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=(x-μ)/σ
Find the p-value of each z-score by looking it up in the z chart.
It'll help you determine the percentile rank if you draw the normal distribution curve.
http://www.mathsisfun.com/data/standard-normal-distribution-table.html
μ = 70 and σ = 20
To find the z-score for each X score, we can use the formula:
z = (X - μ) / σ,
where X is the score, μ is the population mean, and σ is the population standard deviation.
Let's calculate the z-scores for each X score:
For X = 80:
z = (80 - 70) / 20
z = 10 / 20
z = 0.5
For X = 100:
z = (100 - 70) / 20
z = 30 / 20
z = 1.5
For X = 65:
z = (65 - 70) / 20
z = -5 / 20
z = -0.25
For X = 40:
z = (40 - 70) / 20
z = -30 / 20
z = -1.5
To find the percentile rank for each X score, we can use the z-score to look up the corresponding percentile in a standard normal distribution table.
The percentile rank represents the percentage of scores in the population that are below a given value.
Let's calculate the percentile rank for each X score:
For X = 80:
Using the z-score of 0.5, we can find the percentile rank as 0.6915 or 69.15%.
For X = 100:
Using the z-score of 1.5, we can find the percentile rank as 0.9332 or 93.32%.
For X = 65:
Using the z-score of -0.25, we can find the percentile rank as 0.4013 or 40.13%.
For X = 40:
Using the z-score of -1.5, we can find the percentile rank as 0.0668 or 6.68%.
Therefore, the z-scores and percentile ranks for each X score 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%
To find the z-score for a given X score in a population, 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 given X scores:
For X = 80:
z = (80 - 70) / 20
z = 10 / 20
z = 0.5
For X = 100:
z = (100 - 70) / 20
z = 30 / 20
z = 1.5
For X = 65:
z = (65 - 70) / 20
z = -5 / 20
z = -0.25
For X = 40:
z = (40 - 70) / 20
z = -30 / 20
z = -1.5
Now let's calculate the percentile rank for each of these z-scores. The percentile rank represents the percentage of scores that fall below a given value.
To find the percentile rank, you can use a standard normal distribution table or a calculator.
For z = 0.5, the percentile rank is approximately 69.15%. This means that about 69.15% of the scores in the population fall below X = 80.
For z = 1.5, the percentile rank is approximately 93.32%. This means that about 93.32% of the scores in the population fall below X = 100.
For z = -0.25, the percentile rank is approximately 39.44%. This means that about 39.44% of the scores in the population fall below X = 65.
For z = -1.5, the percentile rank is approximately 6.68%. This means that about 6.68% of the scores in the population fall below X = 40.