Respuesta :
Answer:
The score that separates the top 59% from the bottom 41% is 35.6225
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 = 37.4, \sigma = 7.9[/tex]
Find the score that separates the top 59% from the bottom 41%"
This is the value of X when Z has a pvalue of 0.41. So it is X when Z = -0.225.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-0.225 = \frac{X - 37.4}{7.9}[/tex]
[tex]X - 37.4 = -0.225*7.9[/tex]
[tex]X = 35.6225[/tex]
The score that separates the top 59% from the bottom 41% is 35.6225
Answer:
Score that separates the top 59% from the bottom 41% is 35.6
Step-by-step explanation:
We are given that Scores on an English test are normally distributed with a mean of 37.4 and a standard deviation of 7.9, i.e.; [tex]\mu[/tex] = 37.4 and [tex]\sigma[/tex] = 7.9
Now, the z score probability distribution is given by;
Z = [tex]\frac{X - \mu}{\sigma}[/tex] ~ N(0,1)
The bottom 41% area is given by the critical z value of -0.2278 (from z% table)
So, P(Z < [tex]\frac{X-37.4}{7.9}[/tex] ) = 0.41
which means [tex]\frac{X-37.4}{7.9}[/tex] = -0.2278
X - 37.4 = -0.2278 * 7.9
X = 37.4 - 1.79962 = 35.6
Therefore, score of 35.6 separates the top 59% from the bottom 41%.
