Question

The annual per capita consumption of fresh bananas​ (in pounds) in the United States can be approximiated by the normal​ distribution, with a mean of 8.4 pounds and a standard deviation of 4 pounds. Answer the following questions about the specified normal distribution.​ (a) What is the smallest annual per capita consumption of bananas that can be in the top 10​% of​ consumptions?​ (b) What is the largest annual per capita consumption of bananas that can be in the bottom 5​% of​ consumption?

​(a) The smallest annual per captita consumption of bananas that can be in the top 10​% of consumptions is pounds. ​(Round to one decimal place as​ needed.)

​(b) The largest annual per capita consumption of bananas that can be in the bottom 5​% of consumptions is pounds. ​(Round to one decimal place as​ needed.)

Answer 1

Soluton:

Given: mean = 8.4 pound and sd=4 pound
By using the standard normal distribution ,we get
P( Z>z)= 10%
1-P( Z<=z)= 0.10
P( Z<=1.282)= 0.90 ...... ... From the standard normal table.
z= 1.282
Now using the z-score formula, we can find the value of x as follows,
z=(x-mean)/sd
z×sd =( x- mean)
x= mean +( z×sd)
x= 8.4+(1.282×4)
=8.4+5.128
x= 13.528

= 13.5. ........up to 1 decimal place.
Therefore, the smallest per capita consumption of the bananas that can be in the top 10% of the consumption is 13.5
B)
By using the standard normal distribution ,we get
P( Z<z)= 5%
p( Z<z)= 0.05
P( Z< -1.645)= 0.5
From the standard normal table.
z= -1.645
Now using the z-score formula, we can find the value of x as follows,
z=(x-mean)/sd
z×sd =( x- mean)
x= mean +( z× sd)
x=8.4+(-1.645×4)
=30+(-6.58)
x= 2.12

=2.1 ........ Up to 1 decimal palce
Therefore, the largest per capita consumption of the bananas that can be in the top 10% of the consumption is 2.1