Answer :
To find [tex]\( P_{71} \)[/tex], the score that separates the bottom 71% from the top 29% of a normally distributed set of values, we need to follow these steps:
1. Understand the Problem:
- We are given a normal distribution with a mean ([tex]\(\mu\)[/tex]) of 97.6 and a standard deviation ([tex]\(\sigma\)[/tex]) of 23.4.
- We need to find the value, [tex]\( P_{71} \)[/tex], that separates the bottom 71% of this distribution.
2. Use Percentiles with a Standard Normal Distribution:
- A percentile is a value below which a certain percentage of observations fall.
- To find the 71st percentile in a normal distribution, we first need to locate the corresponding z-score from the standard normal distribution table or use a calculator that provides the inverse cumulative distribution function (also known as the percentile point function).
3. Find the Z-score:
- The z-score is the number of standard deviations away from the mean.
- For the 71st percentile ([tex]\( p = 0.71 \)[/tex]), the z-score is approximately 0.553.
4. Convert the Z-score to a Raw Score:
- Use the formula to convert the z-score to the actual score [tex]\( x \)[/tex] in the distribution:
[tex]\[
x = \mu + z \times \sigma
\][/tex]
- Substitute the known values:
[tex]\[
x = 97.6 + 0.553 \times 23.4
\][/tex]
5. Calculate the Result:
- Perform the multiplication and addition:
[tex]\[
x = 97.6 + 12.9362 \approx 110.5
\][/tex]
Hence, the score [tex]\( P_{71} \)[/tex] that separates the bottom 71% from the top 29% is approximately 110.5 when rounded to one decimal place.
1. Understand the Problem:
- We are given a normal distribution with a mean ([tex]\(\mu\)[/tex]) of 97.6 and a standard deviation ([tex]\(\sigma\)[/tex]) of 23.4.
- We need to find the value, [tex]\( P_{71} \)[/tex], that separates the bottom 71% of this distribution.
2. Use Percentiles with a Standard Normal Distribution:
- A percentile is a value below which a certain percentage of observations fall.
- To find the 71st percentile in a normal distribution, we first need to locate the corresponding z-score from the standard normal distribution table or use a calculator that provides the inverse cumulative distribution function (also known as the percentile point function).
3. Find the Z-score:
- The z-score is the number of standard deviations away from the mean.
- For the 71st percentile ([tex]\( p = 0.71 \)[/tex]), the z-score is approximately 0.553.
4. Convert the Z-score to a Raw Score:
- Use the formula to convert the z-score to the actual score [tex]\( x \)[/tex] in the distribution:
[tex]\[
x = \mu + z \times \sigma
\][/tex]
- Substitute the known values:
[tex]\[
x = 97.6 + 0.553 \times 23.4
\][/tex]
5. Calculate the Result:
- Perform the multiplication and addition:
[tex]\[
x = 97.6 + 12.9362 \approx 110.5
\][/tex]
Hence, the score [tex]\( P_{71} \)[/tex] that separates the bottom 71% from the top 29% is approximately 110.5 when rounded to one decimal place.