What is the probability that

Question

The amount of medication in a pill is normally distributed with a mean of 20 µg and a
standard deviation of 0.3 µg.
a. What is the probability that a pill will contain at least 19.9 µg ?
b. What is the probability the pill contains between 19 and 21 µg ?
c. What is amount of medication that corresponds to 90 percentile?
d. What is amount of medication that is exceeded with probability of 0.05?
e. Your prescription calls for you to take 9 pills to cure your sickness with the average
amount of medication in each pill is at least 19.9 µg. What is the probability you
will be cured?

Answer 1

Hi Alhinai

 

The amount of medication in a pill is normally distributed with a mean of 20 µg and a
standard deviation of 0.3 µg.
a. What is the probability that a pill will contain at least 19.9 µg ?

1)We need to calculate the z-score

Z = (x-µ)/σ

Z = (19.9-20)/0.3 = -.3333

2)Look up this value in a z-table

0.3694

3)But this represents the probability that there will be less than 19.9µg, so the probability that there will be at least 19.9µg is

1-.3694 = .6306 = 63.06% 

b. What is the probability the pill contains between 19 and 21 µg ?

1) For this we need two z scores and values

Z₁₉ = (19-20)/0.3 = -3.333

Z₂₀ = (20-20)/0.3 = 0

2)Look up these values in a z table

for Z₁₉ we get 0.0004

for Z₂₀ we get 0.5000

3) Now 0.0004 is the probability that the pill contains less than 19µg and 0.5000 is the probability that the pill contain less than 20µg. So the probability that the pill contains between 19 and 20 µg is the difference between the two.

0.5000 - 0.0004 = 0.4996 = 49.96%

c. What is amount of medication that corresponds to 90 percentile?

1) From the Z-table, to be in the 90th percentile is a z-score of 1.2816...

2) Calculate the value of X that corresponds to that z-score

1.2816... = (X-20)/0.3 

X = 20.3845 µg

d. What is amount of medication that is exceeded with probability of 0.05?

1) From the Z-table, to be exceeded with a probability of 0.05 we need a z-value of 95%.  This corresponds to a z-score of 1.645
2) Calculate the value of X that corresponds to that z-score
1.645= (X-20)/0.3
X = 20.4935 µg

 

e. Your prescription calls for you to take 9 pills to cure your sickness with the average
amount of medication in each pill is at least 19.9 µg. What is the probability you
will be cured?

1) determine the z-score and value for 19.9

(19.9-20)/0.3 = -0.3333...

This gives us a value of 0.0004

2) But this is the probability that the amount of medication will be less than 19.9.  1-0.0004 = 0.9996 is the probability that each pill will be at least 19.9µg

3) Since we need 9 pills, (0.9996)⁹ = .9964 = 99.64%

There is a 99.64% chance you'll be cured.