For 2012, the SAT math test had a mean of 514 and standard deviation 117. The ACT math test is an alternate to the SAT and is approximately

Question

For 2012, the SAT math test had a mean of 514 and standard deviation 117. The ACT math test is an alternate to the SAT and is approximately normally distributed with mean 21 and standard deviation 5.3. If one person took the SAT math test and scored 700 and a second person took the ACT math test and scored 35, who did better with respect to the test they took?

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Maya 2 weeks 2021-10-01T15:17:56+00:00 1 Answer 0

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    2021-10-01T15:19:23+00:00

    Answer:

    The person who took the ACT test had the higher z-score, so he/she did better.

    Step-by-step explanation:

    Problems of normally distributed samples are 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 = \frac{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.

    Who did better with respect to the test they took?

    Whoever had the higher z-score.

    For 2012, the SAT math test had a mean of 514 and standard deviation 117. One person took the SAT math test and scored 700.

    We have to find Z when X = 700, \mu = 514, \sigma = 117[\tex][tex]Z = \frac{X - \mu}{\sigma}

    Z = \frac{700 - 514}{117}

    Z = 1.59

    The ACT math test is an alternate to the SAT and is approximately normally distributed with mean 21 and standard deviation 5.3. Person took the ACT math test and scored 35.

    We have to find Z when X = 35, \mu = 21, \sigma = 5.3[\tex][tex]Z = \frac{X - \mu}{\sigma}

    Z = \frac{35 - 21}{5.3}

    Z = 2.64

    The person who took the ACT test had the higher z-score, so he/she did better.

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