Final answer:
The approximate probability that a single student randomly chosen from all those taking the test scores 25 or higher is approximately 28.23%. For an SRS of 49 students who took the test, the mean is 22 and the standard deviation is 0.7429.
Step-by-step explanation:
(a) To find the approximate probability that a single student randomly chosen from all those taking the test scores 25 or higher, we need to find the z-score associated with the score of 25. The formula to calculate the z-score is:
Z = (X - μ) / σ
where Z is the z-score, X is the score, μ is the mean, and σ is the standard deviation. In this case, X = 25, μ = 22, and σ = 5.2. Plugging these values into the formula, we get:
Z = (25 - 22) / 5.2 = 3 / 5.2 = 0.5769.
Next, we need to find the area under the normal curve to the right of this z-score. Using a standard normal distribution table or a calculator, we find that the area to the right of 0.5769 is approximately 0.2823. Therefore, the approximate probability that a single student randomly chosen from all those taking the test scores 25 or higher is 0.2823, or 28.23%.
(b) To find the mean and standard deviation of an SRS of 49 students who took the test, we can use the fact that for a random sample from a population with mean μ and standard deviation σ, the mean of the sample will be approximately equal to the population mean, and the standard deviation of the sample will be approximately equal to the population standard deviation divided by the square root of the sample size. In this case, μ = 22 and σ = 5.2. Plugging these values into the formulas, we get:
Mean = μ = 22
Standard Deviation = σ / √(n) = 5.2 / √(49) = 5.2 / 7 = 0.7429.