Respuesta :
Using the normal distribution, it is found that the indicated probabilities are given as follows:
a) P(8≤x≤ 12) = 0.1593 = 15.93%.
b. P(x ≥30) = 0.0004 = 0.04%.
Normal Probability Distribution
The z-score of a measure X of a normally distributed variable with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex] is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
- The z-score measures how many standard deviations the measure is above or below the mean.
- Looking at the z-score table, the p-value associated with this z-score is found, which is the percentile of X.
Item a:
The mean and the standard deviation are given, respectively, by:
[tex]\mu = 15, \sigma = 3.2[/tex].
The probability is the p-value of Z when X = 12 subtracted by the p-value of Z when X = 8, hence:
X = 12:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{12 - 15}{3.2}[/tex]
Z = -0.94
Z = -0.94 has a p-value of 0.1736.
X = 8:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{8 - 15}{3.2}[/tex]
Z = -2.19
Z = -2.19 has a p-value of 0.0143.
0.1736 - 0.0143 = 0.1593.
Item b:
The mean and the standard deviation are given, respectively, by:
[tex]\mu = 20, \sigma = 3[/tex]
The probability is one subtracted by the p-value of Z when X = 30, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{30 - 20}{3}[/tex]
Z = 3.33
Z = 3.33 has a p-value of 0.9996.
1 - 0.9996 = 0.0004.
More can be learned about the normal distribution at https://brainly.com/question/27893812
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