To find the first quartile [tex]\( Q_1 \)[/tex] of a normally distributed set of IQ scores with a given mean and standard deviation, we follow these steps:
1. Understand the Distribution:
- The IQ scores are normally distributed with a mean ([tex]\( \mu \)[/tex]) of 96.3 and a standard deviation ([tex]\( \sigma \)[/tex]) of 19.2.
2. Define the First Quartile:
- The first quartile ([tex]\( Q_1 \)[/tex]) is the value below which 25% of the data falls. In terms of a normal distribution, this corresponds to the 25th percentile.
3. Find the Corresponding Z-Score:
- For a standard normal distribution, we want to find the z-score that corresponds to the 25th percentile. The z-score is a value from the standard normal distribution with mean 0 and standard deviation 1.
4. Convert the Z-Score to an IQ Score:
- Once the z-score is found, we convert it to the corresponding IQ score using the mean and standard deviation of the given IQ distribution.
5. Interpret the Percentile and Convert:
- The 25th percentile value in the standard normal distribution typically corresponds to a z-score of approximately -0.674.
However, for a more precise calculation, this specific z-score can be precisely obtained using statistical tables or functions that give the percent point function (inverse of the cumulative distribution function).
6. Apply the Z-Score Formula:
- Once we have the accurate z-score for the 25th percentile, we use the formula for converting a z-score to a raw score:
[tex]\[
Q_1 = \mu + (z \times \sigma)
\][/tex]
- Plugging in the values:
[tex]\[
Q_1 = 96.3 + (-0.674 \times 19.2)
\][/tex]
7. Calculate the Value:
- Performing the multiplication,
[tex]\[
-0.674 \times 19.2 \approx -12.947
\][/tex]
- Adding this to the mean,
[tex]\[
Q_1 = 96.3 - 12.947 \approx 83.353
\][/tex]
8. Round the Result:
- Rounding to one decimal place,
[tex]\[
Q_1 \approx 83.3
\][/tex]
Therefore, the IQ score separating the bottom 25% from the top 75% is approximately:
The first quartile is [tex]\( 83.3 \)[/tex].
