Respuesta :
Answer:
(a) The probability that the mean of a month’s reviews will take Yoonie from 3.5 to 4.25 hours is 0.7492.
(b) The values representing the 95th percentile for the mean time to complete one month's reviews is 4.50 hours.
Step-by-step explanation:
The random variable X is defined as the time it takes her to complete one review.
The random variable X follows a Normal distribution with parameters μ = 4 hours and σ = 1.2 hours.
A random sample of n = 16 reviews are selected as a set.
(a)
Compute the probability that the mean of a month’s reviews will take Yoonie from 3.5 to 4.25 hours as follows:
[tex]P(3.5<\bar X<4.25)=P(\frac{3.5-4}{1.2/\sqrt{16}}<\frac{\bar x-\mu}{\sigma/\sqrt{n}}<\frac{4.25-4}{1.2/\sqrt{16}})[/tex]
[tex]=P(-1.67<Z<0.83)\\=P(Z<0.83)-P(Z<-1.67)\\=0.7967-0.0475\\=0.7492[/tex]
Thus, the probability that the mean of a month’s reviews will take Yoonie from 3.5 to 4.25 hours is 0.7492.
(b)
The pth percentile is a data value such that at least p% of the data set is less than or equal to this data value and at least (100 - p)% of the data set are more than or equal to this data value.
So, the 95th percentile for the mean time to complete one month's reviews can be represented as:
[tex]P(\bar X<a)=0.95[/tex]
This implies that:
[tex]P(\bar X<a)=0.95\\P(\frac{\bar X-\mu}{\sigma/\sqrt{n}}<\frac{a-\mu}{\sigma/\sqrt{n}})=0.95\\P(Z<z)=0.95[/tex]
The value of z is:
z = 1.645
Compute the value of a as follows:
[tex]z=\frac{a-4}{1.2/\sqrt{16}}\\1.645=\frac{a-4}{0.3}\\z=4+(0.3\times 1.645)\\z=4.4935\\z\approx4.50[/tex]
Thus, the values representing the 95th percentile for the mean time to complete one month's reviews is 4.50 hours.
The probability that the mean of a month’s reviews will take Yoonie from 3.5 to 4.25 hours is 0.7492.
How to compute the probability?
The probability that the mean of a month’s reviews will take Yoonie from 3.5 to 4.25 hours will be calculated thus:
P(3.5 < z < 4.5) = P[(3.5 - 4)/(1.2/✓1.6)] < z < [(4.25 - 4)/(1.2/✓16)]
= P(-1.67 < z < 0.83)
= P(z < 0.83) - P(z < -1.67)
= 0.7967 - 0.0475
= 0.7492
The 95th percentile for the mean time to be complete will be calculated thus:
The z value is given as 1.645. The value will be:
z = (a - 4)/(1.2 /✓16)
1.645 = (a - 4)/0.3
a = 4 + (0.3 × 1.645).
a = 4.50
In conclusion, the value is 4.50 hours.
Learn more about probability on:
https://brainly.com/question/24756209
