91Ó°ÊÓ

Data consistent with summary quantities in the article referenced in Exercise \(9.3\) on total calorie consumption on a particular day are given for a sample of children who did not eat fast food on that day and for a sample of children who did eat fast food on that day. Assume that it is reasonable to regard these samples as representative of the population of children in the United States. $$\begin{aligned} &\text { No Fast Food }\\\ &\begin{array}{llllllll} 2331 & 1918 & 1009 & 1730 & 1469 & 2053 & 2143 & 1981 \\ 1852 & 1777 & 1765 & 1827 & 1648 & 1506 & 2669 & \end{array} \end{aligned}$$ $$\begin{aligned} &\text { Fast Food } \\ &\begin{array}{rrrrrrrr} 2523 & 1758 & 934 & 2328 & 2434 & 2267 & 2526 & 1195 \\ 890 & 1511 & 875 & 2207 & 1811 & 1250 & 2117 & \end{array} \end{aligned}$$ a. Use the given information to estimate the mean calorie intake for children in the United States on a day when no fast food is consumed. b. Use the given information to estimate the mean calorie intake for children in the United States on a day when fast food is consumed. c. Use the given information to produce estimates of the standard deviations of calorie intake for days when no fast food is consumed and for days when fast food is consumed.

Step by step solution

01

Calculate the Mean for No Fast Food

Firstly, add up all the calorie intake present in the 'No Fast Food' category and then divide by the total number of values. This will provide the average (or mean) calorie intake for a child who did not eat fast food on that day.

02

Calculate the Mean for Fast Food

Similarly, add up all the calorie intake present in the 'Fast Food' category and then divide by the total number of values. This will provide the average (or mean) calorie intake for a child who ate fast food on that day.

03

Calculate the Standard Deviation for No Fast Food

Use the formula for standard deviation. Subtract the mean from each of the datapoints, square the result and sum them up. Divide this by the total number of data points and then take the square root.

04

Calculate the Standard Deviation for Fast Food

Repeat step 3, but with the 'Fast Food' data. Subtract the mean from each of the datapoints, square the result and sum them up. Divide this by the total number of data points and then take the square root.

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Key Concepts

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

Mean Calculation

In statistics, the mean, also known as the average, is a key metric in understanding data sets. To calculate the mean, you sum up all the values in a data set and then divide by the number of values.
A simple formula to represent this is: \[ \text{Mean} = \frac{\sum_{i=1}^{n} x_i}{n} \]
where \(x_i\) are the data points, and \(n\) is the number of data points.
Calculating the mean gives us a central value that acts as a summary for the entire data set.

• For example, in the problem presented, we would sum all the calorie values for both 'No Fast Food' and 'Fast Food' groups separately.
• Then, we divide the total by the number of children in each group.

This calculation can provide valuable insights, such as understanding typical calorie consumption patterns for different dietary habits.

Standard Deviation

Standard deviation is a measure of how spread out the numbers in a data set are. It indicates the variation or dispersion from the mean.
The formula for calculating standard deviation is:\[ s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2} \]
where \(\bar{x}\) is the mean and \(x_i\) are the data points.

• The steps involve subtracting the mean from each data point and squaring the result.
• These squared differences are then averaged, and the square root of this average is the standard deviation.

This measure helps us understand the degree of variability in the calorie intake among children. For instance, a large standard deviation indicates more spread in the data, meaning calorie intake varies widely among children. Conversely, a smaller standard deviation indicates that most children's calorie intakes are close to the mean.

Data Analysis

Data analysis involves the process of cleaning, transforming, and modeling data to discover useful information and inform conclusions. In the context of this problem, it helps in understanding children's calorie consumption.

• Firstly, through mean calculation, we can identify the average calorie intake for children eating fast food versus those who do not.
• Standard deviation provides insight into the consistency of these calorie intakes in each group.
• These basic descriptive statistics are media for more complex data analysis tasks such as determining the health impacts of fast food consumption.

Data analysis in this scenario supports the identification of patterns indicating whether children who eat fast food might have higher or more variable calorie consumption compared to those who do not. It is an essential tool for converting raw numbers into actionable insights and forming health recommendations.