Respuesta :
Using the normal distribution, it is found that 58.97% of students would be expected to score between 400 and 590.
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.
The mean and the standard deviation are given, respectively, by:
[tex]\mu = 502, \sigma = 115[/tex]
The proportion of students between 400 and 590 is the p-value of Z when X = 590 subtracted by the p-value of Z when X = 400, hence:
X = 590:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{590 - 502}{115}[/tex]
Z = 0.76
Z = 0.76 has a p-value of 0.7764.
X = 400:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{400 - 502}{115}[/tex]
Z = -0.89
Z = -0.89 has a p-value of 0.1867.
0.7764 - 0.1867 = 0.5897 = 58.97%.
58.97% of students would be expected to score between 400 and 590.
More can be learned about the normal distribution at https://brainly.com/question/27643290
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