the random variable x is normally distributed with a mean of 75 and a standard deviation of 15.0. For this distribution, what is the twenty-third percentile?
23% of the population is below x = 64, if that is what you mean.
See http://psych.colorado.edu/~mcclella/java/normal/accurateNormal.html
for how I got that.
Z = (score-mean)/SD
If you do not have a computer available, find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion to help you find the above Z score and then the specific score.
To find the twenty-third percentile for a normally distributed random variable with a mean of 75 and a standard deviation of 15.0, you can use the z-score formula and standard normal distribution.
Step-by-step solution:
1. Calculate the z-score for the given percentile using the formula: z = (x - μ) / σ, where x is the desired percentile, μ is the mean, and σ is the standard deviation.
In this case, the x-value is the twenty-third percentile, μ is the mean (75), and σ is the standard deviation (15.0).
z = (x - 75) / 15.0
2. Use a z-table, calculator, or statistical software to find the z-value corresponding to the desired percentile. In this case, we want to find the z-value for the twenty-third percentile.
3. Once you have the z-value, you can find the corresponding x-value using the formula: x = μ + (z * σ).
Substitute the values into the equation: x = 75 + (z * 15.0). This will give you the value at the twenty-third percentile.
Note: If you're using a z-table, look up the z-value in the table and convert it back to the original x-value using step 3.
By following these steps, you should be able to find the twenty-third percentile for a normally distributed random variable with a mean of 75 and a standard deviation of 15.0.