Respuesta :
Answer:
a) 3 standard deviations above 16
b) More than 2 standard deviations of the mean, so yes, 22 inches is faw away from the mean of 16 inches.
c) Less than 2 standard deviations, so not far away.
Step-by-step explanation:
Z-score:
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.
If Z < -2 or Z > 2, X is considered to be far away from the mean.
In this question, we have that:
[tex]\mu = 16[/tex]
(a) Suppose the data come from a sample whose standard deviation is 2 inches. How many standard deviations is 22 inches from 16 inches?
This is Z when [tex]X = 22, \sigma = 2[/tex].
So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{22 - 16}{2}[/tex]
[tex]Z = 3[/tex]
So 22 inches is 3 standard deviations fro 16 inches.
(b) Is 22 inches far away from a mean of 16 inches?
3 standard deviations, more than two, so yes, 22 inches is far away from a mean of 16 inches.
(c) Suppose the standard deviation of the underlying data is 4 inches. Is 22 inches far away from a mean of 16 inches?
Now [tex]\sigma = 4[/tex]
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{22 - 16}{4}[/tex]
[tex]Z = 1.5[/tex]
1.5 standard deviations from the mean, so 22 inches is not far away from the mean.
