# 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?

Describe the organizational setup of bureau of statics government of sindh give its past achievement in brief
Life Table and its construction
Awais
only asking what is hospital statistics
What's the meaning Of statistic
The mean is the average of the numbers. It is easy to calculate: add up all the numbers, then divide by how many numbers there are. In other words it is the sum divided by the count.
pls I want to know the characterize of mean
I want solution to; x,x+1,3.5,3.4 and 2.5 is 3.1. find the value of x
Trixie
mean is nothing but average.
NAMDEV
(x+x+1+3.5+3.4+2.5)/5=3.1 2x+10.4=3.1*5 x=(15.5-10.4)/2 x=2.55
NAMDEV
How. Pls explain or break it down thanks
Icecool
the sum of all five divided by the 5 which is the average is equal to 3.1 so you sum them, divide and equate to 3.1. then solve for X by making X the subject
Pascale
Think of it as your grade, or gpa. how do you find the average of it?
La
great
Rushikesh
nicely explained
Rushikesh
What is the meaning of statistics and it's relevance in the social sciences
statistics is the branch of mathematics that using math& statistics theories to interpret data to information which is supporting the decision makers in social sciences and applied too. statistics is decision making science now aday
Ishag
I want to slove this math
I'm not seeing the Math problem?
Debra
the data below where from 1 miles. miles 1:8380, 8210,3836,7840,7910,7510,7720,7750,8100,7690 test wherever their difference is significance at 5% level with population of #8000.
Enny
1*1
Enny
me too
New
enny i don't see any math problems
New
hello my friend. how can you help me to solve math
ott
help me the interquartile range and semi interquartile range of grouped data
Ishraaq
what is the difference between macro and micro data?
Macrodata consist of 'headcounts', Microdata contain data on individuals,
Cherry
Mukhtar
macro means whole or big and micro means part or small therefore in relation to data, macro data will be big or whole data and micro data will be small or individual part of a data. for example taking the whole data of an organisation or economy that will be macro data and taking the data on employe
Pascale
employees will be micro data
Pascale
How to determine appropriate statistics tool for my data
hi
Juliana
hi
Emy
You cannot know everything! Statistics is a vast subject so don’t be afraid to say that you don’t know. Ask others for help or look up information on the internet to help. Don’t assume that the student knows anything about the technique they are suggesting!
Kalu
well
Tahir
my class starts at February 4th
Elaine
P is the probability that a given die shows even number. To test H0 : P =1/2 against H1 : P=1/3, following procedure is adopted. Toss the die twice and accept H0 if both times it shows even number. Find the probabilities of type I and type II errors
what is statistics?
what is data
natnael
It Is the collection, collation,Analysing Interpretation,and presentation of data....
Tosayo
statistics is a science that studies numerical values assigned to objects, attributes or variables in order to characterize and make the variables more meaningful.
Okosa
statistics is the science of collection, analysis, Interpretation & presentation of numerical data
Kishan
Numeric data
Tosayo
if you think about normal number and operations of math such as fractions and percent values, statisitics is when it comes down to it a lot of just that - counting and then fractions, then that data and those counts (and ratios)/fractions that were used or counted in one group will represent a pop.
Jesus
An example is, lets say you are in charge of making large orders of supplies and that based on your need of X bricks = X feet of building - now thats the math, well stats is using that math in a way the next person making orders can base those orders of what they will need based on saying "in this
Jesus
..case, when Bilal needed 75 feet of building walls that 150 bricks were used" - now, that helps to get one job done but if we then watch those jobs one at a time, and count that 5/5 times the count was 150 bricks total - the new person could order 150 bricks with a high probability of being right
Jesus
If X is a Uniform random variable in [ -2, 2 ], find the pdf of Y X  and E Y[ ].
could you resend the question it is not clear after " the PDF of---" ?
Anji
ap kis year ki student ho
Shahjee
Mastrs ist year
Anaya
I'm from 2nd year BHU
abhishek
ap kis year ki student ho
Shahjee
I'm student of bs(hons) 1st year
bilal
tell me good definition of statistics?
bilal
I want to know statistics
why is data so important in statistics
want summary statistic on gender, age group, weight, and weight loss
Trixie
are you asking question or looking for Solution
Arun
solution pls
Trixie
1st convert gender and group to factor than use summary function It will give mean median and mode with other details
Arun
its a bit complicated could u bring it to my level of under standing
Trixie
u know the question was put in a tabular form where we were to find the variable type, summary statistics and graph type of the given variables that's the gender,age group, weight and weight loss
Trixie
if you see, gender and group are not numerical due to which they will not give you correct statistics
Arun
how you denote gender m or f
Arun
or t
Arun
ok tnk u
Trixie
these are not numerical so you have to convert they as f=1; m=2; t=3 same thing you have to do with group or any variable which is character else you should drop them
Arun
Arun
oh OK tnk u
Trixie
so pls why is data important in statistics
Trixie
not only data correct data is imp
Arun
statistics works on data only
Arun
without data you can not summarize, can not predict future, can not establish relationship between two and more variables, can not prepare reports and make decisions on it
Arun
so I'll give example. suppose you want to open a restaurant and you have to choose one best location out of 5. then how you will decide which location is best for you
Arun
awww thank you pls
Trixie
pls I want a brief note on observation, survey and experimentation way of obtaining data
Trixie
hi
Abdiwahab
please Tell me difference parameters and non parameter
Abdiwahab
can you tell me about the scopes of statistics?
Minhal
Minhal
Methods of Collecting Data Observation Observational studies allow researchers to document behavior in a natural setting and witness events that could not be produced in a lab.
Arun
Key Points Observation differs from most other forms of data collection in that the researcher does not manipulate variables or directly question participants. The advantages of observation include observing natural behavior, refining hypotheses, and allowing for observation of behavior that canno
Arun
be produced in an artificial environment for ethical or practical reasons. The disadvantages of observation are that these studies do not produce quantitative data, do not allow for cause and effect statements, may be very time consuming, and can be prone to researcher bias.
Arun
Key Terms observational research: Research focusing on the observation of behavior outside of a laboratory setting. external validity: In research, whether or not study findings can be generalized to real world scenarios.
Arun
Surveys and Interviews Surveys are a low-cost option for gathering a large amount of data, but they are also susceptible to reporting bias.
Arun
Key Points The survey method of data collection is likely the most common of the four major research methods. The benefits of this method include low cost, large sample size, and efficiency.
Arun
The major problem with this method is accuracy: since surveys depend on subjects’ motivation, honesty, memory, and ability to respond, they are very susceptible to bias. A researcher must have a strong understanding of how to properly frame survey questions in order to gather reliable and relevant
Arun
Key Terms reliability: The degree to which a measure is likely to yield consistent results each time it is used. validity: The degree to which a measure is actually assessing the concept it was designed to measure. survey: A method for collecting qualitative and quantitative information about ind
Arun
individuals in a population.
Arun
Interviews Interviews are a type of qualitative data in which the researcher asks questions to elicit facts or statements from the interviewee. Interviews used for research can take several forms:
Arun
Informal Interview: A more conversational type of interview, no questions are asked and the interviewee is allowed to talk freely. General interview guide approach: Ensures that the same general areas of information are collected from each interviewee. Provides more focus than the conversational ap
Arun
approach, but still allows a degree of freedom and adaptability in getting the information from the interviewee. Standardized, open-ended interview: The same open-ended questions are asked to all interviewees. This approach facilitates faster interviews that can be more easily analyzed and compared
Arun
Closed, fixed-response interview (Structured): All interviewees are asked the same questions and asked to choose answers from among the same set of alternatives.
Arun
experiments An experiment involves the creation of a contrived situation in order that the researcher can manipulate one or more variables whilst controlling all of the others and measuring the resultant effects.
Arun
Boyd and Westfall1 have defined experimentation as: "...that research process in which one or more variables are manipulated under conditions which permit the collection of data which show the effects, if any, in unconfused fashion."
Arun
Experiments can be conducted either in the field or in a laboratory setting. When operating within a laboratory environment, the researcher has direct control over most, if not all, of the variables that could impact upon the outcome of the experiment
Arun
When experiments are conducted within a natural setting then they are termed field experiments. The variety test carried out by United Fruits on their Gros Michel and Valery bananas is an example of a field experiment.
Arun
parameter Parameters are factors or limits which affect the way that something can be done or made
Arun
Arun
pls can u use mean n mode at the statistical summary pls
Trixie
yes, statistical summary itself gives all value
Arun
but u didn't tell me the advantage and disadvantage of the experimental method
Trixie
but you didn't tell me the advantage and disadvantage of experimental method
Trixie
tell me to make a hosrogram of dicrete values
Husnain
histogram
Husnain
noted!
Abdullahi
by using a sampling distribution? how to estimate the population mean using a ramdom variable n?
what is the important of statistics
Shamsuddeen