Respuesta :
Answer:
The middle 60 students fall between 63.48 inches and 68.52 inches.
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 = 66, \sigma = 3[/tex]
Between which two heights (in inches) do the middle 60 students fall?
The normal probability distribution is symmetric. So the middle 60% fall from a pvalue of 0.50 - 0.60/2 = 0.20(lower bound) to a pvalue of 0.50 + 0.60/2 = 0.80(upper bound)
Lower bound
X when Z has a pvalue of 0.20.
So X when [tex]Z = -0.84[/tex]
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-0.84 = \frac{X - 66}{3}[/tex]
[tex]X - 66 = -0.84*3[/tex]
[tex]X = 63.48[/tex]
Upper bound
X when Z has a pvalue of 0.80.
So X when [tex]Z = 0.84[/tex]
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]0.84 = \frac{X - 66}{3}[/tex]
[tex]X - 66 = 0.84*3[/tex]
[tex]X = 68.52[/tex]
The middle 60 students fall between 63.48 inches and 68.52 inches.
