Respuesta :
Answer:
Step-by-step explanation:
Problems of normally distributed samples are 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 = 100, \sigma = 15[/tex]
Find the probability that a randomly selected adult has an IQ between 88 and 112.
This is the pvalue of Z when X = 112 subtracted by the pvalue of Z when X = 88.
X = 112
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{112 - 100}{15}[/tex]
[tex]Z = 0.8[/tex]
[tex]Z = 0.8[/tex] has a pvalue of 0.7881
X = 88
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{88 - 100}{15}[/tex]
[tex]Z = -0.8[/tex]
[tex]Z = -0.8[/tex] has a pvalue of 0.2119
0.7881 - 0.2119 = 0.5762
57.62% probability that a randomly selected adult has an IQ between 88 and 112.
