How do I answer: A normal distribution has a standard deviation equal to 39. What is the mean of this normal distribution if the probability of scoring above x = 209 is 0.0228

http://davidmlane.com/hyperstat/z_table.html

To find the mean of a normal distribution, we need to use a statistical table or calculator that provides the cumulative probability. Here's how you can find the mean in this case:

Step 1: Convert the probability to a z-score
The z-score represents the number of standard deviations an observation is from the mean. We can use the formula: z = (x - μ) / σ, where x is the given value (209), μ is the mean, and σ is the standard deviation. Rearranging the formula, we have μ = x - z * σ.

To obtain the z-score, we need to look up the corresponding value in a standard normal distribution table or use a calculator that provides the cumulative probability. Since the given probability is for scoring above 209, we subtract it from 1 to get the area under the curve to the left of 209. So, we have 1 - 0.0228 = 0.9772.

Step 2: Look up the z-score in the standard normal distribution table
Using a standard normal distribution table, find the z-score that corresponds to an area of 0.9772. Look for the closest value in the table and its corresponding z-score. In this case, the closest value is 0.978 and the corresponding z-score is approximately 2.05.

Step 3: Calculate the mean
Now that we have the z-score, we can substitute it into the rearranged formula obtained in Step 1. Hence, μ = 209 - 2.05 * 39.

Calculating this, we get:
μ = 209 - 2.05 * 39 ≈ 122.55

Therefore, the mean of the normal distribution is approximately 122.55.