A regional sales manager records data on the number of clients a salesperson contacts in a week (x) and the total sales generated that week (y). The data from 10 salespeople is shown below. Find the equation of the regression line and use it to predict sales if the salesperson contacts (a) 6 clients; (b) 40 clients
Table of contents
- 1. Intro to Stats and Collecting Data55m
- 2. Describing Data with Tables and Graphs1h 55m
- 3. Describing Data Numerically1h 45m
- 4. Probability2h 16m
- 5. Binomial Distribution & Discrete Random Variables2h 33m
- 6. Normal Distribution and Continuous Random Variables1h 38m
- 7. Sampling Distributions & Confidence Intervals: Mean1h 53m
- 8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- 9. Hypothesis Testing for One Sample2h 19m
- 10. Hypothesis Testing for Two Samples3h 22m
- 11. Correlation1h 6m
- 12. Regression1h 4m
- 13. Chi-Square Tests & Goodness of Fit1h 20m
- 14. ANOVA1h 0m
12. Regression
Linear Regression & Least Squares Method
Problem 9.2.7
Textbook Question
"In Exercises 7-12, match the description in the left column with its symbol(s) in the right column.
7. The y-value of a data point corresponding to x;
a. \hat{y}_i
b. y_i
c. b
d. (\bar{x}, \bar{y})
e. m
f. \bar{y}"

1
Step 1: Understand the problem. The question asks to match the description 'The y-value of a data point corresponding to x' with the correct symbol from the provided options.
Step 2: Recall the meaning of each symbol: (a) \( \hat{y}_i \) represents the predicted y-value for a given x-value in a regression model, (b) \( y_i \) represents the actual observed y-value for a given x-value, (c) \( b \) is often used to denote the y-intercept in a regression equation, (d) \( (\bar{x}, \bar{y}) \) represents the mean of x and y values, (e) \( m \) is the slope of the regression line, and (f) \( \bar{y} \) is the mean of the y-values.
Step 3: Focus on the description 'The y-value of a data point corresponding to x.' This refers to the actual observed y-value for a specific x-value in the dataset.
Step 4: Match the description to the correct symbol. Based on the definitions, the symbol \( y_i \) (option b) represents the actual observed y-value for a given x-value.
Step 5: Conclude that the correct match for the description 'The y-value of a data point corresponding to x' is option (b) \( y_i \).

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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Dependent Variable
In statistics, the dependent variable is the outcome or response that is measured in an experiment or study. It is often denoted as 'y' and represents the value that changes in response to the independent variable. In the context of the question, the y-value of a data point corresponding to x refers to the specific value of the dependent variable for a given independent variable.
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Data Point Notation
Data points in statistics are often represented using specific notations. For instance, 'y_i' denotes the actual observed value of the dependent variable for the ith observation, while 'ar{y}' represents the mean of all observed y-values. Understanding these notations is crucial for interpreting data and matching them with their corresponding descriptions.
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Regression Coefficients
In regression analysis, coefficients such as 'b' represent the slope of the regression line, indicating the relationship between the independent and dependent variables. The symbol 'ar{y}' is used to denote the average of the dependent variable, while 'ar{x}' represents the average of the independent variable. Recognizing these symbols helps in understanding how data points relate to the overall trend in the dataset.
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