For a population of μ = 70 and σ = 20, find the z-score and percentile rank for each of the following X scores: 80, 100, 65, 40.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability below each Z score.
To find the z-score for each X score, you can use the formula:
z = (X - μ) / σ
Where:
- X is the individual score
- μ is the mean of the population
- σ is the standard deviation of the population
Let's calculate the z-score for each of the provided X scores:
1. For X = 80:
z = (80 - 70) / 20
= 10 / 20
= 0.5
2. For X = 100:
z = (100 - 70) / 20
= 30 / 20
= 1.5
3. For X = 65:
z = (65 - 70) / 20
= -5 / 20
= -0.25
4. For X = 40:
z = (40 - 70) / 20
= -30 / 20
= -1.5
Now, to find the percentile rank for each z-score, we can use a standard normal distribution table or a statistics calculator. Since the values are not provided in the question, I'll refer to a standard normal distribution table.
1. For z = 0.5:
The percentile rank corresponding to a z-score of 0.5 can be found by looking it up in the standard normal distribution table. The percentile rank is approximately 0.6915, or 69.15%.
2. For z = 1.5:
The percentile rank corresponding to a z-score of 1.5 can be found in the table. The percentile rank is approximately 0.9332, or 93.32%.
3. For z = -0.25:
The percentile rank corresponding to a z-score of -0.25 can be found in the table. The percentile rank is approximately 0.4013, or 40.13%.
4. For z = -1.5:
The percentile rank corresponding to a z-score of -1.5 can be found in the table. The percentile rank is approximately 0.0668, or 6.68%.
So, the z-scores and percentile ranks for the given X scores are as follows:
X = 80: z = 0.5, percentile rank = 69.15%
X = 100: z = 1.5, percentile rank = 93.32%
X = 65: z = -0.25, percentile rank = 40.13%
X = 40: z = -1.5, percentile rank = 6.68%