Algebra 2 Honors - Topic 11 (Statistics)

11-1 Edpuzzle Statistical Variables

Definitions

• Statistical Question: it can be answered by collecting many pieces of information, or data, and summarizing the data

  • Ex:

    • “What month is your birthday” only needs 1 piece of information.

    • “What month has the most birthdays of students at your school” is a statistical question because it can be answered by collecting pieces of information and summarizing the data

• Statistical Variable: a quantity of quality that can be measured or counted, and for which data are expected to differ from one observation to another.

  • Two main types are categorical and quantitative

  • Categorical variable: data that falls into categories, used for qualitative data and generally not numerical data

    • Ex: “Favorite Olympic Sport”

  • Quantitative variable: a quantity that can be counted

    • Ex: Average number of students that came on Monday

• Population: the set of all members of a group you want to know more about

• Sample: a subset of the population that is being studied to answer a statical question about the population

  • Ex: The population is all the students at school while the sample is the group of students that were surveyed

• Parameter: a measure that describes the entire population

• Statistic: a measure that describes a sample of the population

16-3 Variance-Std Dev-TI-84 (Edpuzzle) and Variation (Physical Note)

Definitions

• Range: distance between smallest and largest scores

• Mean Deviation: the average distance/deviation from each data point to the mean, x̄

  • Formula: $\frac{\left\vert x̄-x\right\vert^{}}{n}$ where $\left\vert x̄-x\right\vert^{}$ is done to every value in the data set. Add them all together before dividing by n (n=number of data points)

• Standard Deviation: the square root of the variance

  • Formula: $\sigma=\sqrt{\sigma^2}$

• Variance: the average of the squares of the distance from each data point to the mean

  • Formula: $\sigma^2=\frac{\left\vert x̄-x\right\vert^2}{n}$ where $\left\vert x̄-x\right\vert^2$ is done to every value in the data set. Add them all together before dividing by n (n=number of data points)

• IMPORTANT: Use the calculator stat to find these values

• Put them into the list and go to CALC, then do 1-Var Stats

  • Press vars and 5: Statistics for the variables

• IMPORTANT: Use population deviation as a default mostly

Population specific definitions

• Population mean = μ

• Population variance = $\sigma^2$

• Population standard deviation = $\sigma$

Sample specific definitions

• Sample mean = x̄

• Sample variance = s²

• Sample standard deviation: s

11-3 Data Distributions

How to identify an official outlier

The value must be one of either to be an official outlier

1. value < Q1 - 1.5(Interquartile Range)

2. value > Q3 + 1.5(Interquartile Range)

Different Data Distributions

• Skewed Left

  • tail is longer on the left side, pulling the mean to the left

  • $x̄$ < Median < Mode

• Normal/Symmetric

  • no tail

  • $x̄$ = Median = Mode

• Skewed Right

  • tail is longer on the right side, pulling the mean to the right

  • $x̄$ > Median > Mode

• IMPORTANT:

  • This is how to measure the middle and spread/variability of data

    • When your data is skewed, use median and IQR

    • When your data is symmetric use the $x̄$ and $\sigma$

11-4 Pearson/Normal Distributions (Rogers Edpuzzle and Physical Notes)

Normal distribution

• 68% of the data is within 1 standard deviation

• 95% of the data is within 2 standard deviations

• 99.7% of the data is within 3 standard deviations

Z-score

• Z-score: the z-score of a data point is the number of standard deviations the data is above or below the mean

  • They change a normal distribution with a given mean and standard deviation into a standard normal distribution with x̄=0 and $\sigma$ = 1

• Formula: $z=\frac{x-x̄}{\sigma}$

Probabilities

• the area below a standard normal distribution is always equal to ONE

• Cumulative probability is calculated through normalcdf

How to calculate the probability on the TI-84 Plus CE:

1. Press 2nd vars, then pull up normalcdf

2. Put in the lower and upper bound area

• if doing z < x, lower = -1e99

• if doing z > x, upper = 1e99

3. Put in the mean and the standard deviation (if it involves z-score x̄=0 and $\sigma$ = 1)

4. Calculate

16-5 and 11-2 Sampling (Edpuzzle)

Types of Sampling

• Random sample: a sample selected so that each object/person in the population has an equal chance of being selected and each object/person is chosen independently of any other objects in the sample

• Stratified sampling: when a population is divided into groups with similar characteristics and a sample is randomly chosen from each group

• Cluster sampling: when a population is divided into convenient clusters, and entire clusters are chosen at random as the sample

• Systematic sampling: when you start with one member chosen at random then use a rule, such as “every 3rd member of the population” to select members of the sample

• Convenience sampling: only choosing subjects that are in close proximity or easy to get to

• Self selected sampling: using a sample made up of volunteers

Sampling & Bias

Methods where care must be taken to avoid bias:

• Stratified sampling

• Cluster sampling

• Systematic sampling

Methods with a high risk of bias:

• Convenience sampling

• Self-selected sampling

Types of Studies

• Experiment: involves randomly dividing a population into two groups, applying a different treatment to each group, and measuring the difference between the treatments

• Sample survey: asks every member of a randomly selected sample the same set of questions and records the answers to try and draw a conclusion about the population

• Observational study: you don’t assign treatments to the subjects being studied. The subjects are already affected by the treatments under investigation

11-PH & 16.6 Testing Hypothesis

Significance/Hypothesis Test

• Significance/Hypothesis test: a formal procedure for using observed data to decide between two competing claims (called hypotheses). The claims are statements about the Null and Alternate hypotheses

• Null Hypothesis ($H_{O}$): what we observed occurred by chance

• Alternate Hypothesis ($H_{A}$): what we observed did not occur by chance

Chi-Square Test Statistic

• Formula: $\chi2=\Sigma\frac{\left(observed-\exp ected\right)^2}{\exp ected}$

• On Calculator: First, put the observed values in one list, and the expected values in another. Go to stat, TEST, and x²GOF-Test (alpha D). Put in Observed, Expected, and degrees of freedom (df)

  • Degrees of Freedom (df): Calculated as the number of categories minus one, df = n - 1

• You get the $\chi2$ and the p-value (p)

How to Test A Hypothesis

1. Find the p-value

2. Compare it to the level of significance

   • If the p-value > level of significance, REJECT null hypothesis

   • If the p-value < level of significance, FAIL TO REJECT null hypothesis

3. Decide to reject of fail to reject the null hypothesis

   • When you reject null hypothesis:

     • For alternate hypothesis you say yes; there’s evidence that it didn’t occur by chance

   • When you fail to reject null hypothesis:

     • For alternate hypothesis you say no; I have no evidence to believe that it didn’t occur by chance

• Important:

  • Default level of significance is 5%.

IMPORTANT: When to Use What on Calculator

When to Use normalcdf

• Use normalcdf when:

  • you want to know the probability (cumulative probability) that a random variable will fall between two specific values

  • when you want to find the percentage of data points below or above a certain value

  • when you’re calculating a p-value for a null hypothesis

When to use invNorm

• Use invNorm when:

  • you need to find a specific value within a normal distribution that corresponds to a given percentile or area

  • you need to determine the x-value for a given probability