Respuesta :
Answer:
47.06% of the population has an IQ between 85 and 105.
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]
What percent of the population has an IQ between 85 and 105?
This is the pvalue of Z when X = 105 subtracted by the pvalue of Z when X = 85. So
X = 105
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{105 - 100}{15}[/tex]
[tex]Z = 0.33[/tex]
[tex]Z = 0.33[/tex] has a pvalue of 0.6293.
X = 85
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{85 - 100}{15}[/tex]
[tex]Z = -1[/tex]
[tex]Z = -1[/tex] has a pvalue of 0.1587
So 0.6293 - 0.1587 = 0.4706 = 47.06% of the population has an IQ between 85 and 105.
