Respuesta :
Answer:
The probability that X is between 1.48 and 15.56 is [tex]P(1.48 \leq X \leq 15.56) = 0.919[/tex]
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 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.
X is a normally distributed random variable with a mean of 8 and a standard deviation of 4.
This means that [tex]\mu = 8, \sigma = 4[/tex]
The probability that X is between 1.48 and 15.56
This is the pvalue of Z when X = 15.56 subtracted by the pvalue of Z when X = 1.48. So
X = 15.56
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{15.56 - 8}{4}[/tex]
[tex]Z = 1.89[/tex]
[tex]Z = 1.89[/tex] has a pvalue of 0.9706
X = 1.48
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{1.48 - 8}{4}[/tex]
[tex]Z = -1.63[/tex]
[tex]Z = -1.63[/tex] has a pvalue of 0.0516
0.9706 - 0.0516 = 0.919
Write out the probability notation for this question.
[tex]P(1.48 \leq X \leq 15.56) = 0.919[/tex]
The probability that X is between 1.48 and 15.56 is [tex]P(1.48 \leq X \leq 15.56) = 0.919[/tex]
