Respuesta :
Answer:
a. The percentage of data within 1 standard deviation of the mean is 68.26%.
b. The percentage of data to the right of 1.5 standard deviations below the mean is of 93.32%.
c. The percentage of data more than 0.5 standard deviations away from the mean is of 61.7%.
Step-by-step explanation:
When the distribution is normal, we use 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.
a. Within 1 standard deviation of the mean
Between [tex]Z = -1[/tex] and [tex]Z = 1[/tex]
This is the pvalue of Z = 1 subtracted by the pvalue of Z = -1.
Z = 1 has a pvalue of 0.8413
Z = -1 has a pvalue of 0.1587
0.8413 - 0.1587 = 0.6826
The percentage of data within 1 standard deviation of the mean is 68.26%.
b. To the right of 1.5 standard deviations below the mean
Greater than [tex]Z = -1.5[/tex], that is, 1 subtracted by the pvalue of Z = -1.5.
Z = -1.5 has a pvalue of 0.0668
1 - 0.0668 = 0.9332
The percentage of data to the right of 1.5 standard deviations below the mean is of 93.32%.
c. More than 0.5 standard deviations away from the mean
Below [tex]Z = -0.5[/tex] or above [tex]Z = 0.5[/tex]. These percentages are the same, so we find one and multiply by 1.
Below Z = -0.5
This is the pvalue of Z = -0.5, which is 0.3085
0.3085*2 = 0.617
The percentage of data more than 0.5 standard deviations away from the mean is of 61.7%.
