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The daily high and low temperatures for 14 cities around the world are shown (The Weather Channel, April 22,2009 ). $$\begin{array}{lcccc} \text { City } & \text { High } & \text { Low } & \text { City } & \text { High } & \text { Low } \\ \text { Athens } & 68 & 50 & \text { London } & 67 & 45 \\ \text { Beijing } & 70 & 49 & \text { Moscow } & 44 & 29 \\ \text { Berlin } & 65 & 44 & \text { Paris } & 69 & 44 \\ \text { Cairo } & 96 & 64 & \text { Rio de Janeiro } & 76 & 69 \\ \text { Dublin } & 57 & 46 & \text { Rome } & 69 & 51 \\ \text { Geneva } & 70 & 45 & \text { Tokyo } & 70 & 58 \\ \text { Hong Kong } & 80 & 73 & \text { Toronto } & 44 & 39 \end{array}$$ a. What is the sample mean high temperature? b. What is the sample mean low temperature? c. What is the correlation between the high and low temperatures? Discuss.

Step by step solution

01

Calculate the Sum of High Temperatures

To find the sample mean high temperature, add all the high temperatures for the 14 cities together. For cities: 68, 70, 65, 96, 57, 70, 80, 67, 44, 69, 76, 69, 70, 44. Sum = 68 + 70 + 65 + 96 + 57 + 70 + 80 + 67 + 44 + 69 + 76 + 69 + 70 + 44 = 915.

02

Calculate the Mean of High Temperatures

Now, divide the sum by the number of cities to find the mean.Mean High = \( \frac{915}{14} \approx 65.36 \).

03

Calculate the Sum of Low Temperatures

Similarly, add all the low temperatures for the 14 cities. For cities: 50, 49, 44, 64, 46, 51, 45, 73, 45, 29, 44, 39, 58, 69. Sum = 50 + 49 + 44 + 64 + 46 + 51 + 45 + 73 + 45 + 29 + 44 + 39 + 58 + 69 = 706.

04

Calculate the Mean of Low Temperatures

Now, divide the sum by the number of cities to find the mean.Mean Low = \( \frac{706}{14} \approx 50.43 \).

05

Calculate the Covariance

To find the correlation, first calculate the covariance between high and low temperatures. Use the formula:Covariance \( = \frac{\sum{(x_i - \bar{x}) (y_i - \bar{y})}}{n} \).Here, \( x_i \) are high temperatures, and \( y_i \) are low temperatures. First compute \( x_i - \bar{x} \) and \( y_i - \bar{y} \) for each pair of high and low, multiply, sum up, and divide by 14.

06

Calculate the Standard Deviations

Find standard deviations for high and low temperatures using:Standard deviation \( = \sqrt{\frac{\sum{(x_i - \bar{x})^2}}{n-1}} \).Compute these for both high and low temperatures.

07

Calculate the Correlation

Finally, calculate the correlation coefficient with:Correlation \( r = \frac{\text{Covariance}}{s_x s_y} \),where \( s_x \) and \( s_y \) are the standard deviations of high and low temperatures.

08

Interpret the Correlation

If the correlation is close to 1, it indicates a strong positive linear relationship; if it's close to -1, a strong negative linear relationship; if around 0, no relationship.

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Key Concepts

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

Sample Mean

The concept of a sample mean is essential when analyzing data sets as it provides a measure of the central tendency of individual data points. To find the sample mean of a given data set, such as temperatures across various cities, you start by summing up all the data points. This process is straightforward: simply add up all the high temperatures collected from the 14 cities. For instance, temperatures of 68, 70, 65, and so on are tallied:

• Step 1: Sum of temperatures = 915.
• Step 2: Divide by the number of observations (14 cities) to get the mean.

Thus, the calculation for the mean high temperature is \[\frac{915}{14} \approx 65.36.\]This result signifies the average high temperature observed across all the cities. Similarly, to find the mean of the low temperatures, follow the same steps with the low temperature data. The mean gives a central value that is useful for comparing with other days or another set of data.

Correlation Coefficient

The correlation coefficient is a statistical measure that quantifies the strength and direction of a linear relationship between two variables. In our exercise, the variables are the high and low temperatures of various cities. A correlation coefficient is a value between -1 and 1.

• If the value is close to 1, it implies a strong positive relationship, meaning as high temperatures increase, low temperatures tend to also increase.
• If close to -1, it suggests a strong negative relationship.
• A value around 0 indicates no linear relationship.

To find this coefficient, the covariance between the high and low temperatures is computed first, then divided by the product of their standard deviations. It is a helpful metric when wanting to understand how temperatures behave in relation to each other across different cities.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. When examining temperatures, for example, the standard deviation helps us understand how spread out the temperature values are from the mean. A small standard deviation indicates that the temperature values are close to the mean, while a larger one indicates a wider spread.

• Calculate each temperature’s deviation from the mean.
• Square these deviations.
• Sum them up across all observations.
• Divide by the number of observations minus one.
• Finally, take the square root of this quotient.

This process results in the standard deviation for both high and low temperatures, which is essential for calculating the correlation coefficient and interpreting data variability.

Temperature Analysis

Temperature analysis involves examining and interpreting temperature data to identify patterns, trends, and anomalies. In urban studies, like the temperatures observed across 14 cities globally, it's a way to understand local climate variations. Through statistical methods, such as calculating mean temperatures and correlation coefficients, we can effectively analyze weather data.

• It can help us compare climatological trends between different regions.
• Provides insights into potential impacts of climate change.
• Prepares urban planners and policy-makers to design adaptable infrastructures.

Such analysis is crucial for making informed decisions in agriculture, urban development, and environmental policy, leveraging the statistical data to better understand how local and global climates interact with one another.