Question

The annual ground coffee expenditures for households are approximately normally distributed with a mean of $ 46.64 and a standard deviation of $ 11.00

a. Find the probability that a household spent less than $30.00.

b. Find the probability that a household spent more than ​$60.00.

c. What proportion of the households spent between​$20.00 and ​$30​.00?

d.97.5​% of the households spent less than what​ amount?

Answer 1

Solution:
Given in the question the annual ground coffee expenditures for households are approximately normally distributed with
Mean ()= 46.64
Standard deviation () = 11
Solution(a)
We need to calculate the probability that a household spent less than $30.00
P(X<30) = ?, Here we will use the standard normal distribution table, First we will calculate Z-score as follows
Z-score = (X- )/ = (30-46.64)/11 = -1.51
From Z table we found a p-value
P(X<30) = 0.0655
So there is a 6.55% probability that a household spent less than $30.00.
Solution(b)
We need to calculate the probability that a household spent more than ​$60.00
P(X>60) = 1-P(X<=60)
Here we will use the standard normal distribution table, First we will calculate Z-score as follows
Z-score = (X- )/ = (60-46.64)/11 = 1.21
From Z table we found a p-value
P(X<30) = 1-0.8869 = 0.1131
So there is an 11.31% probability that a household spent greater than 60.
Solution(c)
We need to calculate the probability that a household spent b/w 20 and 30
P(20<X<30) = P(X<30)-P(X<=20)
Here we will use the standard normal distribution table, First we will calculate Z-score as follows
Z-score = (X- )/ = (30-46.64)/11 = -1.51
Z-score = (X- )/ = (20-46.64)/11 = -2.42
From Z table we found a p-value
P(20<X<30) = P(X<30)-P(X<=20) = 0.0655 - 0.0078 = 0.0577
So there is a 5.77% probability that a household spent b/w 20 and 30.
Solution(d)
here we need to calculate the amount to which 97.5​% of the households spent less than
P-value = 0.975
From Z table we found Z-score = 1.96
So the amount can be calculated as
X = + Z-score * = 46.64 + 1.96*11 = 68.2
So 97.5​% of the households spent less than $68.2
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