Questions: Assume the random variable X is normally distributed with mean μ=50 and standard deviation σ=7. Find the 87th percentile. The 87th percentile is . (Round to two decimal places as needed.)

Assume the random variable X is normally distributed with mean μ=50 and standard deviation σ=7. Find the 87th percentile.

The 87th percentile is . (Round to two decimal places as needed.)

Transcript text: Assume the random variable $X$ is normally distributed with mean $\mu=50$ and standard deviation $\sigma=7$. Find the 87th percentile. The 87th percentile is $\square$ . (Round to two decimal places as needed.)

Solution

Solution Steps

Step 1: Define the Parameters

Let the random variable $ X $ be normally distributed with mean $ \mu = 50 $ and standard deviation $ \sigma = 7 $.

Step 2: Identify the Desired Percentile

We are tasked with finding the 87th percentile, which corresponds to a cumulative probability of $ p = 0.87 $.

Step 3: Calculate the 87th Percentile

To find the 87th percentile, we use the inverse cumulative distribution function (CDF) for a normal distribution. The formula for the inverse CDF is given by:

\[ x = \mu + z \cdot \sigma \]

where $ z $ is the z-score corresponding to the cumulative probability $ p $.

Step 4: Find the Z-Score

Using the inverse CDF, we find the z-score $ z $ such that:

\[ P(X \leq x) = 0.87 \]

Step 5: Compute the 87th Percentile Value

Substituting the values into the formula, we find:

\[ x = 50 + z \cdot 7 \]

After calculating, we find that the 87th percentile is approximately $ 57.88 $.

Step 6: Round the Result

Finally, we round the result to two decimal places, yielding:

\[ \text{The 87th percentile is } 57.88. \]