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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

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

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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.

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

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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.

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

This is a graph of an equation. The x-axis is labeled in intervals of 1 from -5 to 5; the y-axis is labeled in intervals of 1 from 0 - 8. The equation's graph is a parabola, a u-shaped curve that has a minimum value at (0, 0).
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[link] contains real data for the first two decades of AIDS reporting.

Adults and adolescents only, united states
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.

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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?

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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.

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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?

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

12,000 pounds of soil

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What is the y -intercept? Interpret its meaning.

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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.

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

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Questions & Answers

mean is number that occurs frequently in a giving data
Chinedu Reply
That places the mode and the mean as the same thing. I'd define the mean as the ratio of the total sum of variables to the variable count, and it assigns the variables a similar value across the board.
Samsicker
what is mean
John Reply
what is normal distribution
RAHAT Reply
What is the uses of sample in real life
Waqas Reply
change of origin and scale
RAHAT Reply
3. If the grades of 40000 students in a course at the Hashemite University are distributed according to N(60,400) Then the number of students with grades less than 75 =*
Ahmad Reply
If a constant value is added to every observation of data, then arithmetic mean is obtained by
Madiha Reply
sum of AM+Constnt
Fazal
data can be defined as numbers in context. suppose you are given the following set of numbers 18,22,22,20,19,21
Tyasia Reply
what are data
Tyasia Reply
what is mode?
Natasha Reply
what is statistics
Natasha
statistics is a combination of collect data summraize data analyiz data and interprete data
Ali
what is mode
Natasha
what is statistics
Alex Reply
It is the science of analysing numerical data in large quantities, especially for the purpose of inferring proportions in a whole from those in a representative sample.
Bernice
history of statistics
Terseer Reply
statistics was first used by?
Terseer
if a population has a prevalence of Hypertension 5%, what is the probability of 4 people having hypertension from 8 randomly selected individuals?
John Reply
Carpet land sales persons average 8000 per weekend sales Steve qantas the firm's vice president proposes a compensation plan with new selling incentives Steve hopes that the results of a trial selling period will enable him to conclude that the compensation plan increases the average sales per sales
lorenda Reply
Supposed we have Standard deviation 1.56, mean 6.36, sample size 25 and Z-score 1.96 at 95% confidence level, what is the confidence interval?
John Reply

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Source:  OpenStax, Introductory statistics. OpenStax CNX. May 06, 2016 Download for free at http://legacy.cnx.org/content/col11562/1.18
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