Problem 66 - Calculus and its applications

Question

Average temperature. Las Vegas, Nevada, has an average daily high temperature of 104 degrees in July, with a standard deviation of 4.5 degrees. (Source: www.weatherspark .com.) a) In what percentile is a temperature of 112 degrees? b) What temperature would be at the 67 th percentile? c) What temperature would be in the top \(0.5 \%\) of all July temperatures for this location?

Step-by-Step Solution

Answer

112°F is in the 96.2nd percentile, 106°F is the 67th percentile, and 115.61°F is in the top 0.5%.

Step 1: Understanding the Problem

First, summarize what needs to be calculated. We know the average temperature in Las Vegas in July is 104°F with a standard deviation of 4.5°F. We will find the percentile rank of a temperature of 112°F, determine the temperature corresponding to the 67th percentile, and find the temperature that lies in the top 0.5%.

Step 2: Calculate the Z-Score for 112°F

Use the Z-score formula for 112°F: \[ Z = \frac{X - \mu}{\sigma} \]where \(X = 112\), \(\mu = 104\), \(\sigma = 4.5\). Substitute the values:\[ Z = \frac{112 - 104}{4.5} = \frac{8}{4.5} \approx 1.78 \]

Step 3: Find the Percentile of 112°F

Use the Z-score to find the percentile from the standard normal distribution table. A Z-score of approximately 1.78 corresponds to the 96.2nd percentile (from tables or calculator).

Step 4: Determine Temperature at the 67th Percentile

Find the Z-score that corresponds to the 67th percentile, which is approximately 0.44 (using Z-tables or calculator). Use the inverse Z-score formula:\[ X = \mu + Z \cdot \sigma \]Substitute \(Z = 0.44\), \(\mu = 104\), and \(\sigma = 4.5\):\[ X = 104 + 0.44 \times 4.5 \approx 105.98 \]

Step 5: Calculate Z-Score for the Top 0.5%

Find the Z-score for the top 0.5% (or 99.5th percentile) which is approximately 2.58. Calculate the corresponding temperature:\[ X = \mu + Z \cdot \sigma \]Substitute \(Z = 2.58\), \(\mu = 104\), and \(\sigma = 4.5\):\[ X = 104 + 2.58 \times 4.5 \approx 115.61 \]

Step 6: Confirm and Conclude

Confirm all calculations and summarize: - The temperature of 112°F is at the 96.2nd percentile. - A temperature of approximately 106°F is at the 67th percentile. - A temperature in the top 0.5% is approximately 115.61°F.

Key Concepts

Percentiles

Percentiles are a way to understand the relative standing or position of a specific value within a data set. Imagine a set of numbers representing temperatures, and percentiles allow you to determine where a specific temperature lies compared to the rest of the data.
For example:

• A temperature at the 50th percentile means it is higher than 50% of the other temperatures.
• The 96.2nd percentile, like the temperature of 112°F from our example, indicates it's higher than 96.2% of July temperatures in Las Vegas.

Percentiles are particularly useful in statistics for comparing data points to distributions, helping define cut-off scores, and understand data spread.

Z-score

A Z-score measures how many standard deviations a particular value is from the mean of the data set. It essentially tells you how "unusual" or "typical" a particular value is in comparison to others.Here's how you compute a Z-score:1. Subtract the mean from your data point.2. Divide the result by the standard deviation.Mathematically, this is represented as: \( Z = \frac{X - \mu}{\sigma} \), where:

• \( X \) is the value you're evaluating (e.g., 112°F).
• \( \mu \) is the mean (average) temperature (104°F in our case).
• \( \sigma \) is the standard deviation (4.5°F).

A Z-score helps in identifying the percentile of the data point, with positive Z-scores indicating values above the mean and negative ones indicating values below the mean.

Standard Deviation

Standard deviation is a crucial concept in statistics that measures the amount of variation or dispersion in a set of values. A low standard deviation indicates that data points tend to be close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range. In our example, the standard deviation of 4.5°F shows that most temperatures in Las Vegas during July hover around the average of 104°F, but variations of around 4.5°F occur. Every data distribution will have its own standard deviation, and it's vital for comparing different data sets or for calculating Z-scores, which can be used to find percentiles.

Normal Distribution

The normal distribution, often called the "bell curve," is a probability distribution where most data points cluster around a central peak and taper off symmetrically towards each end. It is characterized by its mean, median, and mode being equal. Important features of a normal distribution include:

• Symmetry about the mean.
• The area under the curve represents probabilities which total up to 1.
• The empirical rule: about 68% of data falls within one standard deviation, 95% within two, and 99.7% within three.

In solving statistical problems, we use the properties of a normal distribution to interpret Z-scores and percentiles. For example, by looking at the Z-score for the 99.5th percentile, we can find the corresponding temperature in the Las Vegas July temperatures.