Respuesta :
Answer:
The interval that describes how long it takes for passengers to board the middle 95% of the time is between 40.16 minutes and 55.84 minutes.
Step-by-step explanation:
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score 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. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this problem, we have that:
[tex]\mu = 48, \sigma = 4[/tex].
Which interval describes how long it takes for passengers to board the middle 95% of the time?
This is between the 2.5th percentile and the 97.5th percentile.
So this interval is the value of X when Z has a a pvalue of 0.025 and the value of X when Z has a pvalue of 0.975
Lower Limit
Z has a pvalue of 0.025 when [tex]Z = -1.96[/tex]. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.96 = \frac{X - 48}{4}[/tex]
[tex]X - 48 = -1.96*4[/tex]
[tex]X = 40.16[/tex]
Upper Limit
Z has a pvalue of 0.975 when [tex]Z = 1.96[/tex]. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.96 = \frac{X - 48}{4}[/tex]
[tex]X - 48 = 1.96*4[/tex]
[tex]X = 55.84[/tex]
The interval that describes how long it takes for passengers to board the middle 95% of the time is between 40.16 minutes and 55.84 minutes.
