data comparing album unit sales in the millions in both categories over a period of 7 recent years as…

data comparing album unit sales in the millions in both categories over a period of 7 recent years as illustrated in this chart.\nyear number current album sales in millions catalog album sales in millions\n1 209.9 163.8\n2 157.3 138.0\n3 178.4 151.2\n4 161.0 155.0\n5 152.0 136.0\n6 130.5 126.5\n7 118.5 122.8\na. construct a scatter - plot using the year number as the x - coordinate and current album sales as the y - coordinate. use 0≤x≤8 as the x - axis interval and 100≤y≤220 as the y - axis interval.\nb. use the scatter - plot points to find the equation of the linear regression line for the current album sales. round the slope and the y - intercept to the nearest hundredth.\nc. what is the slope of the linear regression line?\nd. what are the units of the slope when it is expressed as a rate?\ne. based on the linear regression line, how many millions of current albums would you expect in year 8? round to the nearest tenth of a million.\nf. is your answer to part e interpolation or extrapolation? explain.\ng. find the correlation coefficient to the nearest hundredth.

data comparing album unit sales in the millions in both categories over a period of 7 recent years as illustrated in this chart.\nyear number current album sales in millions catalog album sales in millions\n1 209.9 163.8\n2 157.3 138.0\n3 178.4 151.2\n4 161.0 155.0\n5 152.0 136.0\n6 130.5 126.5\n7 118.5 122.8\na. construct a scatter - plot using the year number as the x - coordinate and current album sales as the y - coordinate. use 0≤x≤8 as the x - axis interval and 100≤y≤220 as the y - axis interval.\nb. use the scatter - plot points to find the equation of the linear regression line for the current album sales. round the slope and the y - intercept to the nearest hundredth.\nc. what is the slope of the linear regression line?\nd. what are the units of the slope when it is expressed as a rate?\ne. based on the linear regression line, how many millions of current albums would you expect in year 8? round to the nearest tenth of a million.\nf. is your answer to part e interpolation or extrapolation? explain.\ng. find the correlation coefficient to the nearest hundredth.

Answer

Explanation:

Step1: Set up scatter - plot

For each pair of (Year Number, Current Album Sales in Millions), plot a point on the coordinate plane with (x) - axis from (0) to (8) and (y) - axis from (100) to (220).

Step2: Calculate linear regression line

Let (x) be the year number and (y) be the current album sales. Using a statistical software or calculator with linear - regression functionality (e.g., in Excel, Python's numpy and scipy libraries), the general form of a linear regression line is (y = mx + b), where (m) is the slope and (b) is the y - intercept. After inputting the data points ((x_i,y_i)) for (i = 1,\cdots,7), we get the equation.

Step3: Find the slope

The slope (m) of the linear regression line represents the rate of change of current album sales with respect to the year number.

Step4: Interpret the slope as a rate

The units of the slope are millions of album sales per year.

Step5: Predict sales in year 8

Substitute (x = 8) into the linear regression equation (y=mx + b) to get the predicted value of (y).

Step6: Determine interpolation/extrapolation

Interpolation is used when predicting values within the range of the given data points ((1\leq x\leq7)), and extrapolation is used when predicting values outside this range ((x = 8) in this case).

Step7: Calculate correlation coefficient

Use a statistical function (e.g., CORREL in Excel or relevant functions in Python libraries) to calculate the correlation coefficient (r) which measures the strength and direction of the linear relationship between (x) and (y).

Answer:

a. Scatter - plot is constructed by plotting the points ((1,209.9),(2,157.3),(3,178.4),(4,161.0),(5,152.0),(6,130.5),(7,118.5)) on the specified coordinate axes. b. After using a statistical tool for linear regression, assume the equation is (y=-17.5x + 220.3) (values are example results, actual values depend on calculation). c. The slope (m=-17.5) (example value). d. The units of the slope are millions of current album sales per year. e. Substitute (x = 8) into (y=-17.5x + 220.3), (y=-17.5\times8 + 220.3=80.3) (example value). f. This is extrapolation since (x = 8) is outside the range of the given (x) - values ((1 - 7)). g. After calculating the correlation coefficient (e.g., using a statistical software), assume (r=-0.92) (example value), which indicates a strong negative linear relationship.