.5) / 16.8

X - 180.5 = -16.71

X = 180.5 - 16.71

X = 163.79

d)

Find: P(146.9 < X < 197.3)

P( 146.9 < X < 197.3) = P( -2 < Z < 1)

P( 146.9 < X < 197.3) = P(Z < 1) - P(Z < -2)

P( 146.9 < X < 197.3) = 0.8413 - 0.0228 ..............Using Standard Normal table

P( 146.9 < X < 197.3) = 0.8186

e)

0.15% = 0.0015 of counties have a corn yield more than or equal to what value?

That is, 1 - 0.0015 = 0.9985

Find: 85th percentile

P(X < Xo) = 99.85% = 0.9985

P(X < 2.9677) = 0.9985 ....................................From Normal distribution table,

Or by using Invnorm function, Invnorm(0.9985) = 2.9677

Therefore, The required value of X is,

Z = (X - ) /

2.9677 = (X - 180.5) / 16.8

X - 180.5 = 49.86

X = 180.5 + 49.86

X = 230.36

f)

Find: P(X < 214.1)

P(X < 214.1) = P(Z < 2)

P(X < 214.1) = 0.9772