Z-Tests

• Manipulated variables (IV): 0
• Measured variables (DV): 1
• Population Standard Deviation Known?: Yes

The price of gas is normally distributed across the country, $\mu = 2.60, \sigma = .42$. You want to know if our prices here in Johnstown are significantly different from the population. The mean price of 40 gas stations in Johnstown is 2.29. Is the price of gas in Johnstown significantly different from the national gas price?*

*http://www.pitt.edu/~bertsch/Z%20test%20practice.pdf

Single sample t-test

• Manipulated variables (IV): 0
• Measured variables (DV): 1
• Population Standard Deviation Known?: No

A coffee shop relocates to Italy and wants to make sure that all lattes are consistent. They believe that each latte has an average of 4 oz of espresso. If this is not the case, they must increase or decrease the amount. A random sample of 25 lattes shows a mean of 4.6 oz of espresso and a standard deviation of .22 oz.* [footnote] One Sample T-Test Hypothesis Test By Hand. WordPress. http://www.mathandstatistics.com/learn-stats/hypothesis-testing/one-sample-t-test-hypothesis-test-by-hand[/footnote]

Independent Samples t-test

• Manipulated variables (IV): 1
  • Number of levels: 2
• Measured variables (DV): 1
• Independent samples vs repeated or matched participants: Independent

Let’s say you’re curious about whether New Yorkers and Kansans spend a different amount of money per month on movies. It’s impractical to ask every New Yorker and Kansan about their movie spending, so instead you ask a sample of each—maybe 300 New Yorkers and 300 Kansans—and the averages are $14 and $18. Is there a significant statistical difference between the two samples?*

Repeated Measures or Matched Pairs t-test

• Manipulated variables (IV): 1
  • Number of levels: 2
• Measured variables (DV): 1
• Independent samples vs repeated or matched participants: Repeated

Researchers were interested in whether a new depression medication could increase mood. Participants’ moods were measured on a standard PHQ-9 scale (in which a greater score means more severe depression) before taking the drug and the average score was 14 with a standard deviation of 2. After 6 weeks of taking the new drug, the same participants filled out the PHQ-9 again and the new average score was an 11 with a standard deviation of 2.2. Was there a significant difference in the PHQ-9 scores before and after treatment?

Independent One-Way ANOVA

• Manipulated variables (IV): 1
  • Number of levels: More than 2; In this case 3
• Measured variables (DV): 1
• Independent samples vs repeated or matched participants: Independent

The owner of a plant nursery wanted to know which fertilizer type to recommend to her customers in order to help their azaleas produce more flowers. She decided she wanted to test chemical fertilizer, composted fertilizer, and manure. She tested 10 different azalea plants per condition…

Repeated Measures One-Way ANOVA

• Manipulated variables (IV): 1
• Number of levels: More than 2; In this case 4
• Measured variables (DV): 1
  • Independent samples vs repeated or matched participants: Repeated

Researchers were interested in seeing how the season affects overall mood in those who are not necessarily diagnosed with depression, dysthymia, or seasonal affective disorder. Participants were each asked to fill out a PHQ-9 once in the fall, once in the winter, once in the spring, and one final time in the summer.

Two-Factor (or N-Way) ANOVA

• Manipulated variables (IV): More than 1; in this case 2
  • Number of levels for IV1: 2
  • Number of levels for IV2: 3
• Measured variables (DV): 1

The plant nursery owner decided she wanted to change her experiment. Not only was she curious about fertilizer type, but she was wondering if maybe the amount of water she gives to her azaleas affects the bloom. She kept her original three fertilizers (chemical fertilizer, composted fertilizer, and manure), but she also gave half of the plants in each condition more water than she normally would and the other half the normal amount of water.

Correlation

• Manipulated variables (IV): 0
• Measured variables (DV): 2
• Are we just relating stuff or are we trying to predict stuff?: Relate

The following (imaginary) table provides data collected regarding temperature in Fahrenheit and the number of ice creams sold from the ice cream trucks in town. Determine whether there is any sort of relationship between the two variables. Make sure to describe the strength and direction of the correlation.

Regression

• Manipulated variables (IV): 0
• Measured variables (DV): 2
• Are we just relating stuff or are we trying to predict stuff?: Predict

The following (imaginary) table provides data collected regarding temperature in Fahrenheit and the number of ice creams sold from the ice cream trucks in town. Does knowledge about temperature predict ice cream sales?

$p = \frac{X}{n}$ = Number of items having the characteristic of interest / Sample size

$\sigma_p=\sqrt{\pi(1-\pi)}{n}$

$Z = \frac{p – \pi }{\sqrt{\frac{\pi(1-\pi)}{n}}} $

Data collected from customers at a local restaurant indicated that 58% of customers thought that great customer service was an important attribute in determining which restaurant they would patronage. A random sample of 1000 customers at the local restaurant was polled and the goal is to determine the probability that more than 62% of these customers indicated that customer service was an important attribute in deciding which restaurant to visit.

Parameters vs Statistics and Sampling Error

Sampling Methods

• Simple Random Sampling:
  • Everyone has an equal chance of being selected.
  • The selection method is completely random.
  • One individual’s selection does not impact the likelihood that someone else is subsequently selected.
• Systematic Sampling:
  • In a lost of all the individual, every nth individual is picked.
• Convenience Sample:
  • Using the first individuals a surveyor comes across for the sample.
  • Least likely to get a representative sample.
• Cluster Sampling:
  • Dividing the population into groups (usually geographically).
  • The clusters themselves are randomly selected while the people in them are not.
• Stratified Sampling:
  • Divides the population into groups based on characteristics.
  • A sample is taken from each of these groups so that characteristics that are important are accounted for.

Variable Types and Individuals

Types of Statistics & Types of Studies

• Manipulation: The researcher manipulates one variable by changing its value from one level to another. A second variable is observed to determine whether the manipulation causes changes to occur
• Control: The researcher must exercise control over the research situation to ensure that other, extraneous variables do not influence the relationship being examined. These variables that need to be controlled can be participant variables (characteristics such as age, gender, and intelligence that vary from one individual to the other) or environmental variables (lighting, time of day, weather, etc.). Researchers should control for as many variables as they can, and so spend a lot of time designing an experiment about what variables are important to control for and how to go about doing that.

Constructs

Scale Types

• Nominal: Set of categories; no quantitative distinction (Ravens, Steelers, etc.)
• Ordinal: Categories in an ordered sequence ( 1st place vs. 4th place. We don’t know the differences between each race time, only that this is the order that they came in.)
• Interval: Ordered categories with equal intervals. Arbitrary zero point (ex. Celsius, 0 could have been placed anywhere but we decided to place it at the freezing point of water)
• Ratio: Ordered categories with equal intervals. Absolute zero point. (ex. Height or weight)

Evaluating Rational Expressions

Identifying When a Rational Expression is Undefined

Simplifying Rational Expressions

Multiplying Rational Expressions

Dividing Rational Expressions

Adding and Subtracting Rational Expressions with the Same Denominator

Finding the Least Common Denominator

Writing Equivalent Rational Expressions

Adding and Subtracting Rational Expressions with Different Denominators

Solving Equations Containing Rational Expressions

• $ 28 = 2 \ast 2 \ast 7$
• $ 40 = 2 \ast 2 \ast 2 \ast 5 $
• The greatest common factor is $2 \ast 2 = 4$.

• $x^3 = x \ast x \ast x$
• $x^5 = x \ast x \ast x \ast x \ast x$
• $x^2 = x \ast x$
• The greatest common factor is $x \ast x = x^2$

• $8x + 14$
  • $2 \ast 4 \ast x + 2 \ast 7$
  • $2 \ast 4 \ast x + 2 \ast 7$; 2 is the greatest common factor
• The final answer is $2(4x+7)$

Factoring by Grouping

• Step 1: Group the terms in two groups of two terms so that each group has a common factor.
• Step 2: Factor out the GCF from each group.
• Step 3: If there is a common binomial factor, factor it out.
• Step 4: If not, rearrange the terms and try these steps again.
• Step 5: Make sure to check your work by multiplying (FOIL-ing) your answer to see if you can get back to the original problem.

• $(xy + 2x) + (3y + 6)$
• $x(y+2) + 3(y + 2)$
• $x(y+2) + 3(y + 2)$

Factoring Trinomials of the Form $x^2+bx+c$

Factoring Trinomials of the Form $ax^2+bx+c$

• $3x^2 + 11x + 6 = (3x + ?)(x + ?)$

• $6 = 1 \ast 6$ or $6 = 2 \ast 3$

• $(3x + 1)(x + 6) = 3x^2 + 19x + 6$
• $(3x \ast 6) + (1 \ast x) = 19x$

• $(3x + 6)(x + 1) = 3x^2 + 9x + 6$

• $(3x + 2)(x + 3) = 3x^2 + 11x + 6$

Factoring Out the GCF with Polynomials

Factoring Trinomials of the Form $ax^2+bx+c$ by Grouping

Factoring Perfect Square Trinomials and the Difference of Two Squares

Factoring the Difference of Two Squares

Solving Quadratic Equations by Factoring

• $x^2 - 9x - 22 = 0$
• $(x - 11)(x + 2) = 0$
• $(x - 11) = 0$, and so $x = 11$
• $(x + 2) = 0$, and so $x = -2$

• $3x^3 - 12x = 0$
• $3x(x^2 - 4) = 0$
• $3x(x + 2)(x - 2) = 0$
• $3x = 0$, means $x = 0$
• $(x + 2) = 0$, means $x = -2$
• $(x - 2) = 0$, means $x = 2$

Using the Product Rule

• $(-1 \ast 3)(a^7 \ast a^1)(b^4 \ast b^9)$
• $(-3)(a^8)(b^{13})$

Power of a Quotient Rule

Quotient Rule for Exponents

Zero Exponent

Negative Exponents

Scientific Notation

What is a Polynomial?

• Variables: the letters in the equation
• Constants: In the example above, the constant would be the 11. It’s a number that does not contain any modifiable variables.
• Exponents: These are the numbers that you’ll typically find attached to variables
• Addition, subtraction, multiplication, and division

What isn’t Considered a Polynomial?

Evaluating Polynomials

• Evaluate $8x^2 + 4x + 7$ when $x = 2$
• Step 1: Substitute
  • $8 \ast 2^2 + 4 \ast 2 + 7$
• Step 2: PEMDAS
  • $8 \ast 4 + 4 \ast 2 + 7$
• Step 3: PEMDAS
  • $32 + 8 + 7$
• Step 4: PEMDAS
  • $47$

Adding Polynomials

Subtracting Polynomials

Polynomials with Two Variables

Multiplying Monomials

Multiplying Monomials by Polynomials

Using FOIL

Multiplying Binomials

Multiplying Larger Polynomials

Dividing Polynomials by Monomials

Dividing a Polynomial by Another Polynomial

$\frac{8x^3 + 4x^2 +0x +7}{2x + 3}$

Thus, $\frac{4x^2 + 7 +8x^3}{2x + 3} = 4x^2 - 4x + 6 + \frac{-11}{2x + 3}$

1. Find the range
2. Order the table from highest score to lowest score, not skipping scores that might not have shown up in the data set
3. In the next column, document how many times this score shows up in the data set

1. The grouped frequency table should have about 10 intervals. A good strategy is to come up with some widths according to Guideline 2 and divide the total range of numbers by that width to see if there are close to 10 intervals.
2. The width of the interval should be a relatively simple number (like 2, 5, or 10)
3. The bottom score in each class interval should be a multiple of the width (0-9, 10-19, 20-19, etc.)
4. All intervals should be the same width.

Proportions and Percentages

Real Limits

Frequency Distribution Graphs

• Symmetrical: When the shape of the distribution is, at least for the most part, mirrored on both sides if you were to view the mean as the flipping point.
• Asymmetrical: When the shape of the distribution is not mirrored on both sides for whatever reason (usually because of skew).
• Positively Skewed: This is when there is what looks like a tail of data trailing off to the right. I like to remember this is as the P in Positive having fallen on its back.
• Negatively Skewed: This is when there is what looks like a tail of data trailing off to the left.
• Unimodal: This literally means having a buildup of data around what looks to be one number, so one mode. Your typical bell curve is unimodal.
• Bimodal: This is when there is data clustering around two different numbers or spots on the distribution, so having two modes. This can often look like camel humps.
• Multimodal: When a distribution has two or more “humps” in the graph.

• Mean: sum of all scores divided by the number of scores in the data, also referred to as the average.
• Median: the midpoint of the scores in a distribution when they are listen in order from smallest to largest. It divides the scores into two groups of equal size. With an even number of scores, you compute the average of the two middle scores.
• Mode: the most frequently occurring number(s) in a data set.

Computing Central Tendency Measures

Central Tendency and How they Relate to Distribution Shape

• Normal: The mean, median, and mode are all in the same spot
• Bimodal: The mean and median are together in the middle, while the two modes are on either side, represented by the two humps
• Skewed: The mean is going to be closest to the tail, median is between mean and mode (closer to the tail than in a normal distribution, but not as close as the mean), and the mode is found by the hump. This means that a positively skewed distribution will have a mean larger than its median and a median larger than its mode, while a negatively skewed distribution will have a mode larger than its median and a median lager than its mean.

When to Use Each Measure

• Range: The distance between the lowest and the highest score in a distribution. Can be described as one number or represented by writing out the lowest and highest number together (ex. values 4-10). Calculated by subtracting the highest score from the lowest score. If you’re working with continuous variables, it’s the upper real limit for Xmax minus the lower real limit for Xmin.
• Standard deviation: The average distance between the scores in a data set and the mean. This value is also the square root of the variance. Here’s a video to help you conceptualize this.

• Variance: Measures the average squared distance from the mean. This number is good for some calculations, but generally we want the standard deviation to determine how spread out a distribution is. Calculated with this equation:

Sample Variance and Degrees of Freedom

Biased vs. Unbiased

Inferring Patterns in Data

Z-Scores

Random Sampling

• Independent random sampling: Probabilities must stay constant from one selection to the next if more than one individual is selected. In other words, selecting one individual shouldn’t affect the probability of another person being selected; their chances are independent of one another.
• Random sampling with replacement: Each individual in the population has an equal chance of being selected, meaning that to keep the denominator of the probability equation (X/N) the same for each draw, the first draw needs to be returned to the population pool.

Proportions in Frequency Distributions

Central Limit Theorem

Standard Error

Inferential Statistics

Steps of Hypothesis Testing

• Step 1: State the Hypothesis
  • Null Hypothesis (H0): states that in the general population there is no change, no difference, or no relationship, or in the context of an experiment, it predicts that the independent variable has no effect on the dependent variable.
  • Alternative Hypothesis (H1): states that there is a change, a difference, or a relationship for the general population, or in the context of an experiment, it predicts that the independent variable has an effect on the dependent variable.
• Step 2: Set the Criteria for a Decision
  • Alpha Level: Also known as Level of Significance, is a probability value that is used to define the concept of “very unlikely” in a hypothesis test. We chose an alpha level in order to separate the most unlikely sample means from the most likely sample means. Ex. $\alpha = 0.05,$ that means that we’re separating the most unlikely 5% from the most likely 95%. The largest permissible value of alpha is 0.05, although some researchers like to use more conservative alpha levels to reduce the risk that a false report is published. But you don’t want the value to be too conservative because otherwise, you might run the risk of a Type II error, in which case the hypothesis test demands more evidence from the research results, in which case you might be throwing out evidence that a treatment will work.
  • Critical Region: Composed of the extreme sample values that are very unlikely to be obtained if the null hypothesis is true. Determined by alpha level. If sample data fall in the critical region, the null hypothesis is rejected, because it’s very unlikely they’ve fallen there by chance.
• Step 3: Collect Data and Compute Sample Statistics
  • After collecting the data, we find the sample mean. Now we can compare the sample mean with the null hypothesis by computing a z-score that describes where the sample mean is located relative to the hypothesized population mean. We use the z-score formula.
• Step 4: Make a Decision
  • We decided previously what the two z-score boundaries are for a critical score. If the z-score we get after plugging the numbers in the aforementioned equation is outside of that critical region, we reject the null hypothesis. Otherwise, we would say that we failed to reject the null hypothesis.

Regions of the Distribution

Factors that Influence and Assumptions of Hypothesis Testing

• Random sampling: it is assumed that the participants used in the study were selected randomly so that we can confidently generalize our findings from the sample to the population.
• Independent observation: two observations are independent if there is no consistent, predictable relationship between the first observation and the second. The value of σ is unchanged by the treatment; if the population standard deviation is unknown, we assume that the standard deviation for the unknown population (after treatment) is the same as it was for the population before treatment. There are ways of checking to see if this is true in SPSS or Excel.
• Normal sampling distribution: in order to use the unit normal table to identify the critical region, we need the distribution of sample means to be normal (which means we need the population to be distributed normally and/or each sample size needs to be 30 or greater based on what we know about the central limit theorem).

• The variability of the scores, which is measured by either the standard deviation or the variance. The variability influences the size of the standard error in the denominator of the z-score.
• The number of scores in the sample. This value also influences the size of the standard error in the denominator.

Directional Hypotheses and Tailed Tests

• H0: The test scores are not increased/decreased (the treatment doesn’t work)
• H1: The test scores are increased/decreased (the treatment works as predicted)

Effect Size

Statistical Power

Degrees of Freedom

Shape of a t-distribution

• The degrees of freedom: The greater n is, and subsequently df, the more like a normal distribution the t-distribution will begin to look (this also follows the 30 rule)
• The sample variance: The bottom half of the equation deals with the estimated standard error, which changes when the standard deviation changes. Because the ESE is dependent on the sample variance, each sample can create a different ESE

Hypothesis Testing with a Single Sample t-test

1. State the hypotheses and select an alpha level
2. Locate the critical region
3. Calculate the test statistic
4. Make a decision regarding the null hypothesis

• The values in the sample must consist of independent observations.
• The population that is sampled must be normal.

Effect Size for Single Sample t-tests

Confidence Intervals

• Gives the opportunity to conduct an experiment with very little contamination by extraneous factors.
• Lowers the chance of participants suffering from boredom after a long series of tests as well as skewing the results by becoming more accomplished through practice experience.

• Can be complex.
• Requires a large number of participants.
• Needs a new group for every treatment and manipulation.
• Confounding variables brought in by the individuals in the study can weaken results.

Hypothesis Testing with Independent t-tests

1. Calculate the estimated standard error by calculating pooled variance and figuring out the degrees of freedom for each group.
2. Subtract the two means from one another (we assume that the difference between the population means will be 0 given the null hypothesis) and then divide by the standard error.
3. Determine the critical region based on your alpha level and whether you’re running a one or two-tailed test. Then decide whether your calculated t-test falls within the critical region or not.
4. Make a decision about the null hypothesis based on this comparison.

• The observations within each sample must be independent.
• The two populations from which the samples are selected must be normal.
• To justify using the pooled variance, the two populations from which the samples are selected must have equal variances (homogeneity of variance); essentially the standard deviation of after treatment should be very similar to the standard deviation presented before treatment. This can be confirmed using SPSS or Excel. This can also be done using Hartley’s F-max test, which is described later on in this chapter.

Estimated Standard Error and Pooled Variance

1. Find the sum of squares of each sample.
2. Calculate the pooled variance given the sums of squares you just found and the degrees of freedom ($n-1$ for each sample).
3. Calculate the estimated standard error using that pooled variance.
4. Plug the estimated standard error into the t-test formula and solve for $t$.

Effect Size of Independent Samples t-test

Hartley’s F-Max Test

Mean Difference and Estimated Standard Error of the Mean Difference

Hypothesis Testing with Repeated Measures t-tests

1. State the null and alternative hypothesis
2. Locate the critical region (remember that the $df$ is $n-1$)
3. Calculate the t statistic using the t formula after calculating the estimated standard error of the mean difference.
4. Make a decision.

• Fewer subjects needed
• Is well-suited for studying changes over time (developmental, learning, studying)
• Reduces or eliminates caused by individual differences within the participants by either linking participants up based on characteristics or by using the same person twice.

• Increases the likelihood that outside factors that change over time may be responsible for changes in the participants’ scores.
• Participation in the first treatment could affect scores in the second treatment (practice, fatigue, etc.).

Effect Size for Repeated Measures t-tests

Variability as a Measure of Consistency

Degrees of Freedom

Experiment-wise Alpha

Hypothesis Testing with One-way ANOVAs

Variability in One-Way ANOVAs

• The observations within each sample must be independent.
• The population from which the samples are selected must be normal.
• The populations from which the samples are selected must have equal variance (homogeneity of variance).

Effect Size in One-Way ANOVAs

Post-Hoc Test

Parameters vs Statistics and Sampling Error

Sampling Methods

• Simple Random Sampling:
  • Everyone has an equal chance of being selected.
  • The selection method is completely random.
  • One individual’s selection does not impact the likelihood that someone else is subsequently selected.
• Systematic Sampling:
  • In a lost of all the individual, every nth individual is picked.
• Convenience Sample:
  • Using the first individuals a surveyor comes across for the sample.
  • Least likely to get a representative sample.
• Cluster Sampling:
  • Dividing the population into groups (usually geographically).
  • The clusters themselves are randomly selected while the people in them are not.
• Stratified Sampling:
  • Divides the population into groups based on characteristics.
  • A sample is taken from each of these groups so that characteristics that are important are accounted for.

Variable Types and Individuals

Types of Statistics & Types of Studies

• Manipulation: The researcher manipulates one variable by changing its value from one level to another. A second variable is observed to determine whether the manipulation causes changes to occur
• Control: The researcher must exercise control over the research situation to ensure that other, extraneous variables do not influence the relationship being examined. These variables that need to be controlled can be participant variables (characteristics such as age, gender, and intelligence that vary from one individual to the other) or environmental variables (lighting, time of day, weather, etc.). Researchers should control for as many variables as they can, and so spend a lot of time designing an experiment about what variables are important to control for and how to go about doing that.

Constructs

Scale Types

• Nominal: Set of categories; no quantitative distinction (Ravens, Steelers, etc.)
• Ordinal: Categories in an ordered sequence ( 1st place vs. 4th place. We don’t know the differences between each race time, only that this is the order that they came in.)
• Interval: Ordered categories with equal intervals. Arbitrary zero point (ex. Celsius, 0 could have been placed anywhere but we decided to place it at the freezing point of water)
• Ratio: Ordered categories with equal intervals. Absolute zero point. (ex. Height or weight)

• Individual differences among participants do not influence outcomes or influence them very little because everyone is either paired up on important participant characteristics or they are the same person in multiple conditions.
• A smaller number of subjects needed to test all the treatments.
• Ability to assess an effect over time.

• Increases the likelihood that outside factors that change over time may be responsible for changes in the participants’ scores.
• Participation in the first treatment could affect scores in the second treatment (practice, fatigue, etc.).

Hypothesis Testing with Repeated Measures One-Way ANOVA

• The observations within each treatment condition must be independent.
• The population distribution within each treatment must be normal
• The variances of the population distribution for each treatment should be equivalent

Effect Size

Post Hoc Tests

A farmer wants to know the best combination of products to use to maximize her crop yield. She decides to test out three different fertilizer brands (A, B, and C) and two different kinds of seeds (Y and Z). Each product is paired once with another for a total of 6 conditions: AY, BY, CY, AZ, BZ, CZ.

Meaning of the Equation Before the ANOVA

Hypothesis Testing with Two-Factor ANOVAs

• First Stage:
  • Is identical to independent samples ANOVA
  • Compute the $SS_{total}$, $SS_{between treatments}$, and $SS_{within treatments}$
• Second Stage:
  • Calculate $df_{total}$, $df_{within treatments}$, $df_{between treatments}$, $df_A$, $df_B$, and $df_{error}$.
• Third Stage:
  • The goal is to partition the $SS_{between treatments}$ into three separate components
  • Calculate $SS_A$, $SS_B$, and $SS_{AxB}$. In other words, calculate the parts of $SS_{between treatments}$ which can be attributed to main effect one, main effect two, and an interaction.
• Fourth Stage:
  • Calculate $MS_A$, $MS_B$, $MS_{AxB}$, and $MS_{within treatments}$.
• Fifth Stage:
  • Calculate $F_A$, $F_B$, and $F_{AxB}$
  • These are the numbers which you use to determine if there is a main effect one $(F_A)$, main effect two $(F_B)$, and/or an interaction $(F_{AxB})$.
• Sixth Stage:
  • If there is a significant main effect that has three or more levels, this is when you would conduct a post-hoc analysis for that factor alone. This would be no different than one done for a one-way ANOVA.

• The observations within each sample must be independent of each other.
• The populations from which the samples are selected must be normally distributed.
• The populations from which the samples are selected must have equal variances.

Effect Size

Pearson Correlation

1. Correlation does not demonstrate causation. This is something very important to remember; just because two variables have a correlation doesn’t mean that one is causing the other. There may be another factor $(Z)$ that we haven’t measured which may be the real reason. Take into account that when ice cream sales go up, so do the number of drownings in an area. Does that mean that ice creams are causing people to drown? Consider that maybe increased temperatures result in more ice cream consumption as well as an increase in the number of people who are going out swimming. You can never know if there’s a third lingering factor with just a correlation.
2. The value of the correlation is affected by the range of scores in the data. For example, if you’re looking at how height and age correlate, if your sample is just made up of people who are 20 or older, you probably will get a weak correlation, as most adults no longer grow. However, if your sample is 17 or younger, you’re likely to find a decent positive correlation.
3. Extreme points (outliers) have an impact. Data points which vary greatly from the others may sometimes need to be removed as their presence affects the correlation.
4. Correlation cannot be interpreted as a proportion.

Coefficient of Determination

Other Types of Correlation

• Partial correlation: A partial correlation measures the relationship between two variables while controlling the influence of a third variable by holding it constant.
• Spearman correlation: Used when both variables are measured on an ordinal scale; Used when the relationship is consistently directional but may not be linear.
• Point-biserial correlation: Measures relationship between two variables when one variable has only two values (dichotomous value)
• Phi-coefficient: Both variables are dichotomous. Both variables are re-coded to values 0 and 1

Standard Error of the Estimate

The Probability Distribution for a Discrete Variable

Expected Value for a Discrete Variable

Standard Deviation and Variance of a Discrete Variable

Binomial Distribution

1. The sample must consist of a fixed number of observations, $n$
2. Each and every observation can be categorized into one of two mutually exclusive and collectively exhaustive categories
3. The probability of an event of interest, $p$, is constant across all observations. Therefore the probability of a non-event of interest, $1 – p$ (sometimes called $q$) is constant for all observations.
4. Observations are all independent. This simply means the probability of occurrence of any observation is not dependent on any other observation.

Example: There are 10 golf balls in a bag, consisting of 6 orange balls and 4 yellow balls. If we define success as the likelihood of picking an orange ball and therefore failure as not picking an orange ball (and therefore picking a yellow ball), we can illustrate the three distinct possibilities that may be encountered in calculations.

If 6 golf balls are to be selected at random (without replacement):

• What is the probability of picking exactly 4 orange balls?

$P (X = x | n, p) = \frac{n!}{x! (n – x)!} p^x  {(1-p)}^{n-x}$

And

$P(X = x | n, p)$ = probability that $X = x$ events of interest, when $n$ and $p$

$N$ = number of observations = 6

$P$ = probability of an event of interest = 6/10 = 0.6

$I – p(q)$ = prob. of not having an event of interest = 0.4

$X$ = number of events of interest (no. of successes) in the sample ($X$ = 0,1,2, …, $n$) = 4

$P (X = x | n, p) = \frac{6!}{4! (6 – 4)!} 0.6^4 \ast (1-0.6)^{(6-4)} = 0.3110$

• What is the probability of picking at least 4 orange balls?

This equates to: prob. of 4 orange + prob. of 5 orange + prob. of 6 orange

$P (X = x | n, p) = \frac{6!}{4! (6 – 4)!} 0.6^4 \ast (1-0.6)^{(6-4)} $

$+ \frac{6!}{5! (6 – 5)!} 0.6^5 \ast (1-0.6)^{(6-5)} $

$+ \frac{6!}{6! (6 – 6)!} 0.6^6 \ast (1-0.6)^{(6-6)}$

$= 0.31104 + 0.186624 + 0.046656 $

$= 0.5443$

• What is the probability of picking less than 4 orange balls?

This equates to: prob. of 0 orange + prob. of 1 orange + prob. of 2 orange + prob. of 3 orange

$= 1 – $ Prob. of at least 4 orange balls $= 1 – 0.5443 = 0.4557$

Mean of the Binomial Distribution

Standard Deviation of the Binomial Distribution

Poisson Distribution

1. The area of opportunity must be defined by time, length, surface area etc. Per the Poisson distribution, we can determine the number of times a particular event occurs in a given area of opportunity.
2. The probability that an event occurs in a given area of opportunity must be the same for all the areas of opportunity.
3. The number of events that occur in each and every area of opportunity is independent of the number of events that occur in any area of opportunity
4. The probability that two or more events will occur in any area of opportunity approximates to zero as the area of opportunity becomes smaller.

Example: Imagine that the mean number of cars that pass an intersection in a 1-minute interval is 5.0.

• What is the probability that in a given minute, exactly four cars will arrive?

$P (X = 4 | \lambda = 5) = \frac{e^{-5} \ast 5.0^{4}!}{4!} = 0.1755$

• What is the probability that more than four cars will arrive in a given minute?

The probability that more than four cars will arrive:

$P (X > 4) = P (X = 4) + P (X = 5) + P (X = 6) + \cdots +$

Since all probabilities in a distribution sum to 1:

$P (X > 4) = 1 – P(X <= 4) $

$= 1 – [P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4)]$

$= 1 – (0.0067 + 0.0337 + 0.0842 + 0.1404 + 0.1755)$

$= 1 – 0.4405$

$= 0.5595$

Mean vs. Median

Range and Interquartile Range vs Standard Deviation

Boxplot

Histogram

Empirical Rule

• Two-thirds of the data lie within ±1 standard deviations of the mean
• Four-fifths of the data lie within ±1.28 standard deviations of the mean
• 19 of every 20 data points lie within ± 2 standard deviations of the mean

Normal Probability Plot (Q-Q Plot)

• Step 1: Order the data from least to greatest
• Step 2: Find the expected quantile z-scores. This is done using the following equation: $\frac{1}{n+1} \ast q$ where $n$ is the number of data points and $q$ is the ordered value. The smallest number would have an ordered value of 1, the second smallest would have an ordered value of 2, and so on and so forth.
• Step 3: Finally, you can highlight the data and the z-scores (make sure the two columns are next to each other), go to Insert, and create a scatterplot with the data. If the line is straight, then the data is normally distributed. If the line curves upward, the data is positively skewed. If the line curves downward, the data is negatively skewed.

Strategies for Problem Solving

1. Understand the Problem: You may need to read a problem several times before you can conceptualize it. Don’t become frustrated, and take a walk if you need to. It might take some time to click.
2. Devise a Plan: There may be more than one way to solve the problem. Find the way which is most comfortable for you or the most practical.
3. Carry Out the Plan: Try it out. You may need to adjust your plan if you run into roadblocks or dead ends.
4. Look Back and Check: Make sure your answer gives sense given the context.

• Tables and Charts: Sometimes you’ll be working with a lot of data or computing a problem with a lot of different steps. It may be best to keep it organized in a table or chart so you can refer back to previous work later.
• Working Backward: Sometimes you’ll be given a word problem where they describe a series of algebraic functions that took place and then what the end result is. Sometimes you’ll have to work backward chronologically.
• Using Trial and Error: Sometimes you’ll know what mathematical function you need to use but not what number to start with. You may need to use trial and error to get the exact right number.
• Guessing and Checking: Sometimes it will appear that a math problem will have more than one correct answer. Be sure to go back and check your work to determine if some of the answers don’t actually work out.
• Considering a Similar, Simpler Problem: Sometimes you can use the strategy you think you would like to use on a simpler, hypothetical problem first to see if you can find a pattern and apply it to the harder problem.
• Drawing a Sketch: Sometimes—especially with geometrical problems—it’s more helpful to draw a sketch of what is being asked of you.
• Using Common Sense: Be sure to read questions very carefully. Sometimes it will seem like the answer to a question is either too obvious or impossible. There is usually a phrasing of the problem which would lead you to believe that the rules are one way when really it’s describing something else. Pay attention to literal language.

• Words: a set of odd numbers less than 6
• Listing: {1,3,5}
• Set-builder notation: {x|x is an odd counting number less than 6}

Important Definitions for Sets

• Natural numbers or Counting numbers are all integers starting at 1: {1, 2, 3, 4,…}
• Whole numbers are all integers starting at 0: {0, 1, 2, 3, 4…}
• Integers are all whole numbers from -∞ to ∞: {… -3, -2, -1, 0, 1, 2, 3…}
• Rational numbers are numbers that can be created by dividing two integers (like 1/2 or 9/10 or 4/1).
• Real Numbers are any number that isn’t imaginary, so the typical integer, fraction, or decimal we’re used to. An imaginary number is when a negative number is square rooted (√-1)
• Irrational Numbers are decimals that can’t be expressed as the fraction of two integers. The square root of 2 would be an example of this because the decimals are ongoing and there is no discernible pattern to the decimals.
• A cardinal number is the number of elements in a set and it is written as n(A) and spoken as “n of A.” So if I was given set Z = {1, 5, 7, 2, 9, 10}, I would say n(Z) is 6.
• A finite set is a set that has a whole number as its cardinal number. So we could technically count how many elements are in a set.
• An infinite set is a set where the number of elements is infinite and we couldn’t possibly count them.
• Two sets are equal if two conditions are met: (1) every element of the first set is an element of the second set and (2) if every element of the second set is an element of the first one. That being said, it does not matter if the elements are written in a different order ({a, b, c, d} = {a, c, d, b}) and repeating elements doesn’t add a new element ({a, b, a, c, d, d} = {a, b, c, d}).

Venn Diagrams and Subsets

Subsets of a Set

Set Operations