Answer:
You expect to lose money 2.28% of the time.
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:
[tex]\mu = 12, \sigma = 6[/tex]
What percentage of the time do you expect to lose money?
This is the pvalue of Z when X = 0. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{0 - 12}{6}[/tex]
[tex]Z = -2[/tex]
[tex]Z = -2[/tex] has a pvalue of 0.0228.
So you expect to lose money 2.28% of the time.
