Question

A specific chemical reaction to break down a compound follows a normal distribution with a mean time of 160 seconds and a standard deviation of 4 seconds.

(a)

What reaction time would result in a z-score of −1.5?

(b)

Using the Empirical Rule, what reaction time would have 84% of all values fall below it?

(c)

A reaction time that is shorter than 150 seconds or longer than 172 seconds will be re-run for quality control purposes. What is the probability that a reaction will need to be re-run?

Answer 1

Solution:
Given in the question
A specific chemical reaction to break down a compound follows a normal distribution with
Mean () = 160
Standard deviation () = 4
Solution(a)
We need to calculate reaction time at a Z-score = -1.5 which can be calculated as
Z-score = (X-)/
X= +Z-score* = 160 - 1.5*4 = 160 -6= 154 seconds
Solution(b)
We need to calculate Using the Empirical Rule, what reaction time would have 84% of all values fall below it
Given P-value = 0.84
From Z table we found Z-score = 1
So reaction time can be calculated as
X= +Z-score* = 160 + 1*4 = 160 + 4 = 164 seconds
So 164 seconds is the reaction time would have 84% of all values fall below it.
Solution(c)
Here we need to calculate the probability that a reaction time that is shorter than 150 seconds or longer than 172 seconds i.e. P(X<150) + P(X>172) which can be calculated as
Z-score = (X-)/ = (150-160)/4 = -2.5
Z-score = (172-160)/4 = 3
From Z table we found a p-value
P(X<150) = 0.0062
P(X>172) = 0.0014
So P(X<150) + P(X>172) = 0.0062+0.0014 = 0.0076
So there is a 0.76% probability that A reaction time that is shorter than 150 seconds or longer than 172 seconds will be re-run for quality control purposes.