Final answer:
To calculate the 90th percentile of a normal distribution with a mean of 80 and a standard deviation of 5, use the invNorm function to find the z-score that corresponds to 0.90 on the standard normal distribution, then apply the formula k = mean + z*standard deviation. In this example, if the z-score is 1.28, the 90th percentile would be 86.4.
Step-by-step explanation:
When calculating the 90th percentile of a normally distributed random variable such as x, we use the z-score table or a statistical calculator. The mean (μ) is 80 and the standard deviation (σ) is 5. To find the 90th percentile, we look for the z-score that corresponds to an area of 0.90 to the left of it on the standard normal distribution curve. Let's denote the 90th percentile as k.
First, we would use a statistical calculator (such as the invNorm function on a graphing calculator) to find the z-score associated with the 90th percentile. The invNorm function will take the probability (in this case, 0.90) and the mean and standard deviation of the distribution. Once we obtain the z-score, we use the following formula to find the 90th percentile: k = μ + z*σ.
If the z-score for the 90th percentile is, for example, 1.28 (which is a hypothetical value for illustrative purposes – the exact value may differ slightly), then we calculate the 90th percentile as follows:
k = 80 + (1.28 * 5)
k = 80 + 6.4
k = 86.4
Thus, the 90th percentile of x is 86.4.