Final answer:
To find the probability that both individuals scored above 1450 on the SAT, we calculated the z-score for a score of 1450 and used it to find the individual probability of scoring above 1450. Since the scores are independent, the joint probability is the square of the individual probability, which results in approximately 33%.
Step-by-step explanation:
To calculate the probability that both randomly chosen people scored above 1450 on the SAT, we first need to find the z-score for a score of 1450. We know the mean SAT score in 2010 was 1509 with a standard deviation of 312. The z-score can be calculated as follows:
Z = (X - μ) / σ
Where X is the SAT score, μ is the mean, and σ is the standard deviation.
Z = (1450 - 1509) / 312 ≈ -0.1889
Next, we use the standard normal distribution table to find the probability corresponding to the z-score. The probability for a z-score of -0.1889 is approximately 0.5744. This is the probability of one person scoring above 1450. Since we are looking for the probability that both people scored above 1450 and their scores are independent, we multiply the individual probabilities:
P(both > 1450) = P(one > 1450)^2
P(both > 1450) = 0.5744 × 0.5744 ≈ 0.3300
Therefore, the probability that both individuals scored above 1450 is approximately 0.33 or 33%.