Respuesta :
Answer:
The mean is [tex]\mu = 131[/tex]
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]\sigma = 39[/tex]
What is the mean of this normal distribution if the probability of scoring above x = 209 is 0.0228?
This means that when X = 209, Z has a pvalue of 1-0.0228 = 0.9772. So when X = 209, Z = 2.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]2 = \frac{209 - \mu}{39}[/tex]
[tex]209 - \mu = 2*39[/tex]
[tex]\mu = 209 - 78[/tex]
[tex]\mu = 131[/tex]
The mean is [tex]\mu = 131[/tex]
