Answer:
Q1 = 89.1525
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 = 101.1, \sigma = 17.7[/tex]
Find the first quartile Upper Q 1Q1, which is the IQ score separating the bottom 25% from the top 75%
This is the value of X when Z has a pvalue of 0.25. So it is X when Z = -0.675.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-0.675 = \frac{X - 101.1}{17.7}[/tex]
[tex]X - 101.1 = -0.675*17.7[/tex]
[tex]X = 89.1525[/tex]
So
Q1 = 89.1525
