Respuesta :
Answer:
Approximately 16% of adult women have feet longer than 27.2 cm.
Step-by-step explanation:
Normal Probability Distribution
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score 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 p-value, we get the probability that the value of the measure is greater than X.
The feet of the average adult woman are 24.6 cm long
This means that [tex]\mu = 24.6[/tex]
16% of adult women have feet that are shorter than 22 cm
This means that when [tex]X = 22[/tex], Z has a p-value of 0.16, so when [tex]X = 22, Z = -1[/tex]. We use this to find [tex]\sigma[/tex]
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1 = \frac{22 - 24.6}{\sigma}[/tex]
[tex]-\sigma = -2.6[/tex]
[tex]\sigma = 2.6[/tex]
Approximately what percent of adult women have feet longer than 27.2 cm?
The proportion is 1 subtracted by the p-value of Z when X = 27.2. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{27.2 - 24.6}{2.6}[/tex]
[tex]Z = 1[/tex]
[tex]Z = 1[/tex] has a p-value of 0.84.
1 - 0.84 = 0.16
0.16*100% = 16%.
Approximately 16% of adult women have feet longer than 27.2 cm.
