Math Problem Statement

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Topic 7 Homework (Nonadaptive) Question 7 of 16 (1 point)|Question Attempt: 1 of Unlimited

Dennis

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Question 7 You are the owner of Fast Break, a popular local place that sells drinks, snacks, and sandwiches. For inventory management purposes, you are examining how the weather affects the amount of hot chocolate sold in a day. You are going to gather a random sample of 9 days showing that day's high temperature (denoted by x, in °C) and the amount of hot chocolate sold that day (denoted by y, in liters). You will also note the product ·xy of the temperature and amount of hot chocolate sold for each day. (These products are written in the row labeled "xy"). (a)Click on "Take Sample" to see the results for your random sample.

Take Sample High temperature, x (in °C) Amount of hot chocolate sold, y (in liters) xy

Send data to calculator

Based on the data from your sample, enter the indicated values in the column on the left below. Round decimal values to three decimal places. When you are done, select "Compute". (In the table below, n is the sample size and the symbol Σxy means the sum of the values xy.)

n: x: y: sx: sy: Σxy:

Compute Sample correlation coefficient (r): Slope (b1): y-intercept (b0):

(b)Write the equation of the least-squares regression line for your data. Then on the scatter plot for your data, graph this regression equation by plotting two points and then drawing the line through them. Round each coordinate to three decimal places. Regression equation: y=

Amount of hot chocolate sold (in liters) y24681012141618202224x3691215182124273033360

High temperature (in °C)

(c)Use your regression equation to predict the amount of hot chocolate sold on a day with a high temperature of 19 °C. Round your answer to the nearest whole number. Predicted amount of hot chocolate sold: liters

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Submit Assignment © 2024 McGraw Hill LLC. All Rights Reserved. Terms of Use | Privacy Center | Accessibility Clears your work. Undoes your last action. Tool to erase one or more objects from the graph. Tool to plot one or more points on the graph by using the arrow keys to navigate. Tool to plot one or more points on the graph by using the arrow keys to navigate. Tool to draw a line on the graph. Tool to plot one or more points on the graph by entering their coordinates.

Solution

This appears to be a homework problem focused on linear regression, where you're analyzing the relationship between high temperature (in °C) and the amount of hot chocolate sold (in liters). Let's walk through the process of solving it based on the data you gather from the sample.

Steps to solve:

  1. Gather the data: Use the "Take Sample" button to gather a random sample of data points that include temperature (x), hot chocolate sales (y), and the product of x and y (xy).

  2. Calculate the summary statistics: You need the following values:

    • nn = sample size (9 days in this case).
    • xx = the sum of all temperatures.
    • yy = the sum of all sales of hot chocolate.
    • sxs_x = standard deviation of the temperature values.
    • sys_y = standard deviation of the hot chocolate sales.
    • ∑xy\sum xy = the sum of the product of each temperature and corresponding hot chocolate sales.

  3. Compute the sample correlation coefficient rr using: r=n∑xy−(∑x)(∑y)(n∑x2−(∑x)2)(n∑y2−(∑y)2)r = \frac{n \sum xy - (\sum x)(\sum y)}{\sqrt{(n \sum x^2 - (\sum x)^2)(n \sum y^2 - (\sum y)^2)}} This will give you the correlation between temperature and hot chocolate sales.

  4. Calculate the slope b1b_1 and intercept b0b_0** of the regression line**:

    • Slope: b1=r(sysx)b_1 = r \left( \frac{s_y}{s_x} \right)
    • Intercept: b0=yˉ−b1xˉb_0 = \bar{y} - b_1 \bar{x} Where xˉ\bar{x} and yˉ\bar{y} are the means of the temperature and hot chocolate sales, respectively.

  5. Form the equation of the least-squares regression line in the form: y=b0+b1xy = b_0 + b_1 x This equation shows the predicted hot chocolate sales (y) based on the temperature (x).

  6. Predict the amount of hot chocolate sold when the temperature is 19°C by plugging x=19x = 19 into the regression equation.

If you can provide the data, I can help you with the calculations. If you're able to take the sample and input the necessary values, I'd be happy to help with the computations.

Relative Questions:

  1. What is the meaning of the correlation coefficient rr in this context?
  2. How does an increase in temperature affect hot chocolate sales based on the regression equation?
  3. How is the slope b1b_1 interpreted in the context of this problem?
  4. How do the standard deviations of xx and yy influence the regression equation?
  5. Why is it important to round your results to three decimal places in regression analysis?

Tip: Always check if your regression model fits the data well by analyzing the correlation coefficient rr. A strong correlation (close to 1 or -1) indicates a good fit.

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Math Problem Analysis

Mathematical Concepts

Statistics
Linear Regression
Correlation
Prediction

Formulas

Sample correlation coefficient formula: r = (nΣxy - ΣxΣy) / √[(nΣx² - (Σx)²)(nΣy² - (Σy)²)]
Slope of the regression line: b1 = r(sy/sx)
Y-intercept of the regression line: b0 = ȳ - b1x̄
Least-squares regression line: y = b0 + b1x

Theorems

Least-Squares Regression Theorem
Pearson Correlation Coefficient

Suitable Grade Level

College/University Level (Statistics)

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