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Sagot :
Answer:
The middle 90% of all freshman biology majors' GPAs lie between 2.31 and 3.43.
Step-by-step explanation:
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score 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 p-value, we get the probability that the value of the measure is greater than X.
Mean 2.87 and standard deviation .34.
This means that [tex]\mu = 2.87, \sigma = 0.34[/tex]
Middle 90% of scores:
Between the 50 - (90/2) = 5th percentile and the 50 + (90/2) = 95th percentile.
5th percentile:
X when Z has a pvalue of 0.05. So X when Z = -1.645.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.645 = \frac{X - 2.87}{0.34}[/tex]
[tex]X - 2.87 = -1.645*0.34[/tex]
[tex]X = 2.31[/tex]
95th percentile:
X when Z has a pvalue of 0.95. So X when Z = 1.645.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.645 = \frac{X - 2.87}{0.34}[/tex]
[tex]X - 2.87 = 1.645*0.34[/tex]
[tex]X = 3.43[/tex]
The middle 90% of all freshman biology majors' GPAs lie between 2.31 and 3.43.
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