If a distribution of test scores has μ = 65 with σ = 6, and you scored at the 89th percentile, what is your score?
To find your score at the 89th percentile, you need to convert the percentile to a z-score and then use the z-score to find the corresponding raw score.
To start, let's find the z-score associated with the 89th percentile. A z-score represents the number of standard deviations away from the mean a particular value is.
To find the z-score, we can use the standard normal distribution table or a calculator that provides z-score calculations. Since a standard normal distribution has a mean of 0 and a standard deviation of 1, we need to convert our original distribution into a standard normal distribution.
To convert the original distribution to a standard normal distribution, we can use the formula: z = (x - μ) / σ, where x is the raw score, μ is the mean, and σ is the standard deviation.
Plugging in the values, we have: z = (x - 65) / 6.
Next, we find the z-score corresponding to the 89th percentile. In a standard normal distribution, the 89th percentile corresponds to a z-score of approximately 1.23. This can be obtained from a standard normal distribution table or a calculator.
Now, we can solve for the raw score (x) using the z-score formula:
1.23 = (x - 65) / 6
To isolate x, we can multiply both sides of the equation by 6:
1.23 * 6 = x - 65
7.38 = x - 65
To solve for x, we can add 65 to both sides of the equation:
7.38 + 65 = x
x = 72.38
Therefore, if you scored at the 89th percentile in a distribution with a mean of 65 and a standard deviation of 6, your score would be approximately 72.38.