# 12.1 Linear equations  (Page 2/4)

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## References

Data from the Centers for Disease Control and Prevention.

Data from the National Center for HIV, STD, and TB Prevention.

## Chapter review

The most basic type of association is a linear association. This type of relationship can be defined algebraically by the equations used, numerically with actual or predicted data values, or graphically from a plotted curve. (Lines are classified as straight curves.) Algebraically, a linear equation typically takes the form y = mx + b , where m and b are constants, x is the independent variable, y is the dependent variable. In a statistical context, a linear equation is written in the form y = a + bx , where a and b are the constants. This form is used to help readers distinguish the statistical context from the algebraic context. In the equation y = a + bx , the constant b that multiplies the x variable ( b is called a coefficient) is called as the slope . The slope describes the rate of change between the independent and dependent variables; in other words, the rate of change describes the change that occurs in the dependent variable as the independent variable is changed. In the equation y = a + bx , the constant a is called as the y -intercept. Graphically, the y -intercept is the y coordinate of the point where the graph of the line crosses the y axis. At this point x = 0.

The slope of a line is a value that describes the rate of change between the independent and dependent variables. The slope tells us how the dependent variable ( y ) changes for every one unit increase in the independent ( x ) variable, on average. The y -intercept is used to describe the dependent variable when the independent variable equals zero. Graphically, the slope is represented by three line types in elementary statistics.

## Formula review

y = a + bx where a is the y -intercept and b is the slope. The variable x is the independent variable and y is the dependent variable.

Use the following information to answer the next three exercises . A vacation resort rents SCUBA equipment to certified divers. The resort charges an up-front fee of \$25 and another fee of \$12.50 an hour.

What are the dependent and independent variables?

dependent variable: fee amount; independent variable: time

Find the equation that expresses the total fee in terms of the number of hours the equipment is rented.

Graph the equation from [link] .

Use the following information to answer the next two exercises . A credit card company charges \$10 when a payment is late, and \$5 a day each day the payment remains unpaid.

Find the equation that expresses the total fee in terms of the number of days the payment is late.

Graph the equation from [link] .

Is the equation y = 10 + 5 x – 3 x 2 linear? Why or why not?

Which of the following equations are linear?

a. y = 6 x + 8

b. y + 7 = 3 x

c. y x = 8 x 2

d. 4 y = 8

y = 6 x + 8, 4 y = 8, and y + 7 = 3 x are all linear equations.

Does the graph show a linear equation? Why or why not?

[link] contains real data for the first two decades of AIDS reporting.

 Year # AIDS cases diagnosed # AIDS deaths Pre-1981 91 29 1981 319 121 1982 1,170 453 1983 3,076 1,482 1984 6,240 3,466 1985 11,776 6,878 1986 19,032 11,987 1987 28,564 16,162 1988 35,447 20,868 1989 42,674 27,591 1990 48,634 31,335 1991 59,660 36,560 1992 78,530 41,055 1993 78,834 44,730 1994 71,874 49,095 1995 68,505 49,456 1996 59,347 38,510 1997 47,149 20,736 1998 38,393 19,005 1999 25,174 18,454 2000 25,522 17,347 2001 25,643 17,402 2002 26,464 16,371 Total 802,118 489,093

Use the columns "year" and "# AIDS cases diagnosed. Why is “year” the independent variable and “# AIDS cases diagnosed.” the dependent variable (instead of the reverse)?

The number of AIDS cases depends on the year. Therefore, year becomes the independent variable and the number of AIDS cases is the dependent variable.

Use the following information to answer the next two exercises . A specialty cleaning company charges an equipment fee and an hourly labor fee. A linear equation that expresses the total amount of the fee the company charges for each session is y = 50 + 100 x .

What are the independent and dependent variables?

What is the y -intercept and what is the slope? Interpret them using complete sentences.

The y -intercept is 50 ( a = 50). At the start of the cleaning, the company charges a one-time fee of \$50 (this is when x = 0). The slope is 100 ( b = 100). For each session, the company charges \$100 for each hour they clean.

Use the following information to answer the next three questions . Due to erosion, a river shoreline is losing several thousand pounds of soil each year. A linear equation that expresses the total amount of soil lost per year is y = 12,000 x .

What are the independent and dependent variables?

How many pounds of soil does the shoreline lose in a year?

12,000 pounds of soil

What is the y -intercept? Interpret its meaning.

Use the following information to answer the next two exercises . The price of a single issue of stock can fluctuate throughout the day. A linear equation that represents the price of stock for Shipment Express is y = 15 – 1.5 x where x is the number of hours passed in an eight-hour day of trading.

What are the slope and y -intercept? Interpret their meaning.

The slope is –1.5 ( b = –1.5). This means the stock is losing value at a rate of \$1.50 per hour. The y -intercept is \$15 ( a = 15). This means the price of stock before the trading day was \$15.

If you owned this stock, would you want a positive or negative slope? Why?

Example of discrete variable
Gbenga
hi
Kachalla
what's up here ... am new here
Kachalla
sorry question a bit unclear...do you mean how do you analyze quantitative data? If yes, it depends on the specific question(s) you set in the beginning as well as on the data you collected. So the method of data analysis will be dependent on the data collecter and questions asked.
Bheka
how to solve for degree of freedom
saliou
Quantitative data is the data in numeric form. For eg: Income of persons asked is 10,000. This data is quantitative data on the other hand data collected for either make or female is qualitative data.
Rohan
*male
Rohan
Degree of freedom is the unconditionality. For example if you have total number of observations n, and you have to calculate variance, obviously you will need mean for that. Here mean is a condition, without which you cannot calculate variance. Therefore degree of freedom for variance will be n-1.
Rohan
data that is best presented in categories like haircolor, food taste (good, bad, fair, terrible) constitutes qualitative data
Bheka
vegetation types (grasslands, forests etc) qualitative data
Bheka
I don't understand how you solved it can you teach me
solve what?
Ambo
What is the end points of a confidence interval called?
lower and upper endpoints
Bheka
Class members write down the average time (in hours, to the nearest half-hour) they sleep per night.
how we make a classes of this(170.3,173.9,171.3,182.3,177.3,178.3,174.175.3)
Sarbaz
6.5
phoenix
11
Shakir
7.5
Ron
why is always lower class bundry used
Caleb
Assume you are in a class where quizzes are 20% of your grade, homework is 20%, exam _1 is 15%,exam _2 is 15%, and the final exam is 20%.Suppose you are in the fifth week and you just found out that you scored a 58/63 on the fist exam. You also know that you received 6/9,8/10,9/9 on the first
quizzes as well as a 9/11,10/10,and 4.5/7 on the first three homework assignment. what is your current grade in the course?
Diamatu
Abdul
if putting y=3x examine that correlation coefficient between x and y=3x is 1.
what is permutation
how to construct a histogram
You have to plot the class midpoint and the frequency
Wydny
ok so you use those two to draw the histogram right.
Amford
yes
Wydny
ok can i be a friend so you can be teaching me small small
Amford
how do you calculate cost effectiveness?
George
Hi everyone, this is a very good statistical group and am glad to be part of it. I'm just not sure how did I end up here cos this discussion just popes on my screen so if I wanna ask something in the future, how will I find you?
Bheka
To make a histogram, follow these steps: On the vertical axis, place frequencies. Label this axis "Frequency". On the horizontal axis, place the lower value of each interval. ... Draw a bar extending from the lower value of each interval to the lower value of the next interval.
Divya
I really appreciate that
I want to test linear regression data such as maintenance fees vs house size. Can I use R square, F test to test the relationship? Is the good condition of R square greater than 0.5
yes of course must have use f test and also use t test individually multple coefficients
rishi
Alright
umar
hi frnd I'm akeem by name, I wanna study economics and statistics wat ar d thing I must do to b a great economist
akeem
Is R square cannot analysis linear regression of X vs Y relationship?
Mok
To be an economist you have to be professional in maths
umar
hi frnds
Shehu
what is random sampling what is sample error
@Nistha Kashyap Random sampling is the selection of random items (or random numbers) from the group. A sample error occurs when the selected samples do not truely represent the whole group. The can happen when most or all of the selected samples are taken from only one section of the group;
Ron
Thus the sample is not truely random.
Ron
What is zero sum game?
A game in which there is no profit & no loss to any of the both player.
Milan
Differences between sample mean & population mean
***keydifferences.com/difference-between-sample-mean-and-population-mean.html
Lucien
Not difference in the formula except the notation, sample mean is denoted by x bar and population mean is denoted by mu symbol. There is formula as well as notation between difference variance and standard deviations
Akash
Likely the difference would be in the result, unless the sample is an exact representation of the population (which is unlikely.)
Ron
what is data
Nii
Nii Avin - Data is just a simple way to refer to the numbers in the population, or in the sample used in your calculations.
Ron
what are the types of data
Nii
Data is the very pale android from the Star Trek Enterprise
Andrew
Am Emmanuel from Nigeria
Emmanuel
Am Qudus from Nigeria
Rasak
am Handson from Cameroon
Handson
what is a mode?
Handson
Nii - data is whatever you are sampling. Such as the number of students in each classroom.
Ron
Handson Ndintek - the mode is the number appearing most frequently. Example: 7 9 11 7 4 6 3 7 2. 7 is the mode. In a group such as 7 9 1 4 6 3, there is no mode because no number appears more often than any other.
Ron
hi I want to know how to find class boundary
Baalisi
give me the two types of data
qualitative and quantitative
phoenix
primary and secondary data
Peace
qualitative and quantitative
Prince