In 2005, 1,475,623 students heading to college took the SAT. The distribution of scores in the math section of the SAT follows a normal distribution with mean [tex][tex]$\mu=520$[/tex][/tex] and population standard deviation [tex][tex]$=115$[/tex][/tex].

What math SAT score is 1.5 standard deviations above the mean? Round your answer to a whole number.



Answer :

To determine the math SAT score that is 1.5 standard deviations above the mean, let's proceed step by step:

1. Identify the Mean (μ) and Standard Deviation (σ):
- The mean (μ) of the math SAT scores is 520.
- The standard deviation (σ) of the math SAT scores is 115.

2. Calculate the score that is 1.5 standard deviations above the mean:
- To find the score that is 1.5 standard deviations above the mean, we can use the following formula:
[tex]\[ \text{Score} = \mu + (k \times \sigma) \][/tex]
where [tex]\( k \)[/tex] is the number of standard deviations above the mean.
- In this case, [tex]\( k = 1.5 \)[/tex]. So,
[tex]\[ \text{Score} = 520 + (1.5 \times 115) \][/tex]

3. Evaluate the expression:
- First, multiply the standard deviation by 1.5:
[tex]\[ 1.5 \times 115 = 172.5 \][/tex]
- Then, add this value to the mean:
[tex]\[ 520 + 172.5 = 692.5 \][/tex]

4. Round the result to a whole number:
- To present the score as a whole number, we round 692.5 to the nearest integer.
[tex]\[ 692.5 \approx 692 \][/tex]

Therefore, the math SAT score that is 1.5 standard deviations above the mean is 692 when rounded to the nearest whole number.