Faculty in the psychology department at APUS consume an average of 5 cups of coffee per day with a standard deviation of 1.5. The distribution is normal. How many cups of coffee would an individual at the 25th percentile drink per day?
A. 4
B. 5
C. 6
D. 7
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion (.25) related to a Z score. Insert the values into the above equation and solve for the score.
To find the number of cups of coffee an individual at the 25th percentile would drink, we need to use the standard normal distribution table or a statistical calculator.
Here's how you can use a standard normal distribution table:
1. Convert the percentile to a z-score. The z-score represents the number of standard deviations a value is away from the mean.
- First, determine the cumulative proportion up to the 25th percentile. The cumulative proportion up to the 25th percentile is 0.25.
- Look up the corresponding z-score in the standard normal distribution table. For a cumulative proportion of 0.25, the z-score is approximately -0.6745.
2. Use the z-score formula to convert the z-score back to the original units (cups of coffee).
- z = (x - μ) / σ
Where:
- z is the z-score
- x is the observed value (number of cups of coffee)
- μ is the mean of the distribution (average cups of coffee)
- σ is the standard deviation of the distribution
Rearrange the formula to solve for x:
- x = (z * σ) + μ
Substitute the z-score (-0.6745), the standard deviation (1.5), and the mean (5) into the formula:
- x = (-0.6745 * 1.5) + 5
- x ≈ 4.98825
Rounding to the nearest whole number, an individual at the 25th percentile would drink approximately 5 cups of coffee per day.
Therefore, the answer is B. 5.