Science Based Statistics | Madison College Outline of Instruction

10804150 Science Based Statistics

Course Information

Description

This is an introductory course with examples and applications chosen from quantitative x-ray analysis, image analysis, metrology and selected biomedical problems. Covers techniques for the collection, presentation, analysis and interpretation of experimental results and develops procedures to deal with uncertainty in inferences and decisions when data are subject to random error. Topics covered include descriptive statistics; basic probability; the hypergeometric, binomial, normal, Students t, chi-square and F distributions; sampling distributions; the Central Limit Theorem; one-way analysis of variance and linear regression.

Total Credits

Course Competencies (Course Outcomes)

Perform basic mathematical operations

Assessment Strategies

in the solution to a problem on a quiz, homework, project or exam

Criteria

you convert fractions or decimals into percentages and vice versa

you compute expressions that involve addition, subtraction, multiplication, division, exponentiation or combinations of these according to the standard order of operation rules

you evaluate algebraic formulas involving signed numbers or measurements using the standard order of operation rules

you use technology to compute expressions that involve addition according to the standard order of operation rules

you use technology to compute expressions that involve subtraction according to the standard order of operation rules

you use technology to compute expressions that involve multiplication according to the standard order of operation rules

you use technology to compute expressions that involve division according to the standard order of operation rules

you use technology to compute expressions that involve exponentiation according to the standard order of operation rules

you use technology to compute expressions that involve combinations of these according to the standard order of operation rules
Construct graphs

Assessment Strategies

in the solution to a problem on a quiz, homework, project or exam

Criteria

you calculate the slope, y and x intercepts of a line and graphs the line given a linear equation in x and y

you interpret information from graphical representations of data

you extract information from graphical representations of data

you generate a graphical representation of data from a table or relationship given by an algebraic formula

you interpret a graphical representation of data from a table or relationship given by an algebraic formula
Evaluate expressions that use summation notation

Assessment Strategies

in the solution to a problem on a quiz, homework, project or exam

Criteria

you interpret expressions that use summation notation

you evaluate expressions that use summation notation

you use algebraic identities to evaluate or simplify expressions that use summation notation

you use technology to compute expressions that use summation notation
Generate frequency distributions.

Assessment Strategies

in the solution to a problem on a quiz, homework, project or exam

Criteria

you generate frequency distributions from a given data set or groups the scores into classes

you calculate the frequency of each class

you calculate the relative frequency of each class

you calculate the cumulative frequency of each class

you calculate the width and midpoint of each class

you interpret the frequency of each class

you interpret the relative frequency of each class

you interpret the cumulative frequency of each class

you interpret the width and midpoint of each class

you calculate the mean of a frequency distribution

you calculate the mode of a frequency distribution

you calculate the median of a frequency distribution

you interpret the mean of a frequency distribution

you interpret the mode of a frequency distribution

you interpret the median of a frequency distribution

you calculate the quartiles of a frequency distribution

you calculate the percentile scores of a frequency distribution

you calculate inter-quartile range of a frequency distribution

you interpret the quartiles of a frequency distribution

you interpret the percentile range of a frequency distribution

you interpret the inter-quartile range of a frequency distribution

you calculate the range of a frequency distribution

you calculate the variance of a frequency distribution

you calculate the standard deviation of a frequency distribution

you interpret the range of a frequency distribution

you interpret the variance of a frequency distribution

you interpret the standard deviation of a frequency distribution

you demonstrate the relationship between the variance and standard deviation

you distinguish the population standard deviation from the sample standard deviation

you use the properties of the mean and standard deviation

you calculate z scores for a frequency distribution

you interpret z scores for a frequency distribution

you use the properties of z scores

you demonstrate the connections between z scores and the properties of the mean

you demonstrate the connections between z scores and the properties of standard deviation

you use both Tchebyshev's inequality and the percentage points for a normal distribution to estimate the fraction of scores beyond a given z score

you use technology to generate a frequency distribution

you use technology to calculate a frequency distribution as to its descriptive statistics
Generate graphs of frequency distributions.

Assessment Strategies

in the solution to a problem on a quiz, homework, project or exam

Criteria

you generate histograms of frequency distributions

you interpret histograms of frequency distributions

you generate a box plot from a frequency distribution

you interpret a box plot from a frequency distribution

you use a histogram or box plot to estimate measures of central tendency and dispersion

you demonstrate the connections between information found in a histogram or a box plot and calculated descriptive statistics
Use the definitions and axioms of probability.

Assessment Strategies

in the solution to a problem on a quiz, homework, project or exam

Criteria

you distinguish theoretical probabilities from empirical probabilities

you generate a representation of a sample space of an experiment by listing all outcomes, drawing Venn diagrams, making tree diagrams, or constructing tables

you use a representation of the sample space of an experiment and the fair experiment or equi-probability model to compute probabilities

you use the fundamental rule of counting, the number of permutations, the number of combinations, and binomial coefficients to compute the number of outcomes in a given event

you recognize computations which count the number of outcomes corresponding to a given event

you formulate computations which count the number of outcomes corresponding to a given event

you evaluate computations which count the number of outcomes corresponding to a given event

you use these results to compute event probabilities

you appropriately use set theory rules to calculate the probability of unions, intersections, and complements of events

you distinguish mutually exclusive events from events with non-null intersection

you recognize conditional probabilities

you formulate conditional probabilities

you evaluate conditional probabilities

you determine if two events are independent

you compare and contrast the intuitive notion of event independence from the formal definition

you distinguish between mutually exclusive and independent events

you recognize events that are neither independent or mutually exclusive

you use a tree diagram to distinguish the probabilities of a false positive and a false negative test

you use a tree diagram to compute the probabilities of a false positive and a false negative test

you recognize computations which calculate the reliability of systems consisting of components arranged in series or parallel

you formulate computations which calculate the reliability of systems consisting of components arranged in series or parallel

you evaluate computations which calculate the reliability of systems consisting of components arranged in series or parallel
Use probability distributions of a random variable.

Assessment Strategies

in the solution to a problem on a quiz, homework, project or exam

Criteria

you generate the probability distribution of a discrete random variable

you graph the probability distribution of a discrete random variable

you generate histograms for the probability distribution of a discrete random variable

you interpret histograms for the probability distribution of a discrete random variable

you compute the mean of a discrete random variable

you compute the variance of a discrete random variable

you compute the standard deviation of a discrete random variable

you demonstrate the connections between the formulas for the mean, variance and standard deviation of a random variable

you demonstrate the connections between these formulas and those used in descriptive statistics

you distinguish a drawing without replacement from a drawing with replacement and formulate and compute probabilities of events associated with the drawing

you distinguish a drawing without replacement from a drawing with replacement

you recognize when the hypergeometric distribution applies to a drawing without replacement

you formulate and compute probabilities of events associated with the hypergeometric distribution

you recognize when an experiment is a succession of simple Bernoulli trials

you formulate event probabilities for an experiment consisting of simple Bernoulli trials using the binomial distribution

you compute event probabilities for an experiment consisting of simple Bernoulli trials using the binomial distribution

you calculate the mean of a hypergeometric or binomial random variable

you calculate the variance of a hypergeometric or binomial random variable

you calculate the standard deviation of a hypergeometric or binomial random variable

you interpret the mean of a hypergeometric or binomial random variable

you interpret the variance of hypergeometric or binomial random variable

you interpret the standard deviation of a hypergeometric or binomial random variable

you demonstrate the connections between the formulas for the mean, variance and standard deviation of hypergeometric and binomial random variables and the corresponding formulas for a generic random variable

you demonstrate the connections between the formulas for the mean, variance and standard deviation of both hypergeometric and binomial distributions

you use the binomial distribution to approximate the hypergeometric distribution for large populations and justify when such an approximation is valid

you justify the validity of the binomial distribution to approximate the hypergeometric distribution for large populations

you use the Poisson distribution to approximate the binomial distribution for large sample sizes and fixed mean number of successes

you justify why and when the Poisson distribution approximate is valid for the binomial distribution for large sample sizes and fixed mean number of successes
Solve applications using probability distributions.

Assessment Strategies

in the solution to a problem on a quiz, homework, project or exam

Criteria

you use the fair experiment to model simple experiments such as coin or die tosses

you compare the theoretical and empirical probability distributions for a random variable associated with the coin or die tosses experiment

you recognize verbally stated applications which require using the hypergeometric, binomial and/or Poisson distributions

you formulate verbally stated applications which require using the hypergeometric, binomial and Poisson distributions

you solve verbally stated applications which require using the hypergeometric, binomial and/or Poisson distributions

you use probability distributions to analyze processes such as games of chance, insurance rates, instrument reliability and medical tests
Apply the properties of a continuous probability distribution.

Assessment Strategies

in the solution to a problem on a quiz, homework, project or exam

Criteria

you distinguish between a continuous and a discrete random variable

you recognize the connection between the area under a probability density curve and the probability of an event

you formulate the connection between the area under a probability density curve and the probability of an event

you estimate the mean of a continuous probability distribution from the graph of its probability density curve

you estimate the standard deviation of a continuous probability distribution from the graph of its probability density curve
Apply the normal distribution.

Assessment Strategies

in the solution to a problem on a quiz, homework, project or exam

Criteria

you use the normal distribution to approximate the binomial distribution for large sample sizes and a fixed probability of success

you justify why and when the normal distribution approximation is valid for the binomial distribution for large sample sizes and a fixed probability of success

you compute the area between two scores in a normal distribution by transforming to z scores and using the standard normal distribution

you determine critical z scores of a standard normal distribution from a stated probability

you recognize verbally stated applications which involve using the normal distribution

you formulate verbally stated applications which involve using the normal distribution

you solve verbally stated applications which involve using the normal distribution
Apply the chi-squared distribution.

Assessment Strategies

in the solution to a problem on a quiz, homework, project or exam

Criteria

you recognize when empirical and theoretical probability distributions are in agreement based on their graphs

you determine critical scores of a chi-squared distribution from a stated probability

you compute the mean of a chi-squared distribution from the degrees of freedom

you compute the standard deviation of a chi-squared distribution from the degrees of freedom

you use the chi-squared distribution at a given level of significance to test whether the disagreement between theoretical and empirical probabilities is not random

you provide valid justification for the use of a chi-squared distribution in testing whether the disagreement between theoretical and empirical probabilities is not random
Apply the Central Limit Theorem.

Assessment Strategies

in the solution to a problem on a quiz, homework, project or exam

Criteria

you identify the sampling distribution of means

you represent the sampling distribution of means

you distinguish the mean and standard deviation of the parent population from the mean and standard deviation of the sampling distribution of means

you calculate the mean of the sampling distribution of means from the mean and standard deviation of the parent population

you calculate the standard deviation of the sampling distribution of means from the mean and standard deviation of the parent population

you verify the content of the Central Limit Theorem by computational simulation

you use the Central Limit Theorem to compute probabilities for a range of values of a sample mean

you calculate at a given level of significance a confidence interval for a sample mean in terms of parent population parameters
Calculate confidence intervals for population parameters based on sample data.

Assessment Strategies

in the solution to a problem on a quiz, homework, project or exam

Criteria

you recognize confidence intervals for population means in terms of sample means and sample standard deviations

you formulate confidence intervals for population means in terms of sample means and sample standard deviations

you calculate confidence intervals for population means in terms of sample means and sample standard deviations

you demonstrate the connection between the Central Limit Theorem and confidence intervals for the population mean based on large sample sizes

you demonstrate the connections between confidence intervals for a population mean and a population proportion

you demonstrate the similarities between confidence intervals for a population mean and a population proportion

you demonstrate the differences between confidence intervals for a population mean and a population proportion

you use the degrees of freedom of a Student's t

you provide justification for the use of a t distribution

you demonstrate the connections between the standard normal distribution and a t distribution

you demonstrate the similarities between the standard normal distribution and a t distribution

you demonstrate the differences between the standard normal distribution and a t distribution

you use the sampling distribution of differences of means

you recognize the pooled estimate of the variance of two populations of equal variance based on the sample variances

you formulate the pooled estimate of the variance of two populations of equal variance based on the sample variances

you compute the pooled estimate of the variance of two populations of equal variance based on the sample variances

you recognize confidence intervals for the difference of two population means in terms of the sample means and sample standard deviations for both large and small independent samples

you formulate confidence intervals for the difference of two population means in terms of the sample means and sample standard deviations for both large and small independent samples

you calculate confidence intervals for the difference of two population means in terms of the sample means and sample standard deviations for both large and small independent samples

you calculate a confidence interval on a difference of two population means using a matched pair experimental design

you calculate a confidence interval on a difference of two population means using a matched pair experimental design
Test hypotheses about population parameters based on sample data.

Assessment Strategies

in the solution to a problem on a quiz, homework, project or exam

Criteria

you recognize alternative and null hypotheses from a written statement of a question to be decided

you translate alternative and null hypotheses from a written statement of a question to be decided

you formulate alternative and null hypotheses from a written statement of a question to be decided

you recognize Type I and Type II errors

you recognize the connection between Type I errors and the level of significance

you recognize one tail and two tail hypotheses tests about population means

you recognize one tail and two tail hypotheses tests about population proportions

you recognize one tail and two tail hypotheses tests about differences of two population means

you distinguish between one tail and two tail hypotheses tests about population means

you distinguish between one tail and two tail hypotheses tests about population proportions

you distinguish between one tail and two tail hypotheses tests about differences of two population means

you formulate one tail and two tail hypotheses tests about population means

you formulate one tail and two tail hypotheses tests about population proportions

you formulate one tail and two tail hypotheses tests about differences of two population means

you test one tail and two tail hypotheses tests about population means

you test one tail and two tail hypotheses tests about population proportions

you test one tail and two tail hypotheses tests about differences of two population means

you calculate significance probabilities (p-values) for hypotheses tests using the standard normal distribution

you interpret significance probabilities (p-values) for hypotheses tests using the standard normal distribution

you estimate significance probabilities for hypotheses tests that use the Student's t or chi-squared distributions

you interpret significance probabilities for hypotheses tests that use the Student's t or chi-squared distributions

you explain the relationship between the stated level of significance and the calculated significance probability

you perform a hypothesis test on a difference of two population means using a matched pair experimental design

you test whether two categorical variables are dependent by performing a contingency table analysis using a chi-squared distribution

you recognize the underlying assumptions involved in testing a given hypothesis

you recognize the underlying limitations involved in testing a given hypothesis
Interpret a one way analysis of variance.

Assessment Strategies

in the solution to a problem on a quiz, homework, project or exam

Criteria

you recognize a one-way analysis of variance (ANOVA) to test whether there are differences in performance due to a single factor varied at more than two levels

you formulate a one-way analysis of variance (ANOVA) to test whether there are differences in performance due to a single factor varied at more than two levels

you perform a one-way analysis of variance (ANOVA) to test whether there are differences in performance due to a single factor varied at more than two levels

you interpret the treatment sum of squares and the error sum of squares

you interpret the null hypothesis for a one-way ANOVA

you explain the null hypothesis for a one-way ANOVA

you relate the null hypothesis to the interpretation of the mean square due to treatment and the mean square due to error

you use the F distribution in a one-way ANOVA

you recognize the underlying assumptions of a one-way ANOVA

you recognize the underlying limitations of a one-way ANOVA
Determine which levels of a single factor make a significant difference in performance.

Assessment Strategies

in the solution to a problem on a quiz, homework, project or exam

Criteria

you test for which treatment means are different by performing multiple pair comparison tests

you interpret the value of the level of significance used in the multiple pair comparisons

you explain the value of the level of significance used in the multiple pair comparisons

you explain the connections between the conclusions of the one-way ANOVA test and the multiple pair comparisons
Set up and calculate a linear regression fit of bivariate data.

Assessment Strategies

in the solution to a problem on a quiz, homework, project or exam

Criteria

you paraphrase the method of least squares and its connection to the linear regression equations

you generate the scatter diagram

you generate the regression slope and the regression intercept on a scatter diagram

you plot the regression line for a set of paired xy data on the scatter diagram
Perform statistical inference on a linear regression fit of bivariate data.

Assessment Strategies

in the solution to a problem on a quiz, homework, project or exam

Criteria

you recognize the underlying assumptions of a linear regression analysis

you recognize the underlying limitations of a linear regression analysis

you perform statistical inference (calculation of a confidence interval or performance of a hypothesis test) on the population slope and population intercept

you calculate confidence intervals for an average y measurement

you interpret confidence intervals for an average y measurement

you calculate confidence intervals for a single y measurement for a given x value

you interpret confidence intervals for a single y measurement for a given x value

you calculate the coefficient of determination and the correlation coefficient for a set of paired xy data

you interpret the coefficient of determination and the correlation coefficient for a set of paired xy data

you demonstrate the connections between the regression slope and the correlation coefficient

you compute the degrees of freedom associated with the error variation and the explained variation

you demonstrate the connections between the explained variation, the error variation, ANOVA and the coefficient of determination

you interpret the results of ANOVA applied to the regression fit

you plot graphs of the residuals

you inspect graphs of the residuals for deterministic deviations from a linear model
Apply statistical methods to a real experiment.

Assessment Strategies

in the solution to a problem on a quiz, homework, project or exam

Criteria

you acquire real data from an experiment

you apply statistical processes and reasoning to questions posed by acquired data

you communicate the hypotheses tested

you communicate the procedure for data acquisition

you communicate the statistical methods used

you communicate the conclusions reached

you communicate any underlying assumptions made about the nature of the data, and the self consistency or reasonableness of such assumptions