Question

Consider the Angus weights model \(N(1152,84)\) one last time. a) What weight represents the 40 th percentile? b) What weight represents the 99 th percentile? c) What's the IQR of the weights of these Angus steers?

Short answer

40th percentile: 1130.75, 99th percentile: 1348.72, IQR: 113.4

Step-by-step solution

Identify the Parameters

The Angus weights model is a normal distribution with mean \(\mu = 1152\) and standard deviation \(\sigma = 84\). This is represented as \(N(1152,84)\).

Define the 40th Percentile

To find the weight at the 40th percentile, we need to find the value \(x\) such that the cumulative distribution function (CDF) for \(N(1152,84)\) at \(x\) is 0.40.

Calculate the Z-Score for the 40th Percentile

We look up the 40th percentile in the standard normal distribution table to find the z-score. The z-score corresponding to the 40th percentile is approximately \(z = -0.253\).

Convert Z-Score to X Value for 40th Percentile

Use the formula \(x = \mu + z\sigma\) to find the x-value. Substitute \(\mu = 1152\), \(z = -0.253\), and \(\sigma = 84\):\[x = 1152 + (-0.253)(84) = 1130.748\]

Define the 99th Percentile

To find the weight at the 99th percentile, we need to find the value \(x\) such that the CDF for \(N(1152,84)\) at \(x\) is 0.99.

Calculate the Z-Score for the 99th Percentile

We look up the 99th percentile in the standard normal distribution table to find the z-score. The z-score corresponding to the 99th percentile is approximately \(z = 2.33\).

Convert Z-Score to X Value for 99th Percentile

Use the formula \(x = \mu + z\sigma\) to find the x-value. Substitute \(\mu = 1152\), \(z = 2.33\), and \(\sigma = 84\):\[x = 1152 + (2.33)(84) = 1348.72\]

Define IQR (Interquartile Range)

The IQR is the difference between the 75th and 25th percentiles. We must first find these percentiles.

Calculate Z-Score for 25th Percentile

The z-score for the 25th percentile (first quartile, \(Q_1\)) is approximately \(z = -0.675\).

Convert Z-Score to X Value for 25th Percentile

Use the formula \(x = \mu + z\sigma\) to find the x-value. Substitute \(\mu = 1152\), \(z = -0.675\), and \(\sigma = 84\):\[x = 1152 + (-0.675)(84) = 1095.3\]

Calculate Z-Score for 75th Percentile

The z-score for the 75th percentile (third quartile, \(Q_3\)) is approximately \(z = 0.675\).

Convert Z-Score to X Value for 75th Percentile

Use the formula \(x = \mu + z\sigma\) to find the x-value. Substitute \(\mu = 1152\), \(z = 0.675\), and \(\sigma = 84\):\[x = 1152 + (0.675)(84) = 1208.7\]

Calculate the IQR

The IQR is given by \(Q_3 - Q_1\). Substitute the values previously calculated:\[IQR = 1208.7 - 1095.3 = 113.4\]

Key concepts

Percentile

In statistics, a percentile indicates the relative standing of a value within a data set. It shows you the percentage of data points that fall below a particular value. For example, if a weight falls in the 40th percentile, it means that 40% of the weights are below this value.
Understanding percentiles is crucial for grasping how data is spread out in a normal distribution. A normal distribution is symmetric, so percentiles can help identify variations in the dataset.

• How to find a percentile: In a normal distribution, utilize the cumulative distribution function (CDF) to find the value corresponding to a specific percentile.
• Example: For the Angus weights, the 40th percentile is calculated using a z-score of -0.253, yielding a weight of about 1130.748 pounds.

Knowing how to compute percentiles can provide valuable insights into where certain values lie in relation to others, making it easier to interpret data sets.

Z-Score

The z-score is a measure that describes a value's position relative to the mean of a group of values, measured in units of standard deviation. It tells us how many standard deviations away a particular value is from the mean.
The ability to calculate a z-score is essential for understanding the normal distribution, as it allows you to determine how typical or atypical a particular data point is.

• Z-score formula: The formula is given by \( z = \frac{x - \mu}{\sigma} \), where \( x \) is the data point, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
• Application: For the Angus weights, using a z-score of 2.33 for the 99th percentile allows us to find the weight of 1348.72 pounds, which is very high relative to the average.

Mastering z-scores is key to interpreting and comparing different data sets within the context of a normal distribution pattern.

Interquartile Range (IQR)

The interquartile range (IQR) measures the middle 50% spread of data, providing a sense of variability. It's the difference between the 75th percentile (Q3) and the 25th percentile (Q1).
This concept is crucial for identifying the concentration of data around the median and for understanding how data is distributed across a range.

• IQR calculation: Using the formulas for Q3 and Q1 with their corresponding z-scores helps us compute the IQR. For instance, in the Angus weights, the IQR is calculated as \( Q3 - Q1 = 1208.7 - 1095.3 = 113.4 \) pounds.
• Significance: A smaller IQR suggests less variability, while a larger IQR indicates more spread within the data.

Understanding the IQR helps you describe the spread of data and identify potential outliers in a dataset structured by a normal distribution.