Respuesta :
Answer:
The value of ROE that will be exceeded by 78% of the firms is -1.77%.
Step-by-step explanation:
Problems of normally distributed samples can be 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:
The mean ROE for the firms studied was 14.93% and the standard deviation was 21.74%. This means that [tex]\mu = 0.1493, \sigma = 0.2174[/tex]
What value of ROE will be exceeded by 78% of the firms?
This is the value of X when Z has a pvalue of 1-0.78 = 0.22.
This is [tex]Z = -0.77[/tex]
So:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-0.77 = \frac{X - 0.1493}{0.2174}[/tex]
[tex]X - 0.1493 = -0.77*0.2174[/tex]
[tex]X = -0.0177[/tex]
The value of ROE that will be exceeded by 78% of the firms is -1.77%.
