Finding Specified Data Values In Exercises 31–38, answer the questions about the specified normal distribution.


Weights of Teenagers In a survey of 18-year old males, the mean weight was 166.7 pounds with a standard deviation of 49.3 pounds. (Adapted from National Center for Health Statistics)


c. What weight represents the first quartile?

Verified step by step guidance

1. Identify the key parameters of the normal distribution: the mean (μ) is 166.7 pounds, and the standard deviation (σ) is 49.3 pounds. The first quartile corresponds to the 25th percentile of the distribution.
2. Recall that for a normal distribution, the z-score corresponding to the 25th percentile can be found using a z-table or statistical software. The z-score for the 25th percentile is approximately -0.674.
3. Use the z-score formula to find the weight (x) corresponding to the first quartile: z = (x - μ) / σ. Rearrange the formula to solve for x: x = μ + z * σ.
4. Substitute the known values into the formula: x = 166.7 + (-0.674) * 49.3. This will give the weight corresponding to the first quartile.
5. Perform the calculation to find the weight. This value represents the first quartile of the distribution, meaning 25% of the weights are below this value.

Key Concepts

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve, defined by its mean and standard deviation. Understanding this concept is crucial for analyzing data that follows this distribution, as it allows for the application of various statistical methods.

Quartiles

Quartiles are values that divide a dataset into four equal parts, each containing 25% of the data points. The first quartile (Q1) is the median of the lower half of the dataset, representing the 25th percentile. Knowing how to calculate quartiles is essential for understanding the distribution of data and identifying outliers or trends within a dataset.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range. In the context of normal distribution, it helps in determining the spread of data points and is vital for calculating quartiles and other statistical measures.