Question

Suppose you can somehow choose two people at random who took the SAT in 2014. A reminder that scores were Normally distributed with mean and stanard deviation of 1497 and 322, respectively. What is the probability that both of them scored above a 1520? Assume that the scores of the two test takers are independent.

Answer 1

Answer:

22.29% probability that both of them scored above a 1520

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.

In this problem, we have that:

$\mu = 1497, \sigma = 322$

The first step to solve the question is find the probability that a student has of scoring above 1520, which is 1 subtracted by the pvalue of Z when X = 1520.

So

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{1520 - 1497}{322}$

$Z = 0.07$

$Z = 0.07$ has a pvalue of 0.5279

1 - 0.5279 = 0.4721

Each students has a 0.4721 probability of scoring above 1520.

What is the probability that both of them scored above a 1520?

Each students has a 0.4721 probability of scoring above 1520. So

$P = 0.4721*0.4721 = 0.2229$

22.29% probability that both of them scored above a 1520

Answer 2

The required probability is $0.2223$

Independent probability:

An event $E$ can be called an independent of another event $F$ if the probability of occurrence of one event is not affected by the occurrence of the other. Suppose two cards are drawn one after the other.

Let $X$ be the scores of the exam.

Now, solving $P(X > 1520)$

$P(X > 1520)=P(\frac{x-\mu}{\sigma} > \frac{1520-1497}{322} )\\=P(z > 0.0714)\\=1-0.5285\\=0.4715$

Now, let $x_1$ and $x_2$ be the scores of those two people where  $x_1$ and $x_2$ are given to be independent.

Then,

$P(x_1 > 1520,x_2 > 1520)=P(x_1 > 1520)\times P(x_2 > 1520)\\=0.4715 \times 0.4715\\=0.2223$

Learn more about the topic of Independent probability:

https://brainly.com/question/26169642