For a normal distribution $Z(\mu, \sigma) \mu=100, \sigma=10$ What percentile is the value $X=85$ write your answer to a decimal place of one tenth percent a. 67 th percentile b. 6.6 percentile c. 85 th percentile d. 6.7 th percentile

Step 1 ：Given a normal distribution \(Z(\mu, \sigma)\) where \(\mu=100\), \(\sigma=10\), we are asked to find the percentile of the value \(X=85\).

Step 2 ：First, we need to calculate the z-score. The z-score is calculated as follows: \(Z = \frac{X - \mu}{\sigma}\) where \(X\) is the raw score, \(\mu\) is the population mean, and \(\sigma\) is the standard deviation of the population.

Step 3 ：In this case, \(X = 85\), \(\mu = 100\), and \(\sigma = 10\). Let's plug these values into the formula and calculate the z-score.

Step 4 ：After calculating the z-score, we can use the cumulative distribution function (CDF) of the standard normal distribution to find the percentile. The CDF will give us the probability that a random variable is less than or equal to a certain value. This probability is the percentile of the value.

Step 5 ：Let's calculate the z-score and then find the percentile.

Step 6 ：Given values are \(X = 85\), \(\mu = 100\), and \(\sigma = 10\).

Step 7 ：Calculate the z-score: \(z = \frac{X - \mu}{\sigma} = -1.5\).

Step 8 ：Calculate the percentile using the cumulative distribution function (CDF) of the standard normal distribution: percentile = \(6.680720126885807\).

Step 9 ：The percentile of the value \(X=85\) in a normal distribution with mean 100 and standard deviation 10 is approximately 6.7. This means that approximately 6.7% of the data in this distribution is less than or equal to 85.

Step 10 ：Final Answer: \(\boxed{d. 6.7 th percentile}\)