Answer: So the probability that both students have reaction times greater than or equal to 9 seconds is approximately 0.0251 or 2.51%.
Explanation:
However, assuming that you are referring to a hypothetical scenario where two students are chosen at random from a larger population, and that their reaction times follow a normal distribution with a mean of μ and a standard deviation of σ, the probability that both students have reaction times greater than or equal to 9 seconds can be calculated as follows:
Let X1 and X2 be the reaction times of the first and second students, respectively. Then, we can write:
P(X1 ≥ 9 and X2 ≥ 9) = P(X1 ≥ 9) * P(X2 ≥ 9 | X1 ≥ 9)
Since the students are chosen at random, we can assume that their reaction times are independent, which means that:
P(X2 ≥ 9 | X1 ≥ 9) = P(X2 ≥ 9)
Now, if we assume that the reaction times follow a normal distribution, we can standardize them using the z-score:
z = (X - μ) / σ
where X is the reaction time, μ is the mean, and σ is the standard deviation. Then, we can use a standard normal distribution table to find the probability that a random variable Z is greater than or equal to a certain value z. In this case, we have:
P(X ≥ 9) = P(Z ≥ (9 - μ) / σ)
Assuming that μ = 8 seconds and σ = 1 second, we can calculate:
P(X ≥ 9) = P(Z ≥ 1)
Using a standard normal distribution table, we can find that P(Z ≥ 1) ≈ 0.1587.
Therefore:
P(X1 ≥ 9 and X2 ≥ 9) = P(X1 ≥ 9) * P(X2 ≥ 9 | X1 ≥ 9)
= P(X ≥ 9) * P(X ≥ 9)
= (0.1587) * (0.1587)
≈ 0.0251
So the probability that both students have reaction times greater than or equal to 9 seconds is approximately 0.0251 or 2.51%.