Respuesta :
Answer:
a) [tex]P(R<95)=P(\frac{R-\mu}{\sigma}<\frac{95-\mu}{\sigma})=P(Z<\frac{95-100}{3.606})=P(Z<-1.387)[/tex]
And we can find this probability using the normal standard table or excel and we got:
[tex]P(z<-1.387)=0.0827[/tex]
b) [tex]P(R>110)=P(\frac{R-\mu}{\sigma}>\frac{110-\mu}{\sigma})=P(Z>\frac{110-100}{3.606})=P(Z>2.774)[/tex]
And we can find this probability using the complement rule and the normal standard table or excel and we got:
[tex]P(z>2.774)=1-P(Z<2.774) = 1-0.9972 = 0.0028[/tex]
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Part a
Let X the random variable that represent the time for the step 1 and Y the time for the step 2, we define the random variable R= X+Y for the total time and the distribution for R assuming independence between X and Y is:
[tex]R \sim N(40+60 = 100,\sqrt{2^2 +3^2}= 3.606 s)[/tex]
Where [tex]\mu=65.5[/tex] and [tex]\sigma=2.6[/tex]
We are interested on this probability
[tex]P(R<95)[/tex]
And the best way to solve this problem is using the normal standard distribution and the z score given by:
[tex]z=\frac{R-\mu}{\sigma}[/tex]
If we apply this formula to our probability we got this:
[tex]P(R<95)=P(\frac{R-\mu}{\sigma}<\frac{95-\mu}{\sigma})=P(Z<\frac{95-100}{3.606})=P(Z<-1.387)[/tex]
And we can find this probability using the normal standard table or excel and we got:
[tex]P(z<-1.387)=0.0827[/tex]
Part b
[tex]P(R>110)=P(\frac{R-\mu}{\sigma}>\frac{110-\mu}{\sigma})=P(Z>\frac{110-100}{3.606})=P(Z>2.774)[/tex]
And we can find this probability using the complement rule and the normal standard table or excel and we got:
[tex]P(z>2.774)=1-P(Z<2.774) = 1-0.9972 = 0.0028[/tex]
Given Information:
Mean = μ = 40 + 60 = 100 minutes
Standard deviation = σ = 2² + 3² = 13 minutes
Required Information:
a. P(X < 95) = ?
b. P(X > 110) = ?
Answer:
a. P(X < 95) = 0.0823
b. P(X > 110) = 0 .0028
Explanation:
a)
Let random variable X represents the time in minutes of wheel throwing and firing.
The probability that a piece of pottery will be finished within 95 minutes means,
P(X < 95) = P(Z < (x - μ)/√σ)
P(X < 95) = P(Z < (95 - 100)/√13)
P(X < 95) = P(Z < -1.39)
The z-score corresponding to -1.39 is 0.0823
P(X < 95) = 0.0823
Therefore, there is 8.23% probability that a piece of pottery will be finished within 95 minutes.
b)
P(X > 110) = 1 - P(X < 110)
P(X > 110) = 1 - P(X < (x - μ)/√σ)
P(X > 110) = 1 - P(X < (110 - 100)/√13)
P(X > 110) = 1 - P(X < 2.77)
The z-score corresponding to 2.77 is 0.9972
P(X > 110) = 1 - 0.9972
P(X > 110) = 0 .0028
Therefore, there is 0.28% probability that a piece of pottery will take longer than 110 minutes.
