# 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?

a random variable is __________ if takes on a values continuous scale.
merits of arithmetic mean
formula of HM
Bikram
hello
umar
hi
Candyda
n/sum(1/xi). this i is in subscript
moniba
this is HM formula
moniba
Thank you
Candyda
So guys u can help me get better in statistics
Lion
class interval and class limit are same ?
what is a differnece between class interval and class boundries ?
Shantal
I don't think so
Plain
interval between the element of class- class interval
Hemanta
class boundary consist of the upper and lower class limits but class class limit is the same decimal values as the data values
Najeeb
x 0 1 2 3 4 5 0.05 0.09 0.17 0.27 0.23 0.19 a) Find the probability that 1 or 2 smart phones have defects (show calculations). b) Find the probability that more than 3 smart phones have defects (show calculations). c) Find the probability that at most 2 smart phones have defects (show calculat
Please expand on the number set data
Jesus
what is proportion?
A proportion is simply a statement that two ratios are equal
Foeor
Two dice are thrown. Let A be the event that the sum of the upper face number is odd and be the event of at least one ace
does proportion alwaus give u a yes or no answer for data
frequency destribution
7.The following data give thenumber of car thefts that occurred in a city in the past 12 days. 63711438726915 Calculate therange, variance, and standard deviation.
express the confidence interval 81.4% ~8.5% in interval form
a bad contain 3 red and 5 black balls another 4 red and 7 black balls, A ball is drawn from a bag selected at random, Find the probability that A is red?
The information is given as, 30% of customers shopping at SHOPNO will switch to DAILY SHOPPING every month on the other hand 40% of customers shopping at DAILY SHOPPING will switch to other every month. What is the probability that customers will switch from A to B for next two months?
Calculate correlation coefficient, where SP(xy) = 144; SS(x) = 739; SS(y) = 58. (2 Points)
The information are given from a randomly selected sample of age of COVID-19 patients who have already survived. These information are collected from 200 persons. The summarized information are as, n= 20; ∑x = 490; s^2 = 40. Calculate 95% confident interval of mean age.
Ashfat
The mode of the density of power of signal is 3.5. Find the probability that the density of a random signal will be more than 2.5.
Ashfat
The average time needed to repair a mobile phone set is 2 hours. If a customer is in queue for half an hour, what is the probability that his set will be repaired within 1.6 hours?
Ashfat
A quality control specialist took a random sample of n = 10 pieces of gum and measured their thickness and found the mean 9 and variance 0.04. Do you think that the mean thickness of the spearmint gum it produces is 8.4
3. The following are the number of mails received in different days by different organizations: Days (x) : 23, 35, 38, 50, 34, 60, 41, 32, 53, 67. Number of mails (y) : 18, 40, 52, 45, 32, 55, 50, 48, 26, 25. i) Fit a regression line of y on x and test the significance of regression. ii) Estimate y