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Chapter 3: Problem 18

Male crickets chirp by rubbing their front wings together, and their chirping is temperature dependent. Crickets chirp faster with increasing temperature and slower with decreasing temperatures. The table below shows the number of chirps per second for a cricket, recorded at 10 different temperatures. $$ \begin{array}{l|rrrrrrrrrr} \text { Chirps per second } & 20 & 16 & 19 & 18 & 18 & 16 & 14 & 17 & 15 & 16 \\\ \hline \text { Temperature } & 88 & 73 & 91 & 85 & 82 & 75 & 69 & 82 & 69 & 83 \end{array} $$ a. Which of the two variables (temperature and number of chirps) is the independent variable, and which is the dependent variable? b. Plot the data using a scatterplot. How would you describe the relationship between temperature and number of chirps? c. Find the least-squares line relating the number of chirps to the temperature. d. If a cricket is monitored at a temperature of 80 degrees, what would you predict his number of chirps would be?

Short Answer

Expert verified
Answer: The relationship between temperature and cricket chirps is positive, meaning that as the temperature increases, the number of chirps per second also increases. Based on the linear regression equation, we can predict that a cricket at 80 degrees would chirp approximately 15.62 times per second.

Step by step solution

01

Identifying the independent and dependent variables

In this problem, temperature affects the number of chirps a cricket makes, so the independent variable is Temperature, and the dependent variable is Chirps per second.
02

Describing the relationship between the variables with a scatterplot

By plotting the data on a graph, we can observe that as the temperature increases, so does the number of chirps per second, and vice versa. This indicates a positive relationship between temperature (independent variable) and chirps per second (dependent variable).
03

Calculating the least-squares line (linear regression)

In this step, we will find the linear regression equation for our dataset. The equation of the linear regression line can be written as: $$ y = mx + b $$ where 'y' represents the dependent variable (Chirps per second), 'x' represents the independent variable (Temperature), 'm' is the slope of the line, and 'b' is the y-intercept. We can use the method of least squares to find the slope 'm' and y-intercept 'b'. We can make use of these formulas to calculate the slope and the y-intercept: $$ m = \frac{n\sum(xy) - \sum x \sum y}{n(\sum x^2) - (\sum x)^2} $$ $$ b = \frac{\sum y - m\sum x}{n} $$ Applying these formulas to our dataset, we have: $$ m = \frac{10\cdot 11644 - 743\cdot 177}{10\cdot7153 - 743^2} \approx 0.202 $$ $$ b = \frac{177 - 0.202\cdot 743}{10} \approx -0.982 $$ Therefore, the linear regression equation is: $$ y = 0.202x - 0.982 $$
04

Predicting the number of chirps based on a given temperature

Now that we have our linear regression line, we can use it to predict the number of chirps at a specific temperature (80 degrees in this case). By substituting the temperature value (x) into the equation, we have: $$ y = 0.202\cdot 80 - 0.982 \approx 15.62 $$ If a cricket is monitored at a temperature of 80 degrees, we would predict his number of chirps would be approximately 15.62 chirps per second.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Independent Variable
In linear regression, understanding the role of variables is crucial. The **independent variable** is the one that is believed to influence or predict changes in another variable. In our crickets' chirping study, the temperature is the independent variable.

This means it's the variable we choose to study its impact on the cricket's chirping behavior. Think of it as the cause or input variable. When you adjust the temperature, the number of chirps changes.
  • Temperature can be measured or controlled in an experiment.
  • It's often plotted on the x-axis in graphs.
  • Other examples include time, distance, and speed in different scenarios.
Dependent Variable
The **dependent variable** is the outcome or effect that is observed in relation to the independent variable. It 'depends' on the alterations made to the independent variable. In our cricket example, the chirps per second are dependent on the temperature.

This variable is what researchers measure in an experiment or observational study. As temperature changes, it influences the frequency of cricket chirping.
  • This variable is plotted on the y-axis in graphs.
  • It can also be thought of as the response variable.
  • Any variations in this variable are due to changes in the independent variable.
Scatterplot
A **scatterplot** is a type of graph used to visualize the relationship between two numerical variables. In our exercise, it's used to depict the relationship between temperature and chirping rate in crickets.

Creating a scatterplot involves plotting each data point from the dataset on a graph, using coordinates that correspond to the two variables.
  • Points often form a cloud, and the pattern can indicate a relationship.
  • An upward trend indicates a positive correlation.
  • Helps in identifying outliers or unusual data points.
Overall, a scatterplot for our cricket data shows that as temperature increases, the chirping rate tends to go up, suggesting a positive correlation.
Least-Squares Method
The **least-squares method** is a mathematical approach to find the line that best fits a set of data points. This method minimizes the sum of the squares of the residuals, which are the distances from each point to the line.

In our cricket example, the goal is to find a line that best matches the trend of crickets' chirp rates as the temperature varies. The formula for the line of best fit is often written as \( y = mx + b \). Here, 'y' is the dependent variable, 'x' is the independent variable, 'm' is the slope, and 'b' is the y-intercept.
  • The slope (m) indicates how much the dependent variable changes for a one-unit change in the independent variable.
  • The y-intercept (b) represents the value of the dependent variable when the independent variable is zero.
  • Helps in making predictions based on the regression equation.
In the context of our study, this method provides a reliable tool for predicting how frequent a cricket will chirp at various temperatures.

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Most popular questions from this chapter

A local chamber of commerce surveyed 126 households within its city and recorded the type of residence and the number of family members in each of the households. The data are shown in the table. $$ \begin{array}{lccc} &&{\text { Type of Residence }} \\ \hline \text { Family Members } & \text { Apartment } & \text { Duplex } & \text { Single Residence } \\ \hline 1 & 8 & 10 & 2 \\ 2 & 15 & 4 & 14 \\ 3 & 9 & 5 & 24 \\ 4 \text { or more } & 6 & 1 & 28 \end{array} $$ a. Use a side-by-side bar chart to compare the number of family members living in each of the three types of residences. b. Use a stacked bar chart to compare the number of family members living in each of the three types of residences. c. What conclusions can you draw using the graphs in parts a and \(b\) ?

Consider this set of bivariate data: $$ \begin{array}{l|llllll} x & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline y & 5.6 & 4.6 & 4.5 & 3.7 & 3.2 & 2.7 \end{array} $$ a. Draw a scatterplot to describe the data. b. Does there appear to be a relationship between \(x\) and \(y\) ? If so, how do you describe it? c. Calculate the correlation coefficient, \(r\). Does the value of \(r\) confirm your conclusions in part b? Explain.

Professor Isaac Asimov was one of the most prolific writers of all time. He wrote nearly 500 books during a 40 -year career prior to his death in \(1992 .\) In fact, as his career progressed, he became even more productive in terms of the number of books written within a given period of time. \(^{8}\) These data are the times (in months) required to write his books, in increments of 100: $$ \begin{array}{l|ccccc} \text { Number of Books } & 100 & 200 & 300 & 400 & 490 \\ \hline \text { Time (in months) } & 237 & 350 & 419 & 465 & 507 \end{array} $$ a. Plot the accumulated number of books as a function of time using a scatterplot. b. Describe the productivity of Professor Asimov in light of the data set graphed in part a. Does the relationship between the two variables seem to be linear?

The demand for healthy foods that are low in fats and calories has resulted in a large number of "low-fat'" and "fat-free" products at the supermarket. The table shows the numbers of calories and the amounts of sodium (in milligrams) per slice for five different brands of fat-free American cheese. $$ \begin{array}{lcc} \text { Brand } & \text { Sodium (mg) } & \text { Calories } \\ \hline \text { Kraft Fat Free Singles } & 300 & 30 \\ \text { Ralphs Fat Free Singles } & 300 & 30 \\ \text { Borden Fat Free } & 320 & 30 \\ \text { Healthy Choice Fat Free } & 290 & 30 \\ \text { Smart Beat American } & 180 & 25 \end{array} $$ a. Draw a scatterplot to describe the relationship between the amount of sodium and the number of calories. b. Describe the plot in part a. Do you see any outliers? Do the rest of the points seem to form a pattern? c. Based only on the relationship between sodium and calories, can you make a clear decision about which of the five brands to buy? Is it reasonable to base your choice on only these two variables? What other variables should you consider?

3.11 A set of bivariate data consists of these measurements on two variables, \(x\) and \(y:\) \(\begin{array}{llllll}(3,6) & (5,8) & (2,6) & (1,4) & (4,7) & (4,6)\end{array}\) a. Draw a scatterplot to describe the data. b. Does there appear to be a relationship between \(x\) and \(y\) ? If so, how do you describe it? c. Calculate the correlation coefficient, \(r\), using the computing formula given in this section. d. Find the best-fitting line using the computing formulas. Graph the line on the scatterplot from part a. Does the line pass through the middle of the points?
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