The scores of a recent standardized test taken by finance majors at a four-year college had an approximately normal distribution with a mean of 225 and a standard deviation of 18. Determine the probability to the nearest integer that a randomly selected test scored between 200 and 245.(1 point)
To find the probability that a randomly selected test scored between 200 and 245, we need to find the z-scores for these two values and then use a z-table to find the area under the normal curve between these two z-scores.
First, let's find the z-score for a test score of 200:
z = (X - μ) / σ
z = (200 - 225) / 18
z ≈ -1.39
Next, let's find the z-score for a test score of 245:
z = (X - μ) / σ
z = (245 - 225) / 18
z ≈ 1.11
Now, we look up these z-scores in a standard normal distribution table (z-table) to find the area between -1.39 and 1.11. The area under the normal curve between these two z-scores represents the probability that a randomly selected test scored between 200 and 245.
From the z-table:
- For a z-score of -1.39, the area to the left of this z-score is approximately 0.0823.
- For a z-score of 1.11, the area to the left of this z-score is approximately 0.8643.
Therefore, the probability that a randomly selected test scored between 200 and 245 is:
0.8643 - 0.0823 ≈ 0.782
Thus, the probability to the nearest integer is 78%.