Question

Suppose that Arturo and Bella take an exam for which the mean score is 70 and standard deviation of scores is 8. Arturo’s score on the exam is 75, and Bella’s score is 1.5 standard deviations above Arturo’s score. What is Bella’s score on the exam? (Hint: z-scores are helpful here)

Answer 1

Bella's score on the exam is 87.

Explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

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.

In this problem, we have that:

`\mu = 70, \sigma = 8`

Arturo's score on the exam is 75

His score is Z standard deviations above the mean

`Z = (X - \mu)/(\sigma)`

`Z = (75 - 70)/(8)`

`Z = 0.625`

Bella’s score is 1.5 standard deviations above Arturo’s score.

So Bella's z-score is 1.5 + 0.625 = 2.125.

Her score is the value of X when Z = 2.125. So:

`Z = (X - \mu)/(\sigma)`

`2.125 = (X - 70)/(8)`

`X - 70 = 8*2.125`

`X = 87`

Bella's score on the exam is 87.