Answer :
To find the first quartile of heights for 10-year-old girls, we use the properties of the normal distribution. The first quartile represents the 25th percentile of the data. Here’s a step-by-step approach to find it:
1. Understand the Normal Distribution:
- We have a normal distribution of heights with a mean (average height, µ) of 54.4 inches and a standard deviation (σ) of 2.7 inches.
2. Locate the First Quartile:
- The first quartile (Q1) corresponds to the 25th percentile. This means 25% of the observations fall below this value.
3. Utilize the Standard Normal Distribution:
- To find this percentile, we use the inverse of the cumulative distribution function (often called the "percent-point function" or "quantile function") for a normal distribution.
4. Apply the Formula:
- The percentile (Q1) for a normal distribution can be calculated using a statistical table or software that provides the inverse function.
- Using such tools, we find that the 25th percentile value, when the data is distributed with a mean of 54.4 and a standard deviation of 2.7, is approximately 52.6 inches.
Therefore, the first quartile of heights for 10-year-old girls is 52.6 inches.
1. Understand the Normal Distribution:
- We have a normal distribution of heights with a mean (average height, µ) of 54.4 inches and a standard deviation (σ) of 2.7 inches.
2. Locate the First Quartile:
- The first quartile (Q1) corresponds to the 25th percentile. This means 25% of the observations fall below this value.
3. Utilize the Standard Normal Distribution:
- To find this percentile, we use the inverse of the cumulative distribution function (often called the "percent-point function" or "quantile function") for a normal distribution.
4. Apply the Formula:
- The percentile (Q1) for a normal distribution can be calculated using a statistical table or software that provides the inverse function.
- Using such tools, we find that the 25th percentile value, when the data is distributed with a mean of 54.4 and a standard deviation of 2.7, is approximately 52.6 inches.
Therefore, the first quartile of heights for 10-year-old girls is 52.6 inches.
