current learning objective; The Standard Normal Distribution Question 19 Prer Hent Score: 0 of 4 points In $$ 2005,1,475,623 $$ students heading to college took the SAT. The distribution of scores in the math section of the SAT followis a normal distribution with mean $$ \mu=500 $$ and standard deviation $$ \sigma=75 $$, (a). Calculate the $$ \% $$-score for an SAT score of 411 . $$ z \approx $$ $$ \square $$ Enter your answer (Enter your answer as a decimal, rounded to two decimal places if necessary.) (b), What math SAT score is 1,13 standard deviations above the meam? $$ \square $$ Enter your answer (Enter your answer as an exact numerical value.) (c). For 2024, the SAT math test had a mean of 600 and a standard deviation of 85 . The ACT math test is an alternative to the SAT and is approximately normally distributed with a mean of 20 and a standard deviation of 6.5 . If one person took the SAT math test and scored 514 and a second person took the ACT math test and scored 19, who did better with respect to the test they took? Enter your answer $$ \square $$ (Enter $$ S $$ if the person who took the SAT math test did better, and $$ A $$ if the person who took the ACT did better.)

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UpStudy AI · Accepted Answer

**(a) Calculation of the $$ z $$-score for an SAT score of 411**

The formula for the $$ z $$-score is  
$$
z = \frac{X - \mu}{\sigma}
$$
where  
$$ X = 411 $$,  
$$ \mu = 500 $$, and  
$$ \sigma = 75 $$.

Substitute the values:
$$
z = \frac{411 - 500}{75} = \frac{-89}{75} \approx -1.19
$$

**(b) Finding the math SAT score that is 1.13 standard deviations above the mean**

We use the formula:
$$
X = \mu + z\sigma
$$
Given  
$$ \mu = 500 $$,  
$$ \sigma = 75 $$, and  
$$ z = 1.13 $$, we have:
$$
X = 500 + 1.13 \times 75 = 500 + 84.75 = 584.75
$$

**(c) Comparing the performance on the SAT and ACT tests**

*For the SAT math test (2024):*  
Given  
$$ \mu = 600 $$,  
$$ \sigma = 85 $$, and  
score $$ X = 514 $$, the $$ z $$-score is:
$$
z_{\text{SAT}} = \frac{514 - 600}{85} = \frac{-86}{85} \approx -1.01
$$

*For the ACT math test:*  
Given  
$$ \mu = 20 $$,  
$$ \sigma = 6.5 $$, and  
score $$ X = 19 $$, the $$ z $$-score is:
$$
z_{\text{ACT}} = \frac{19 - 20}{6.5} = \frac{-1}{6.5} \approx -0.15
$$

Since $$ z_{\text{ACT}} $$ is closer to 0 than $$ z_{\text{SAT}} $$, the ACT score is relatively better compared to its distribution.

**Answer to part (c):** $$ A $$
**(a) $$ z \approx -1.19 $$** **(b) 584.75** **(c) $$ A $$**

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