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Bivariate Data Analysis for Common Core Algebra 1 Homework Answers
If you are taking Common Core Algebra 1, you may encounter some homework problems that involve bivariate data analysis. Bivariate data analysis is the study of the relationship between two variables, such as height and weight, or temperature and ice cream sales. In this article, we will explain what bivariate data analysis is, how to create and interpret scatter plots, how to find and use best fit lines, and how to understand correlation and causation.
What is Bivariate Data Analysis?
Bivariate data analysis is a branch of statistics that deals with two variables, usually denoted by x and y. A variable is a quantity that can change or vary, such as age, height, weight, income, etc. Bivariate data analysis aims to explore how the two variables are related to each other, and whether there is a pattern or a trend in their values.
One way to analyze bivariate data is to create a scatter plot, which is a graph that shows the values of the two variables as points on a coordinate plane. The x-variable is usually plotted on the horizontal axis, and the y-variable is usually plotted on the vertical axis. For example, here is a scatter plot that shows the relationship between height (in inches) and weight (in pounds) for 10 students:
A scatter plot can help us visualize the relationship between the two variables, and see if there is a pattern or a trend in the data. For example, we can see from the scatter plot above that there seems to be a positive relationship between height and weight: as height increases, weight also tends to increase.
How to Find and Use Best Fit Lines?
A best fit line is a line that best represents the trend or pattern in a scatter plot. It is also called a line of best fit or a regression line. A best fit line can help us make predictions or estimates based on the bivariate data. For example, we can use the best fit line to estimate the weight of a student who is 70 inches tall, or the height of a student who weighs 150 pounds.
There are different methods to find a best fit line for a scatter plot, such as using a ruler or a graphing calculator. One common method is called the least squares method, which minimizes the sum of the squared errors between the actual y-values and the predicted y-values from the line. The equation of the best fit line has the form y = mx + b, where m is the slope and b is the y-intercept.
For example, here is a possible best fit line for the scatter plot of height vs weight:
The equation of this best fit line is y = 4.8x – 259.6. We can use this equation to make predictions or estimates based on the bivariate data. For example, we can estimate that a student who is 70 inches tall would weigh about 4.8(70) – 259.6 = 76.4 pounds. Or we can estimate that a student who weighs 150 pounds would be about (150 + 259.6)/4.8 = 85.3 inches tall.
How to Understand Correlation and Causation?
Correlation and causation are two important concepts in bivariate data analysis. Correlation measures how strong or weak the relationship between two variables is. Causation implies that one variable causes or affects another variable.
Correlation can be positive or negative, depending on whether the two variables tend to move in the same direction or in opposite directions. For example, height and weight have a positive correlation: as height increases, weight also tends to increase. On the other hand, temperature and ice cream sales have a negative correlation: as temperature increases, ice cream sales tend to decrease.
Correlation can also be measured by a number called the correlation
To calculate the correlation coefficient, we can use different formulas depending on the type of data we have. One common formula is called Pearson’s r, which is given by:
where x and y are the two variables, x̄ and ȳ are their means, and sx and sy are their standard deviations. The summation sign Σ means to add up all the products of (x – x̄) and (y – ȳ) for each pair of data points, and then divide by n – 1, where n is the sample size.
For example, if we want to calculate the correlation coefficient between height and weight for the 10 students, we can use a table like this:
| x (height in inches) |
y (weight in pounds) |
x – x̄ |
y – ȳ |
(x – x̄)(y – ȳ) |
| 64 |
120 |
-3.6 |
-9.4 |
33.84 |
| 66 |
130 |
-1.6 |
0.6 |
-0.96 |
| 68 |
140 |
0.4 |
10.6 |
4.24 |
| 70 |
150 |
2.4 |
20.6 |
49.44 |
| 72 |
160 |
4.4 |
30.6 |
134.64 |
| 74 |
170 |
6.4 |
40.6 |
259.84 |
| 76 |
180 |
8.4 |
50.6 |
424.64 |
| 78 |
190 |
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60
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627
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84
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x̄ = 67
.
6
|
ȳ = 129
.
4
|
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∑ (x – x̄)(y – ȳ) = 1534
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32
|
To find the standard deviations of x and y, we can use another table like this:
| x (height in inches) |
x – x̄ |
(x – x̄)2 |
y (weight in pounds) |
y – ȳ |
(y – ȳ)2 |
| 64 </
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-3
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6
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12
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96
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120
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-9
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4
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88
.
36
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Causation, on the other hand, implies that one variable causes or affects another variable. For example, smoking causes lung cancer, or exercise causes weight loss. Causation is a stronger and more specific claim than correlation, and it requires more evidence to support it.
However, correlation does not necessarily mean causation. Just because two variables are correlated, it does not mean that one variable causes the other. There are several possible explanations for why two variables might be correlated without having a causal relationship:
- There might be a third variable that causes both variables. For example, ice cream sales and violent crime rates are closely correlated, but they are not causally linked with each other. Instead, hot temperatures, a third variable, affect both variables separately.
- There might be a reverse causality, meaning that the effect is actually the cause and vice versa. For example, low self-esteem and depression are correlated, but it is not clear whether low self-esteem causes depression, or whether depression causes low self-esteem.
- There might be a bidirectional causality, meaning that both variables cause and affect each other. For example, stress and insomnia are correlated, but it is not clear whether stress causes insomnia, or whether insomnia causes stress.
- There might be an indirect causality, meaning that one variable causes another variable that then causes another variable. For example, education and income are correlated, but it is not clear whether education causes income, or whether education causes skills that then cause income.
- There might be a spurious or coincidental correlation, meaning that the correlation is due to chance and has no meaningful relationship. For example, the number of people who drowned by falling into a pool correlates with the number of films Nicolas Cage appeared in, but this is obviously a random coincidence.
To establish causation, we need to use more rigorous methods than just observing correlations. We need to control for other possible factors that might affect the variables, and we need to test whether changing one variable actually leads to changes in another variable.
Causal research In a causal research design, you manipulate one variable to see its effect on another variable. You also control for other variables that might influence the outcome. The most common type of causal research design is an experiment, in which you randomly assign participants to different groups and expose them to different treatments or conditions.
Example: Causal research
You want to test whether vitamin D supplements can reduce depression symptoms. You recruit 100 participants who have been diagnosed with depression and randomly assign them to two groups: one group receives vitamin D pills and the other group receives placebo pills. You measure their depression levels before and after the treatment using a standardized questionnaire. You also control for other factors that might affect depression, such as age, gender, medication use, and sunlight exposure.
By comparing the depression scores of the two groups, you can infer whether vitamin D supplements have a causal effect on depression. If the vitamin D group shows a greater reduction in depression than the placebo group, you can conclude that vitamin D causes a decrease in depression. If there is no difference between the groups, you can conclude that vitamin D has no effect on depression.
Common Core Algebra 1 Homework Answers
If you are looking for homework answers for Common Core Algebra 1, you can find many online resources that provide solutions and explanations for various topics and exercises. Here are some examples of websites that offer free solutions for Algebra 1 Common Core:
- Quizlet provides step-by-step solutions and answers to Algebra 1 Common Core textbook problems, as well as flashcards and quizzes to help you study.
- eMathInstruction offers video lessons, homework sets, answer keys, and assessments for Unit 10 – Statistics of Algebra 1 Common Core.
- FlippedMath features video lessons, practice problems, answer keys, and additional resources for Unit 5 – Bivariate Data of Algebra 1 Common Core.
However, keep in mind that these websites are not official sources of Common Core Algebra 1 curriculum or standards. They may not cover all the topics or exercises that you need to learn or practice. They may also have errors or inaccuracies in their solutions or explanations. Therefore, it is always advisable to check your answers with your teacher or textbook, and to use these websites as supplementary tools rather than substitutes for learning.
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