To estimate the probability of at least 20 students scoring greater than 78 out of a sample of 75 students, we can use the normal distribution and the properties of z-scores.
First, we need to standardize the scores using the formula for z-scores: z = (x - μ) / σ, where x is the individual score, μ is the mean, and σ is the standard deviation.
Let's find the z-score associated with a score of 78:
z = (78 - 68.2) / 10.4
z = 0.9423
Next, we'll find the probability associated with this z-score using a standard normal table or calculator. We want to find P(Z > 0.9423), where Z represents a standard normal random variable. Looking up this value in a standard normal table, we find that the probability is approximately 0.1736.
Now, we need to consider the sample of 75 students. Since the scores are independent, we can use the binomial distribution to model the number of students scoring greater than 78.
Using the binomial distribution equation: P(X ≥ k) = 1 - P(X < k), where X is the number of students scoring greater than 78, and k is the minimum number of students.
We want to find P(X ≥ 20) for our given sample size of 75 students. We can use the binomial probability formula to calculate this:
P(X ≥ 20) = 1 - P(X < 20)
= 1 - sum(C(n, x) * p^x * q^(n-x)) for x = 0 to 19, where n is the sample size, p is the probability of success (probability of scoring greater than 78), and q is the probability of failure (probability of scoring less than or equal to 78).
Using these values:
n = 75, p = 0.1736, q = 1 - p = 0.8264
We calculate:
P(X ≥ 20) = 1 - sum(C(75, x) * (0.1736)^x * (0.8264)^(75-x)) for x = 0 to 19
Calculating this sum may be time-consuming by hand, but you can use statistical software or an online calculator to quickly find the result.