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Chapter 5: Problem 66

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?

Short Answer

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

Step by step solution

01

Understanding the Problem

First, summarize what needs to be calculated. We know the average temperature in Las Vegas in July is 104掳F with a standard deviation of 4.5掳F. We will find the percentile rank of a temperature of 112掳F, determine the temperature corresponding to the 67th percentile, and find the temperature that lies in the top 0.5%.
02

Calculate the Z-Score for 112掳F

Use the Z-score formula for 112掳F: \[ Z = \frac{X - \mu}{\sigma} \]where \(X = 112\), \(\mu = 104\), \(\sigma = 4.5\). Substitute the values:\[ Z = \frac{112 - 104}{4.5} = \frac{8}{4.5} \approx 1.78 \]
03

Find the Percentile of 112掳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).
04

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 \(\sigma = 4.5\):\[ X = 104 + 0.44 \times 4.5 \approx 105.98 \]
05

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 \(\sigma = 4.5\):\[ X = 104 + 2.58 \times 4.5 \approx 115.61 \]
06

Confirm and Conclude

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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掳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掳F).
  • \( \mu \) is the mean (average) temperature (104掳F in our case).
  • \( \sigma \) is the standard deviation (4.5掳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掳F shows that most temperatures in Las Vegas during July hover around the average of 104掳F, but variations of around 4.5掳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.

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