Respuesta :
Answer:
Probability that the sample average time taken is less than 11 minutes for Day 1 is 0.86864.
Probability that the sample average time taken is less than 11 minutes for Day 2 is 0.88877.
Step-by-step explanation:
We are given that the time taken by a randomly selected applicant for a mortgage to fill out a certain form has a normal probability distribution with average time 10 minutes and a standard deviation of 2 minutes.
Also, five individuals fill out the form on Day 1 and six individuals fill out the form on Day 2.
(a) Let [tex]\bar X[/tex] = sample average time taken
The z score probability distribution for sample mean is given by;
Z = [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] ~ N(0,1)
where, [tex]\mu[/tex] = population mean time = 10 minutes
[tex]\sigma[/tex] = standard deviation = 2 minutes
n = sample of individuals fill out form on Day 1 = 5
Now, the probability that the sample average time taken is less than 11 minutes for Day 1 is given by = P([tex]\bar X[/tex] < 11 minutes)
P([tex]\bar X[/tex] < 11 minutes) = P( [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] < [tex]\frac{11-10}{\frac{2}{\sqrt{5} } }[/tex] ) = P(Z < 1.12) = 0.86864
The above probability is calculated by looking at the value of x = 1.12 in the z table which has an area of 0.86864.
(b) Let [tex]\bar X[/tex] = sample average time taken
The z score probability distribution for sample mean is given by;
Z = [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] ~ N(0,1)
where, [tex]\mu[/tex] = population mean time = 10 minutes
[tex]\sigma[/tex] = standard deviation = 2 minutes
n = sample of individuals fill out form on Day 2 = 6
Now, the probability that the sample average time taken is less than 11 minutes for Day 2 is given by = P([tex]\bar X[/tex] < 11 minutes)
P([tex]\bar X[/tex] < 11 minutes) = P( [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }[/tex] < [tex]\frac{11-10}{\frac{2}{\sqrt{6} } }[/tex] ) = P(Z < 1.22) = 0.88877
The above probability is calculated by looking at the value of x = 1.22 in the z table which has an area of 0.88877.
