High School

Assume that adults have scores that are normally distributed with a mean of 97.9 and a standard deviation of 18.6.

Find the first quartile, [tex]Q_1[/tex], which is the IQ score separating the bottom 25% from the top 75%.

(Hint: Draw a graph.)

The first quartile is ______. (Type an integer or decimal rounded to one decimal place as needed.)

Answer :

The first quartile when [tex]\mu = 97.4[/tex] and [tex]\sigma = 17.6[/tex] is [tex]Q_1 = 85.54[/tex] .

In statistics, a quartile is a type of quantile that divides the number of data points into four corridors, or diggings, of more-or-less equal size. The data must be ordered from lowest to largest to cipher quartiles; as similar, quartiles are a form of order statistic.

Let us assume that X is the adult IQ score.

[tex]\mu = 97.4[/tex]

[tex]\sigma = 17.6[/tex]

Let [tex]z_0[/tex] be the z-score for the first quartile.

[tex]P(z\geq z_0 ) = 0.75[/tex]

Now, use the area under the normal curve to find the value of [tex]z_0[/tex] .

[tex]z_0 = -0.674[/tex]

[tex]\frac{X-\mu}{\sigma} = -0.674[/tex]

[tex]\frac{X-97.4}{18.6} = -0.674[/tex]

[tex]X = -0.674\times 18.6 + 97.4[/tex]

[tex]= 85.54[/tex]

Hence, the first quartile is [tex]Q_1 = 85.54[/tex] .

Learn more about quartile here:

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