Question

It is possible to score higher than 1600 on the combined mathematics and reading portions of the SAT, but scores 1600 and above are reported as 1600. The distribution of SAT scores (combining mathematics and reading) in 2013 was close to Normal with mean 1011 and standard deviation 216. What proportion of SAT scores were reported as 1600 (That is, what proportion of SAT scores were actually higher than 1600?)?

Answer 1

Answer:

The proportion of scores reported as 1600 is 0.0032

Step-by-step explanation:

Let X be the score for 1 random person in SAT combining maths and reading. X has distribution approximately N(μ = 1011,σ = 216).

In order to make computations, we standarize X to obtain a random variable W with distribution approximately N(0,1)

$W = \frac{X-\mu}{\sigma} = \frac{X-1011}{216} \simeq N(0,1)$

The values of the cummulative distribution function of the standard Normal random variable, lets denote it $\phi$ are tabulated, you can find those values in the attached file. Now, we are ready to compute the probability of X being bigger than 1600

$P(1600< X) = P(\frac{1600-1011}{216} < \frac{X-1011}{216}) = P(2.7269 < W) = 1- \phi(2.7269)\\= 1- 0.9968 = 0.0032$

Hence, the proportion of scores reported as 1600 is 0.0032.

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