Respuesta :
Answer:
a) 305; b) 179; c) Yes
Step-by-step explanation:
We use z scores for these questions. The formula for a z score is
[tex]z=\frac{X-\mu}{\sigma}[/tex]
The mean, μ, is 3 hours and 48 minutes; this is 3(60)+48 = 180+48 = 228 minutes. The standard deviation, σ, is 52.
For part a,
We look 93%, or 0.93, up in the cells of a z table. The closest we can get to this value is 0.9306, which corresponds to a z score of 1.48:
1.48 = (X-228)/52
Multiply both sides by 52:
52(1.48) = ((X-228)/52)*52
76.96 = X-228
Add 228 to each side:
76.96+228 = X-228+228
304.96 = X
This rounds to 305 minutes.
For part b,
We look 17%, or 0.17, up in the cells of a z table. The closest we can get to this value is 0.1685, which corresponds to a z score of -0.95:
-0.95 = (X-228)/52
multiply both sides by 52:
52(-0.95) = ((X-228)/52)*52
-49.4 = X-228
Add 228 to each side:
-49.4+228 = X-228+228
178.6 = X
This rounds to 179 minutes.
For part c,
Yes, the distribution might be different for Friday than for Saturday. People that work Monday through Friday might not get off work until Friday evening, but might be off all day on Saturday. This might mean they can get to the theater sooner on Saturday than they can on Friday.
Using the normal distribution, we have that:
a) 305 minutes after doors open.
b) 178 minutes after doors open.
c) Yes, as during Friday, must people work during the day, so they should go the show closer to the evening hours.
In a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
- It measures how many standard deviations the measure is from the mean.
- After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.
In this problem:
- Mean of 3 hours and 48 minutes, thus, in minutes, [tex]\mu = 3(60) + 48 = 228[/tex]
- Standard deviation of 52 minutes, thus [tex]\sigma = 52[/tex].
Item a:
This is the 93rd percentile of times, which is X when Z has a p-value of 0.93, so X when Z = 1.475.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.475 = \frac{X - 228}{52}[/tex]
[tex]X - 228 = 1.475(52)[/tex]
[tex]X = 305[/tex]
So, 305 minutes after doors open.
Item b:
This is the 17th percentile of times, which is X when Z has a p-value of 0.17, so X when Z = -0.955.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-0.955 = \frac{X - 228}{52}[/tex]
[tex]X - 228 = -0.955(52)[/tex]
[tex]X = 178[/tex]
So, 178 minutes after doors open.
Item c:
Yes, as during Friday, must people work during the day, so they should go the show closer to the evening hours.
A similar problem is given at https://brainly.com/question/1993757
