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.
There are 200 students in a sample. How many of these students will have scores that fall under the score of 41?
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.