Respuesta :
Answer:
(a) [tex]0.00612[/tex].
(b) [tex]0.49379[/tex].
Step-by-step explanation:
We have been given that the length of life of an instrument produced by a machine has a normal distribution with a mean of 12 months and standard deviation of 2 months.
(a) First of all, we will find z-score corresponding to sample score of 7 months as:
[tex]z=\frac{x-\mu}{\sigma}[/tex], where,
z = Z-score,
x = Sample score,
[tex]\mu[/tex] = Mean,
[tex]\sigma[/tex] = Standard deviation.
Upon substituting our given values in z-score formula, we will get:
[tex]z=\frac{7-12}{2}=\frac{-5}{2}=-2.5[/tex]
Now, we need to find the probability that a z-score is less than [tex]-2.5[/tex].
Using normal distribution table, we will get:
[tex]P(z<-2.5)=0.00621[/tex]
Therefore, the probability that an instrument produced by this machine will last less than 7 months is [tex]0.00612[/tex].
(b) Let us find z-score corresponding to sample score of 12 months.
[tex]z=\frac{12-12}{2}=\frac{0}{2}=0[/tex]
Using formula [tex]P(a<z<b)=P(z<b)-P(z<a)[/tex], we will get:
[tex]P(-2.5<z<0.0)=P(z<0.0)-P(z<-2.5)[/tex]
[tex]P(-2.5<z<0.0)=0.50000-0.00621[/tex]
[tex]P(-2.5<z<0.0)=0.49379[/tex]
Therefore, the probability that an instrument produced by this machine will last between 7 and 12 months is [tex]0.49379[/tex].
Answer:
Step-by-step explanation:
Since the length of life of an instrument produced by a machine has a normal distribution, we would apply the formula for normal distribution which is expressed as
z = (x - µ)/σ
Where
x = length of life of instruments in months.
µ = mean time
σ = standard deviation
From the information given,
µ = 12 months
σ = 2 months
a) We want to find the probability that an instrument produced by this machine will last for less than 7 months. It is expressed as
P(x < 7)
For x = 7,
z = (7 - 12)/2 = - 2.5
Looking at the normal distribution table, the probability corresponding to the z score is 0.0062
b) between 7 and 12 months is expressed as P(7 ≤ x ≤ 12)
For x = 7, the probability is 0.0062
For x = 12,
z = (12 - 12)/2 = 0
Looking at the normal distribution table, the probability corresponding to the z score is 0.5
Therefore,
P(7 ≤ x ≤ 12) = 0.5 - 0.0062 = 0.4938
