Statistics

You are a robotics engineer working for a large tech company. For one of your artificially intelligent robots, you are looking to see if the size of the data file used to train the robot is predictive of the time the robot takes to complete a maze. You are going to collect a random sample of 7 attempts at completing the maze. You will note for each attempt the size of the data file (denoted by x, in MB) and the time (denoted by y, in seconds) to complete the maze. You will also note the product x*y of the size of the data file and the completion time for each attempt. Data file size, x (in MB): 123.8, 99.6, 198.1, 42.5, 61.3, 164.2, 85.4 Completion time, y (in seconds): 134, 253, 85, 386, 423, 157, 226 xy: 16,589.2, 25,198.8, 16,838.5, 16,405, 25,929.9, 25,779.4, 19,300.4 Find: n, x_bar, y_bar, s_x, s_y, sum(xy) Then compute: - Sample correlation coefficient r - Slope b_1 - y-intercept b_0 - Regression equation: y_hat = b_0 + b_1*x

Step-by-step solution with explanation

Final Answer

, MB, sec, , , ; (strong negative linear relationship); ; ; Regression equation:

Step-by-step solution

1

Identify n and compute sum of x, y, xy

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We count the number of data pairs (n = 7) and add up all x values, all y values, and all xy products. These sums are building blocks for every formula that follows.
2

Compute sample means x̄ and ȳ

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The sample mean is just the sum divided by the number of data points. These are the 'center' values for x and y that anchor the regression line.
3

Compute sample standard deviations s_x and s_y

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Standard deviation measures how spread out the data is around the mean. We use the sample formula (divide by n−1) because we have a sample, not the full population.
4

Compute correlation coefficient r

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r measures the strength and direction of the linear relationship. A value near −1 means a strong negative relationship: as file size increases, completion time tends to decrease.
5

Compute slope b₁ of regression line

The slope tells us the expected change in y for each 1-unit increase in x. Here, each additional 1 MB of file size is associated with about 2.056 fewer seconds to complete the maze.
6

Compute y-intercept b₀

The y-intercept is found by plugging the means and slope into the mean-point equation. It gives the predicted completion time when x = 0 MB (though that is outside our data range).
7

Write the regression equation ŷ

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This is the least-squares regression line. Plug in any file size x (in MB) to get the predicted completion time ŷ (in seconds).

Understanding this problem

Learning Insight

The correlation coefficient r measures how tightly the data clusters around a straight line, ranging from −1 to +1. A negative r here means bigger training files are linked to faster maze completion — perhaps larger files give the robot better 'knowledge.' The regression line minimizes the total squared vertical distances from each data point to the line, which is why it is called 'least-squares.'

Quick Tip

Remember: slope = r × (s_y / s_x). If you already have r and both standard deviations, the slope is one quick multiplication. Then get b₀ by plugging the two means into ȳ = b₀ + b₁x̄ and solving for b₀.

Common Mistake

Students often use n instead of n−1 when computing s_x and s_y for a sample. Always use n−1 in the denominator for sample standard deviation; using n gives the population standard deviation and will make your r and slope slightly off.