Question

For the wheat-yield distribution of Exercise 4.3.5, find

(a) The 65th percentile
(b) The 35th percentile
Exercise 4.3.5
In an agricultural experiment, a large uniform field was planted with a single variety of wheat. The field was divided into many plots (each plot being 7 × 100 ft) and the yield (lb) of grain was measured for each plot. These plot yields followed approximately a normal distribution with mean 88 lb and standard deviation 7 lb

Answer 1

a) 90.695 lb

b) 85.305 lb

Explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

`\mu = 88, \sigma = 7`

(a) The 65th percentile

X when Z has a pvalue of 0.65. So X when Z = 0.385.

`Z = (X - \mu)/(\sigma)`

`0.385 = (X - 88)/(7)`

`X - 88 = 7*0.385`

`X = 90.695`

(b) The 35th percentile

X when Z has a pvalue of 0.35. So X when Z = -0.385.

`Z = (X - \mu)/(\sigma)`

`-0.385 = (X - 88)/(7)`

`X - 88 = 7*(-0.385)`

`X = 85.305`