Question

You can monitor every step you take, your speed, your pace, or some other aspect of your daily activity. The data that follows lists the overall rating scores for 14 fitness trackers and their prices. \({ }^{13}\) $$\begin{array}{lcc}\hline \text { Fitness Trackers } & \text { Score } & \text { Price (\$) } \\\\\hline \text { Fitbit Surge } & 87 & 250 \\\\\text { TomTom Spark 3 } & 85 & 250 \\\\\text { Garmin Forerunner 38 } & 85 & 200 \\\\\text { TomTom Spark } & 84 & 200 \\\\\text { Fitbit Charge 2 } & 83 & 150 \\\\\text { Garmin Vivosmart HR } & 83 & 120 \\\\\text { Fitbit Blaze } & 82 & 200 \\\\\text { Huawei Fit } & 82 & 130 \\\\\text { Garmin Vivosmart HR+ } & 79 & 180 \\\\\text { Withings Steel HR } & 79 & 145 \\\\\text { Fitbit Alta } & 78 & 130 \\\\\text { Garmin Vivoactive HR } & 77 & 250 \\\\\text { Samsung Gear Fit 2 } & 76 & 180 \\\\\text { Under Armour Band } & 74 & 80 \\\\\hline\end{array}$$ a. Use a scatterplot of the data to check for a relationship between the rating scores and prices for the fitness trackers. b. Calculate the sample coefficient of correlation \(r\) and interpret its value. c. By what percentage was the sum of squares of deviations reduced by using the least-squares predictor \(\hat{y}=a+b x\) rather than \(\bar{y}\) as a predictor of \(y ?\)

Short answer

Answer: The percentage reduction in the sum of squares of deviations when using the least-squares predictor as opposed to using the mean of y is approximately 7.5%.

Step-by-step solution

Organize the data

List scores and prices in two columns: Scores: \(S = \{87, 85, 85, 84, 83, 83, 82, 82, 79, 79, 78, 77, 76, 74\}\) Prices: \(P = \{250, 250, 200, 200, 150, 120, 200, 130, 180, 145, 130, 250, 180, 80\}\)

Create the scatterplot

Plot the data points with Scores on the x-axis and Prices on the y-axis. Observe if there's a relationship between the two variables.

Calculate the correlation coefficient

Use the formula for the correlation coefficient, which is: $$r = \frac{n\sum (x_i y_i) - \sum x_i \sum y_i}{\sqrt{[n\sum x_i^2 - (\sum x_i)^2][n\sum y_i^2 - (\sum y_i)^2]}}$$ Where \(x_i\) are the score values, \(y_i\) are the price values, and \(n\) is the number of data points (n=14). First, determine \(\sum x_i\), \(\sum y_i\), \(\sum (x_i y_i)\), \(\sum x_i^2\), \(\sum y_i^2\): \(\sum x_i = 1113\), \(\sum y_i = 2415\), \(\sum (x_i y_i) = 243952\), \(\sum x_i^2 = 93489\), \(\sum y_i^2 = 455725\) Now, plug these values into the \(r\) formula: $$r = \frac{14 \times 243952 - 1113 \times 2415}{\sqrt{[14\times 93489 - 1113^2][14\times 455725 - 2415^2]}}$$ $$r \approx 0.213$$

Interpret the correlation coefficient

The value of the correlation coefficient (r) is 0.213, which indicates a weak positive relationship between the rating scores and prices of fitness trackers, meaning that as ratings increase, the price also tends to increase, but the relationship isn't very strong.

Calculate the sum of squared deviations for both predictors

To find the sum of squared deviations, we will first find the sums for both predictors: 1. Calculate the mean of \(y\) (Price) values: \(\bar{y} = \frac{\sum y_i}{n} = \frac{2415}{14} \approx 172.5\) 2. Calculate the sum of squared deviations using \(\bar{y}\): $$\sum (y_i - \bar{y})^2 = (250-172.5)^2 + (250-172.5)^2 + (200-172.5)^2 + \cdots + (80-172.5)^2 \approx 72120$$ Next, we need to find the least-squares predictor, \(\hat{y} = a + bx\), using the following formulas: $$b = \frac{n\sum (x_i y_i) - \sum x_i \sum y_i}{n\sum x_i^2 - (\sum x_i)^2} \approx \frac{14 \times 243952 - 1113 \times 2415}{14\times 93489 - 1113^2} \approx 6.677$$ $$a = \bar{y} - b\bar{x} \approx 172.5 - 6.677 \times \frac{1113}{14} \approx -349.9$$ Now, find the sum of squared deviations using the least-squares predictor: $$\sum (y_i - \hat{y})^2 = (250- (-349.9+ 6.677\times 87))^2 + \cdots + (80-(-349.9+6.677\times 74))^2 \approx 66719 $$

Calculate the percentage reduction

Now that we have both sums of squared deviations, we can find the percentage reduction: $$\text{Percentage Reduction} = \frac{72120 - 66719}{72120} \times 100 \approx 7.5\% $$ The sum of squares of deviations is reduced by approximately 7.5% when using the least-squares predictor \(\hat{y} = a + bx\) rather than \(\bar{y}\).

Key concepts

Correlation Coefficient

Understanding the correlation coefficient is essential for identifying the strength and direction of a linear relationship between two variables. In the context of the fitness tracker data, it's used to examine whether higher ratings are associated with higher prices.

The correlation coefficient, denoted as \(r\), ranges from -1 to 1. A value close to 1 indicates a strong positive correlation, meaning that as one variable increases, the other tends to increase as well. A value close to -1 signifies a strong negative correlation, with one variable decreasing as the other increases. A value around 0 suggests little to no linear relationship between the variables.

In our fitness tracker example, the calculated \(r \approx 0.213\) suggests a weak positive correlation. This information is beneficial as it guides consumers and manufacturers in understanding how much ratings influence price or vice versa. However, it's important to remember that correlation does not imply causation and that other factors may influence both variables.

Least-Squares Predictor

Now let's turn to the least-squares predictor, a method to find the best-fitting line through a set of points on a scatterplot. This line minimizes the sum of the squares of the vertical distances (residuals) between the observed values and the line itself.

Represented by the equation \(\text{\(\hat{y} = a + bx\)}\), where \(\hat{y}\) is the predicted value, \(a\) is the y-intercept, \(b\) is the slope, \(x\) is the independent variable (ratings in this case), this line provides a way to predict the dependent variable (price) given an independent variable's value.

In the fitness tracker scenario, the least-squares predictor helps us predict the price of a tracker based on its rating. The calculations from the solution conclude that the line \(\text{\(\hat{y} = -349.9 + 6.677x\)}\) best fits our data. Using this line for price prediction is statistically more accurate compared to using the mean value of the prices as a predictor, as it takes into account the variation in prices with ratings.

Sum of Squares of Deviations

The sum of squares of deviations is a key concept in statistics used to measure the variability within a dataset. When examining a set of points in relation to a predictor, such as a mean or a least-squares line, we calculate the squared distance each point is from the predictor.

In our exercise, we compare the sum of squared deviations from two different predictors of the price: the sample mean (\(\bar{y}\)) and the least-squares line (\(\hat{y}\)). The sum of the deviations not only provides us insight into the total error but also allows us to measure the effectiveness of different prediction methods.

After calculating, we find a 7.5% reduction in the sum of squared deviations when switching from the mean \(\bar{y}\) as a predictor to the least-squares line. This reduction represents an improvement in the predictive power of our model, meaning the least-squares line leads to less variability and a tighter fit to the data compared to the mean. This concept is pivotal when choosing the method to predict future outcomes; in this case, the least-squares predictor is the better choice.