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?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to that Z score. Multiply by 200.
To find out how many students will have scores that fall under a score of 41, we need to calculate the z-score for this particular score and then use the cumulative distribution function (CDF) of the standard normal distribution.
The formula to calculate the z-score is:
z = (X - μ) / σ
Where:
X = score we want to find the probability for (41 in this case)
μ = mean of the distribution (45 in this case)
σ = standard deviation of the distribution (4 in this case)
Substituting the values, we get:
z = (41 - 45) / 4
z = -4 / 4
z = -1
Now, we can use the CDF function to find the probability of getting a score less than or equal to -1 in a standard normal distribution. This will give us the proportion of students who scored less than or equal to 41.
Using a standard normal distribution table or a statistical calculator, we can find that the cumulative probability at z = -1 is approximately 0.1587.
To calculate the number of students who fall under the score of 41, we multiply the proportion by the total sample size:
Number of students = Proportion * Total sample size
Number of students = 0.1587 * 200
Number of students ≈ 31.74
Since we cannot have fractional students, we round down to the nearest whole number.
Therefore, approximately 31 students out of the 200 will have scores that fall under the score of 41.