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The daily high and low temperatures for 12 U.S. cities are as follows (Weather Channel, January 25,2004 ). $$\begin{array}{lccccc} \text { City } & \text { High } & \text { Low } & \text { City } & \text { High } & \text { Low } \\ \text { Albany } & 9 & -8 & \text { Los Angeles } & 62 & 47 \\ \text { Boise } & 32 & 26 & \text { New Orleans } & 71 & 55 \\ \text { Cleveland } & 21 & 19 & \text { Portland } & 43 & 36 \\ \text { Denver } & 37 & 10 & \text { Providence } & 18 & 8 \\ \text { Des Moines } & 24 & 16 & \text { Raleigh } & 28 & 24 \\ \text { Detroit } & 20 & 17 & \text { Tulsa } & 55 & 38\end{array}$$ a. What is the sample mean daily high temperature? b. What is the sample mean daily low temperature? c. What is the correlation between the high and low temperatures?

Step by step solution

01

Compute the Mean High Temperature

First, we add up all the high temperatures: \( 9 + 32 + 21 + 37 + 24 + 20 + 62 + 71 + 43 + 18 + 28 + 55 = 420 \).Next, we divide this sum by the number of cities, which is 12: \( \frac{420}{12} = 35 \).Thus, the sample mean daily high temperature is \( 35 \) degrees.

02

Compute the Mean Low Temperature

Add all the low temperatures: \( -8 + 26 + 19 + 10 + 16 + 17 + 47 + 55 + 36 + 8 + 24 + 38 = 288 \).Then, divide this sum by the number of cities: \( \frac{288}{12} = 24 \).So, the sample mean daily low temperature is \( 24 \) degrees.

03

Calculate Correlation Coefficient

First, find means of high (\( \bar{x} = 35 \)) and low (\( \bar{y} = 24 \)) temperatures.Next, for each city pair, calculate products of deviations from mean: \((x - \bar{x})(y - \bar{y})\) and sum them.Compute the standard deviations of high and low: \(s_x = \sqrt{\frac{1}{n-1} \sum (x_i - \bar{x})^2}\) and \(s_y = \sqrt{\frac{1}{n-1} \sum (y_i - \bar{y})^2}\).Finally, the correlation is \(r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{(n-1) s_x s_y}\).Performing these calculations yields a correlation coefficient of approximately \( r \approx 0.92 \).

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Key Concepts

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

Mean Temperature

The mean temperature is an average that gives us a central value around which the daily temperatures of different cities are organized. It helps paint a picture of typical climate conditions.
To compute the mean, we add up all temperatures and divide by the number of observations. For daily high temperatures, the sum is 420 degrees from 12 cities. We divide this sum by 12, giving us a mean high temperature of 35 degrees. Similarly, for low temperatures, the total is 288 degrees. Dividing by 12 yields a mean low temperature of 24 degrees.
The calculation of mean temperature involves:

• Add all temperature values.
• Count the number of observations.
• Divide the sum of temperatures by this count.

This process provides a single representative figure that simplifies complex climate data, making it easier to compare and analyze different regions.

Correlation Coefficient

The correlation coefficient is a statistical measure that expresses the extent to which two variables are linearly related. In temperature analysis, it tells us whether there's a relationship between high and low temperatures across various cities.
The value of the correlation coefficient, denoted as \( r \), ranges from -1 to 1. An \( r \) close to 1 indicates a strong positive relationship, meaning as one temperature increases, so does the other. An \( r \) near -1 would suggest an inverse relationship, while an \( r \) around zero indicates no relationship.
To calculate it, we:

• Find the deviations of each temperature from their respective means.
• Multiply these deviations for high and low temperatures.
• Sum these products.
• Normalize by dividing by the product of the number of observations minus one and the standard deviations of both temperature sets.

In our example, the correlation coefficient is approximately 0.92, suggesting a strong positive relationship between high and low temperatures in the cities studied.

Temperature Analysis

Temperature analysis involves understanding temperature characteristics across different regions and their patterns. It can help in weather forecasting, climate monitoring, and even in practical matters like planning agricultural activities.
By calculating mean temperatures, we establish average climatic conditions, while the correlation coefficient informs us of the variability and relationship between different aspects of temperature.
Such analysis helps in:

• Identifying trends and anomalies in temperature patterns.
• Understanding climate zones and their differences.
• Predicting future temperature changes based on historical data.

Temperature analysis thus combines statistical tools like mean calculation and correlation to derive meaningful insights into how we experience warmth and cold across geographies.