a set of data is normally distributed with a mean of 500 and a standard deviation of 100.
a) what would the standard score fora score of 700 be?
b) what % of scores is between 500 and 700?
c) what would the percentile rank for a score of 700?
please show calculations for a-c. want to compare calculations. thank you.
mary
Suppose that IQ scores have a bell-shaped distribution with a mean of 96 and a standard deviation of 17. Using the empirical rule, what percentage of IQ scores are between 79 and 113?
a) To find the standard score (also known as z-score) for a specific score, we can use the formula:
z = (x - μ) / σ
where:
- z is the standard score
- x is the specific score
- μ is the mean of the distribution
- σ is the standard deviation of the distribution
In this case, the mean (μ) is 500, the standard deviation (σ) is 100, and the specific score (x) is 700.
Plugging the values into the formula, we get:
z = (700 - 500) / 100
z = 2
Therefore, the standard score for a score of 700 is 2.
b) To determine the percentage of scores between 500 and 700, we need to find the area under the normal distribution curve between these two scores.
First, let's find the z-scores for both scores using the formula mentioned above:
z1 = (500 - 500) / 100 = 0
z2 = (700 - 500) / 100 = 2
Using a standard normal distribution table or calculator, we can find the area to the left of z2, which represents the percentage of scores less than 700.
The area to the left of z2 is approximately 0.9772.
Next, we find the area to the left of z1, which represents the percentage of scores less than 500. The area to the left of z1 is approximately 0.5.
The difference between these two areas gives us the percentage of scores between 500 and 700:
0.9772 - 0.5 = 0.4772
Therefore, approximately 47.72% of scores are between 500 and 700.
c) The percentile rank represents the percentage of scores that are equal to or below a particular score. To find the percentile rank for a score of 700, we need to calculate the area to the left of that score.
Using the z-score formula from part (a), we have:
z = (x - μ) / σ
z = (700 - 500) / 100
z = 2
Now, using a standard normal distribution table or calculator, we find the area to the left of z = 2, which is approximately 0.9772.
Converting this to a percentile by multiplying by 100, we have:
0.9772 * 100 = 97.72
Therefore, the percentile rank for a score of 700 is approximately 97.72%.
To find the answers to the questions, we can use the standard normal distribution, also known as the z-score. The z-score tells us how many standard deviations a given value is from the mean.
a) To find the standard score for a score of 700, we can use the formula:
z = (x - μ) / σ
where x is the score, μ is the mean, and σ is the standard deviation.
Substituting the values, we'll have:
z = (700 - 500) / 100
z = 2
Therefore, the standard score for a score of 700 is 2.
b) To find the percentage of scores between 500 and 700, we can use the cumulative probability distribution of the standard normal distribution. We need to find the area under the curve between z = 0 (the mean) and z = 2 (the score of 700).
Using a z-score table or a calculator, we find that the area to the left of z = 2 is approximately 0.9772. Since the total area under the curve is 1, the area between z = 0 and z = 2 is:
Area = 0.9772 - 0.5 = 0.4772
Therefore, the percentage of scores between 500 and 700 is approximately 47.72%.
c) The percentile rank represents the percentage of scores that are equal to or below a particular value. To find the percentile rank for a score of 700, we can use the cumulative probability distribution as mentioned before.
Using the z-score table or a calculator, we find that the area to the left of z = 2 is approximately 0.9772. This means that approximately 97.72% of scores are equal to or below a score of 700.
Therefore, the percentile rank for a score of 700 is approximately 97.72%.