solve the problem :score a test are normally distributed with a mean of 68.2 and a standard deviation of 10.4 estimate the probablility that among 75 randomly selected students, at least 20 of them score greater than 78

To solve this problem, we need to use the concept of the normal distribution and the z-score. The z-score measures how many standard deviations an individual value is from the mean.

First, let's calculate the z-score for a score of 78:

z = (x - μ) / σ
z = (78 - 68.2) / 10.4
z = 0.9423

The next step is to find the probability of scoring greater than 78. We can use a standard normal distribution table or a statistical calculator to find this value. For now, let's assume this probability is P(score > 78) = 0.8264 (you can verify this using a standard normal distribution table).

Now, let's calculate the probability of at least 20 out of 75 students scoring greater than 78. We can use binomial probability:

P(X ≥ 20) = 1 - P(X < 20)

Next, let's calculate the mean and standard deviation for the binomial distribution. In this case, the mean (μ) is n * p and the standard deviation (σ) is √(n * p * (1 - p)), where n is the sample size and p is the probability of success.

Mean (μ) = n * p = 75 * 0.8264 = 61.98
Standard Deviation (σ) = √(n * p * (1 - p)) = √(75 * 0.8264 * (1 - 0.8264)) ≈ 5.08

Now, we can calculate the probability using binomial distribution:

P(X ≥ 20) = 1 - P(X < 20)

Using the binomial distribution table, we can find the probability of P(X < 20). From the table, we find P(X < 20) = 0.2875.

P(X ≥ 20) = 1 - P(X < 20)
P(X ≥ 20) = 1 - 0.2875
P(X ≥ 20) ≈ 0.7125

Therefore, the estimated probability that among 75 randomly selected students, at least 20 of them score greater than 78 is approximately 0.7125.