Suppose you administered an anxiety test to a large sample of people and obtained normally distributed scores with a mean of 45 and standard deviation of 4.

To find out how many students will have scores that fall under 41, we need to calculate the z-score for 41 and then use the standard normal distribution table (also known as the z-table).

The z-score formula is calculated as:

z = (x - μ) / σ

Where:
x = individual score
μ = mean score
σ = standard deviation

Let's calculate the z-score for 41 using the given information:

z = (41 - 45) / 4
= -4 / 4
= -1

Now, we need to find the area to the left of the z-score -1 using the z-table. The z-table provides the cumulative probability up to a certain z-score.

Looking up the z-score of -1 in the z-table, we find that the area to the left of -1 is approximately 0.1587.

Since we want to find the number of students with scores under 41, we can multiply the area obtained from the z-table by the total number of students in the sample (200):

Number of students = Area under the curve * Total number of students
= 0.1587 * 200
≈ 31.74

Therefore, approximately 31.74 students would have scores falling under 41. Since the number of students must be a whole number, we can round down to the nearest whole number. Thus, about 31 students would have scores below 41.