Questions: A normal population has a mean of 55 and a standard deviation of 9. What is the 84th percentile of the population? Use the π-84 Plus calculator. Round the answer to at least one decimal place.

A normal population has a mean of 55 and a standard deviation of 9. What is the 84th percentile of the population? Use the π-84 Plus calculator. Round the answer to at least one decimal place.

Transcript text: A normal population has a mean of 55 and a standard deviation of 9. What is the 84th percentile of the population? Use the $\pi$-84 Plus calculator. Round the answer to at least one decimal place.

Solution

Solution Steps

Step 1: Identify Parameters

We are given a normal population with the following parameters:

Mean ($ \mu $): 55
Standard Deviation ($ \sigma $): 9

Step 2: Determine the Z-Score for the 84th Percentile

The z-score corresponding to the 84th percentile is approximately $ Z = 1.0 $.

Step 3: Calculate the 84th Percentile Value

Using the formula for the percentile value in a normal distribution:

\[ X = \mu + Z \cdot \sigma \]

Substituting the known values:

\[ X = 55 + 1.0 \cdot 9 \]

Calculating this gives:

\[ X = 55 + 9 = 64 \]

Step 4: Round the Result

The 84th percentile value is rounded to one decimal place, which is:

\[ X = 64.0 \]

Final Answer

The 84th percentile of the population is \$\boxed{64.0}\$.

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