Data Unit 2 Test

Question 1

Scatter Plots:

Answer 1

Graph used to determine if there is a relationship between two variables. The independent variable is on the horizontal axis and the dependent variable is on the vertical axis.

Question 2

Line of Best Fit (trend line)

Answer 2

A straight line that passes as close as possible to all the points in a scatter plot.

Question 3

The line is drawn with the following criteria in mind:

Answer 3

1. The line passes through as many points as possible.
2. There are evenly distributed points above and below the line (the sum of the perpendicular distances for all the points above the line should equal that of the points below it).
3. Ignore outliers, whenever possible.
4. Consider the origin as a possible point (e.g. extrapolate to time 0)

Question 4

Outliers:

Answer 4

Data or points that lie significantly away from the majority of the other data. They can skew a regression analysis, especially when the collected data is small. More information should be sought about the outlier before including or excluding it from the analysis.

Question 5

Correlation:

Answer 5

in data analysis, one variable may be affected by another variable and when a change in the independent variable affects the dependent variable, there is a correlation between them.

Question 6

To describe a correlation, we use 3 attributes:

Answer 6

1. Linear (clear line), Non-linear (curved), No correlation (scattered points)
2. Positive (positively sloped), Negative (negatively sloped)
3. Strength; Strong (or high), Moderate (or medium), or Weak

Question 7

Linear Correlation:

Answer 7

Variables have a linear correlation if the changes in one variable tend to be
proportional to changes in the other variable.

Question 8

Correlation Coefficient:

Answer 8

Gives a quantitative measure of the strength of a linear correlation or a measure of how closely the points of a scatter plot is to the line of best fit. It is the covariance of the two variables divided by the product of the standard deviations of each variable.

Question 9

Negative Linear Correlation Values

Answer 9

Strong is -1 to -0.67
Moderate is -0.67 to -0.33
Weak is -0.33 to 0

Question 10

Positive Linear Correlation Values

Answer 10

Weak is 0 to 0.33
Moderate is 0.33 to 0.67
Strong is 0.67 to 1

Question 11

Steps for using the Correlation Coefficient Formula

Answer 11

1. Create a chart with 5 columns (x, y, x^2, y^2, xy)
2. Calculate the sums of each column (by adding each value)
3. Plug in the values and solve with BEDMAS

Question 12

Linear Regressions in Desmos

Answer 12

1. Click on the + sign and insert in a Table
2. Type in your x and values
3. On the next line type y1~mx1+b and the r and r^2 will appear

Question 13

You can find the equation of the line of best fit by

Answer 13

subbing in the m and b that Desmos gives you. Just remember to write x after the m because ts y = mX +b.

Question 14

A positive relationship between two variables will sound like this:

Answer 14

as the # of minutes increases, the cost also increases.

Question 15

To find unknown y values that are within our data sets we can either

Answer 15

1) sub that x into the equation and solve or 2) we can hover over the x value on desmos and get the y.

Question 16

Interpolation is

Answer 16

data within our data set.

Question 17

Extrapolation is

Answer 17

data beyond the data set.

Question 18

To find unknown x values that are within our data set we can either

Answer 18

1) sub in the y and solve or 2) hover over the trendline as close to the y value as possible.

Question 19

A point can be rejected as an outlier if

Answer 19

1) There was an error collecting data, or outside factors affected the data,
ex. Kathy collected data on the heights of students and their arm spans.
When recording the data, she mistakenly recorded a high of 181cm for 101cm. (Thus, an error in data collection)

2) If outside factors affect the data.
ex. A company records its total revenue each month.
Last month, the workers at the company went on strike for two weeks.

Question 20

The presence of an outlier can affect the

Answer 20

analysis of data

Question 21

Other factors, such as

Answer 21

sample size and composition also need to be considered when analyzing data.

Question 22

Linear regression is only appropriate

Answer 22

if the data appears to be linear.

Question 23

Non-Linear Regression:

Answer 23

Definition — An analytical technique for finding a curve of best fit for data having a non-linear correlation.

Question 24

Examples of other types of relationships are:

Answer 24

1) Quadratic: Equation of the form: y= ax^2 + bx +c
2) Cubic: Equation of the form: y= ax^3 + bx^2 + cx +d
3) Quartic: Equation of the form: y= ax^4 + bx^3 + cx^2 + dx + e
4) Exponential: Equation of the form: y = a(b)^x

Question 25

Quadratic: Equation

Answer 25

y= ax^2 + bx +c

Question 26

Cubic: Equation

Answer 26

y= ax^3 + bx^2 + cx +d

Question 27

Quartic: Equation

Answer 27

y= ax^4 + bx^3 + cx^2 + dx + e

Question 28

Exponential: Equation

Answer 28

y = a(b)^x

Question 29

Any and all of these relationships can be tested to see

Answer 29

which best fits the data.

Question 30

When fitting a curve, the measure that is used is called

Answer 30

the Coefficient of Determination (r^2).

Question 31

Coefficient of Determination (r^2)

Answer 31

- Measures how closely the data is to the curve. - It is the Correlation Coefficient squared. - It can have values from 0 to 1, with 1 being a perfect fit. - The higher the r^2 value, the stronger the relationship. The stronger the relationship, the more accurate interpolated and extrapolated predictions are. Conversely, the lower the r^2 value, the less accurate it is in making predictions.

Question 32

Value Scale for Coefficient of Determination (r^2)

Answer 32

- Weak is 0 to 0.33 - Moderate is 0.33 to 0.67 - Strong is 0.67 to 1

Question 33

Criteria for Analyzing the Best Regression Model:

Answer 33

- See if a curve is present - Is there a negative y-intercept, does that fit the context? Can there be negative bacteria? - Is the y-intercept higher or lower than it should be? Does it fit within the values of the chart? - Look at the full chart, does the curve start to drop randomly? Even though the values continue to skyrocket? If the actual diagram does not fit the context of the data, it is not the right curve - The r^2 must be strong, the y-int must match the context, and the curve must be visually representative of the data for it to be the right curve.

Question 34

Degrees of Causal Relationships

Answer 34

- Correlation studies between variables are mainly used to find evidence of a cause-and-effect relationship. - A strong correlation does not prove that a change in one variable caused a change in the other. = This does not necessarily mean a cause-and-effect relationship -- There are various types and degrees of relationships between variables.-

Question 35

1. Cause-and-Effect Relationship

Answer 35

- A change in X causes a change in Y. - Ex: An increase in the speed of a production line produces an increase in the output of items produced - ie. more studying = better test scores

Question 36

2. Common-Cause Factor

Answer 36

- An external variable causes two variables to change in the same way. - Ex: A hot summer causes more people to go to the beach and increases the sales of water = the beach caused both of those

Question 37

3. Reverse Cause-and-Effect Relationship

Answer 37

- The dependent and independent variables are reversed in the process of determining the relation - Ex: A research project attempts to show that people drinking coffee get nervous but finds nervous people drink more coffee = the more coffee you drink the more anxious you are but anxious people tend to drink coffee - reverse - Ex: An increase in the number of police officers means more charges laid. The increase in crime has increased the number of police officers. = More crime happens because more police are there to notice the crime but they’re being hired due to more crime in the first place - Ex: Average amount of traffic in a city and the number of roads built. = More roads = more cars on the roads = more traffic - Constant cycle - has an intention and the reverse happened

Question 38

4. Accidental Relationship

Answer 38

- A correlation exists without any causal relationship between the variables (by accident). - Ex: An increase in SUV sales and an increase in sales of red pens. - Just a fluke - strong correlation with no connection

Question 39

5. Presumed Relationship

Answer 39

- A correlation seems apparent or present although it can’t be proven - a guess or assumption - Ex: Active people will like a new sports streaming service = Just because you go to the gym and value wellness doesn’t mean you necessarily love sports - Ex: The earth’s average air temperature and the concentration of CO2 in the atmosphere = So many other factors are involved - An assumption that there’s a correlation

Question 40

Extraneous Variable (Outside Factors)

Answer 40

- Variables that affect the determination of a causal relationship. - It can affect the dependent or independent variable. - Ex. There is a strong correlation between students’ term mark and their final exam marks. However, extraneous variables that could affect the degree of this correlation are things like a student’s exam schedule, interrupted study time, the pressure of writing an exam, etc. - In order to reduce the effect of extraneous variables, researchers often compare an experimental group to a control group. These two groups should be as similar as possible, so that extraneous variables will have about the same effect on both groups. The researcher may vary the independent variable for the experimental group but not the control group. Any difference in the dependent variables for the two groups can then be attributed to changes in the independent variable.

Question 41

Subtle wording that changes the meaning of information.

Answer 41

e.g. Last year, the unemployment rate was 8.5%. This year, unemployment has had a 1.5 percentage-point increase. - 8.5 -> 10% (+1.5%) - Highlights a “small” increase - Trying to cover up the 10% rate (by making people do the math in their head)

Question 42

The use of large numbers that can lead to misunderstandings about the significance of data.

Answer 42

e.g. Annual healthcare spending in Ontario will increase by $80 million. - from what? - $80 million out of how much in total budget? (Ontario’s budget could be billions, so $80 million just seems big) - how does it compare with education, infrastructure, etc

Question 43

Comparisons that are made where the items are not weighted equally.

Answer 43

e.g. The following chart shows unemployment statistics, by province, for Canada. Instead of numbers of thousands of unemployed people, it should show the unemployment rate The graph does not take into account the population of each province Obviously, Quebec and Ontario have much larger populations Should be a %, not the total amount (the unemployment rate %)

Question 44

Small samples are used to represent larger populations, which distort the data.

Answer 44

e.g. A manager uses a systematic sample to choose every 7" employee from a roster of 30 employees to test how an aptitude test compares to employee productivity. He concludes that the company should hire only applicants who do well on the aptitude test. - Only 4 pieces of data follow the trend line (those 4 just happened to follow this trend when not everyone might follow it - 30 people is small enough to have done an entire population of data

Question 45

The method(s) of presentation may not give the whole picture

Answer 45

e.g. - Pyramids all the same size - misleading - Months shown in the wrong order (reversed), ales are actually decreasing not increasing

Question 46

When data follows a general trend with no significant outliers,

Answer 46

this may indicate the presence of extraneous variables.

Question 47

Extraneous Variables

Answer 47

These are variables which influence the outcome of an experiment, though they are not the variables of interest.

Question 48

If possible, extraneous variables must be eliminated or at least accounted for as they

Answer 48

tend to weaken the correlation.

Question 49

Ways to eliminate extraneous variables:

Answer 49

1) Remove the variable ex: if classroom noise was a factor, make sure the experiment was done in a quiet room. 2) Use a control group ex: the experimental group gets the real pill, the control group gets the placebo pill.

Question 50

When data does not follow a general trend and has an unusual pattern such as two distinct clusters, it is not the work of extraneous variables but rather

Answer 50

hidden variables.

Question 51

When data follows an unusual trend, this may indicate the presence of a

Answer 51

hidden variable.

Question 52

These are variables which can skew statistical results so much that they can

Answer 52

invalidate the statistical results.

Question 53

If possible, hidden variables must be

Answer 53

researched and eliminated.

Question 54

How to eliminate hidden variables:

Answer 54

Separate your data into 2 clusters: 1) Before the event that caused the clusters 2) After the year that caused the clusters (include that year) 3) Then you can write the equation, the r, and the r^2

Question 55

Extraneous Variables Traits

Answer 55

Data follows trend Can be accounted for

Question 56

Hidden Variables Traits

Answer 56

Unusual Pattern Can invalidate results Therefore must be eliminated

Question 57

Similarities between Extraneous and Hidden

Answer 57

Can weaken correlation No outliers Can be eliminated