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Statistics II

Course Number: MTH 244
Transcript Title: Statistics II
Created: September 1, 2012
Updated: December 19, 2014
Total Credits: 4
Lecture Hours: 40
Lecture / Lab Hours: 0
Lab Hours: 0
Satisfies Cultural Literacy requirement: No
Satisfies General Education requirement: Yes
Grading options: A-F (default), P-NP, audit

Prerequisites

MTH 243

Course Description

Topics include confidence interval estimation; tests of significance including z-tests, t-tests, ANOVA, and chi-square; and inference for linear regression. Applications are investigated from science, business, and social science perspectives. TI graphing calculator with advanced statistical programs required and/or computer software, see instructor. Prerequisites: MTH 243 and its prerequisite requirements. Audit available.

Intended Outcomes

Upon successful completion students should be:

  1. Critically analyze the data from observational studies, such as surveys and experiments where treatments are deliberately imposed on the subjects, and using appropriate statistical methods and technology, judge if the results are reasonable, and then interpret and clearly communicate the results.
  2. Interpret studies in scholarly and scientific publications and make sense of statistical information provided by the media.
  3. Appreciate probability and statistics concepts that are encountered in the real world, understand and be able to communicate the underlying mathematics involved to help another person gain insight into the situation.
  4. Have sufficient command of the science of reasoning from data and correct mathematical terminology, notation, and symbolic processes in order to engage in work, study, and other applications that require the use of and an understanding of the concepts of statistics in a data-based setting.

Outcome Assessment Strategies

Assessment must include:

  1. At least two in-class or proctored examinations, and
  2. At least two of the following additional measures:
    1. Take-home examinations
    2. Graded homework/worksheets
    3. Quizzes
    4. Writing assignments
    5. Group/individual projects
    6. In-class activities

Course Activities and Design

All activities will follow the premise that formal definitions and procedures evolve from the investigation of practical problems. Concepts will be introduced using lecture, group activities, calculator programs, and computer laboratory explorations. Students will communicate their results verbally and in writing.

Course Content (Themes, Concepts, Issues and Skills)

1.0 RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS
The instructional goal is to explore and analyze various random variables and probability distributions.

1.1 Explore probability distributions:

1.1.1 Normal.
1.1.2 Student’s t.
1.1.3 F.
1.1.4 Chi-square.

2.0 ESTIMATION: ONE SAMPLE
The instructional goal is to estimate a population parameter by calculating a confidence interval.  

2.1 Identify and describe terminology:

2.1.1 Confidence coefficient.
2.1.2 Confidence level.
2.1.3 Point estimator.

2.2 Check the conditions associated with a confidence interval for a population mean.
2.3 Check the conditions associated with a confidence interval for a population proportion.
2.4 Calculate and interpret a confidence interval for a population mean.
2.5 Calculate and interpret a confidence interval for a population proportion.
2.6 Calculate a sample size to attain a desired margin of error and confidence level.
2.7 Using technology, input a sample and execute the commands to create a confidence interval; interpret the output.

3.0 STATISTICAL INFERENCE: ONE SAMPLE
The goal is to utilize sample information to test whether a population parameter is less than, not equal to, or greater than a specified value.

3.1 Perform a two-sided test using:

3.1.1 A test of significance.
3.1.2 A confidence interval.

3.2 Calculate sample size.
3.3 Identify and describe terminology:

3.3.1 Null and alternative hypotheses.
3.3.2 Test statistic.
3.3.3 Type I and Type II errors,  and .
3.3.4 Observed significance level: P-value.

3.4 Check the conditions associated with a test of significance about a population mean.
3.5 Check the conditions associated with a test of significance about a population proportion.
3.6 Construct and interpret a z-test about a population mean.
3.7 Construct and interpret a t-test about a population mean.
3.8 Construct and interpret a z-test about a population proportion.
3.9 Using technology, input a sample and execute the commands to perform a t-test or a z-test; interpret the output.
3.10 Calculate and interpret the power of a z-test.

4.0 ESTIMATION AND STATISTICAL INFERENCE: TWO SAMPLES
The instructional goal is to utilize sample information to infer whether a difference exists between two population means or two population proportions.

4.1 Perform a two-sided test using:

4.1.1 A test of significance.
4.1.2 A confidence interval.

4.2 Check the conditions associated with a confidence interval or test of significance about the difference between two population means using two independent samples.
4.3 Check the conditions associated with a confidence interval or test of significance about the difference between two population proportions using two independent samples.
4.4 Check the conditions associated with a confidence interval or test of significance about the mean difference between two populations using paired samples.
4.5 Construct and interpret a confidence interval about the difference between two population means using two independent samples.
4.6 Construct and interpret a t-test about the difference between two population means using two independent samples.
4.7 Construct and interpret a confidence interval about the difference between two population proportions using two independent samples.
4.8 Construct and interpret a z-test about the difference between two population proportions using two independent samples.
4.9 Construct and interpret a confidence interval about the mean difference between two populations using paired samples.
4.10 Construct and interpret a t-test about the mean difference between two populations using paired samples.
4.11 Using technology, input two independent samples and execute the commands to perform a two-sample difference of means confidence interval; interpret the output.
4.12 Using technology, input two independent samples and execute the commands to perform a two-sample difference of means test; interpret the output.
4.13 Using technology, input two independent samples and execute the commands to perform a two-sample difference of proportions confidence interval; interpret the output.
4.14 Using technology, input two independent samples and execute the commands to perform a two-sample difference of proportions test; interpret the output.
4.15 Using technology, input two paired samples and execute the commands to perform a one-sample confidence interval; interpret the output.
4.16 Using technology, input two paired samples and execute the commands to perform a one-sample t-test; interpret the output.

5.0 ANALYSIS OF VARIANCE (ANOVA)
The instructional goal is to design and analyze a sampling experiment to compare the means of more than two populations.

5.1 Identify and describe terminology:

5.1.1 Response (dependent) variable.
5.1.2 Factor (independent variable, stimulus).
5.1.3 Levels (treatments) of a factor.
5.1.4 Sum of squares for treatments (SST) and error (SSE).
5.1.5 Mean square for treatments (MST) and error (MSE).

5.2 Check the conditions associated with an ANOVA test.
5.3 Compare the treatment means.
5.4 Summarize the results of the F test in an ANOVA table.
5.5 Using technology, input sample data and execute the commands to perform ANOVA; interpret the output.
5.6 Use a multiple comparisons method to determine which pairs of means differ; interpret the results.

6.0 CHI-SQUARE TESTS AND CONTINGENCY TABLES
The instructional goal is to explore a non-parametric procedure on categorical variables.

6.1 Identify and describe terminology:

6.1.1 Multinomial probabilities.
6.1.2 Contingency table.
6.1.3 Marginal probabilities.

6.2 Check the conditions associated with a goodness-of-fit test.
6.3 Check the conditions associated with a test of independence.
6.4 Check the conditions associated with a test of homogeneity.
6.5 Perform a goodness-of-fit test about the probability distribution of a random variable.
6.6 Determine whether two classifications of nominal data are independent using a contingency table, multinomial probabilities, and a chi-square test.
6.7 Using technology, input sample data, choose commands to perform an appropriate chi-square test; interpret the output.

7.0 SIMPLE LINEAR REGRESSION AND CORRELATION
The instructional goal is to explore a straight-line relationship between two random variables, and use the least-squares line as a basis for inference about a population from which our observations are a sample.

7.1 Identify the explanatory variable and the response variable.
7.2 Check the conditions associated with constructing a least-squares linear regression model.
7.3 Construct a scatterplot of the sample data.
7.4 Identify the least-squares estimates of the intercept and the slope (the parameters) of the population regression model.
7.5 Specify the probability distribution of the random error term, and estimate the standard deviation of this distribution.
7.6 Evaluate the utility of the model:

7.6.1 Conduct a test of significance to determine whether the data provide sufficient evidence to indicate that the explanatory variable contributes information for the linear prediction of the response variable.
7.6.2 Construct and interpret a confidence interval to estimate the slope of the population regression model.
7.6.3 Calculate and interpret the sample correlation coefficient .
7.6.4 Calculate and interpret the coefficient of determination .

7.7 Use the least-squares line for estimation and prediction:

7.7.1 Construct and interpret a confidence interval for the mean value of the response value when the explanatory variable takes on a specific value.
7.7.2 Construct and interpret a prediction interval for an individual value of the response value when the explanatory variable takes on a specific value.

7.8 Using technology, input sample data and execute the commands to produce a least-squares regression equation, a fitted line, a residual plot, and ; interpret the output.

ADDENDUM TO COURSE CONTENT:
Recommended:

1.0 PLUS FOUR ESTIMATES
The instructional goal is to explore alternative procedures for confidence intervals for proportions when sample sizes are small.

1.1 Check the conditions associated with plus four estimates of a single population proportion and plus four estimates of a difference in two population proportions
1.1 Calculate and interpret a plus four estimate of a single population proportion, and a plus four estimate of the difference in two population proportions.
If time permits, the instructor may supplement the core required competencies with no more than two of the following topics.

2.0 TWO-WAY ANOVA
The instructional goal is to classify several populations according to two categorical variables and compare the means. Reduce the residual variation in a model by including a second factor thought to influence the response.

2.1 Compare the treatment means using:

2.1.1 Randomized block design.
2.1.2 Factorial design.

2.1.2.1 Create and interpret an interaction plot.
2.1.2.2 Conduct and interpret a significance test on interaction.
2.1.2.3 Conduct and interpret significance tests on the main effects of each factor on the response variable.

2.2 Using technology, input sample data and execute the commands to perform two-way ANOVA; interpret the output.

3.0 STATISTICAL PROCESS CONTROL
The instructional goal is to explore process-control techniques to improve quality and reduce waste.

3.1 Identify and describe terminology:

3.1.1 Upper and lower control limits (UCL and LCL).
3.1.2 Consumer and producer risks.

3.2 Construct control charts for measurement data, such as time elapsing from the start of a process to its end.

3.2.1 Plot and interpret an x-bar chart.
3.2.2 Plot and interpret an s-chart.
3.2.3 Plot and interpret an R-chart.

3.3 Construct control charts for attribute (count) data, such as the number of purchase orders that contain one or more errors by plotting and interpreting a p-chart.
3.4 Explore acceptance sampling by constructing and interpreting an OC curve.
3.5 Using technology, input sample data and execute the commands to produce quality-control charts; interpret the output.

4.0 TIME-SERIES ANALYSIS
The instructional goal is to analyze data that are collected over time and to explore methods for forecasting.

4.1 Compute and interpret an index number.

4.1.1 Laspeyres.
4.1.2 Paasche.
4.1.3 Fisher.

4.2 Study the Consumer Price Index.
4.3 Perform time-series analysis.

4.3.1 Identify and describe terminology:

4.3.1.1 Long-term trend.
4.3.1.2 Seasonal, cyclical, and irregular variation.

4.3.2 Construct moving averages to smooth out random variation.
4.3.3 Determine the linear trend equation to show steady upward (downward) movement.
4.3.4 Determine a set of seasonal indexes.
4.3.5 Deseasonalize data using seasonal indexes.
4.3.6 Determine a seasonally adjusted forecast.

4.4 Input sample data and execute the commands to display a time-series plot, determine a linear trend equation, and display estimates for future time periods; interpret the output.

5.0 DECISION THEORY
The instructional goal is to explore the economic consequences and the probabilities of realizing them to determine which of several courses of action is best.

5.1 Identify and describe terminology:

5.1.1 Decision alternatives (actions).
5.1.2 Payoff values (utility).
5.1.3 Expected payoff values.
5.1.4 Minimax, maximin, minimum, and maximax strategies.
5.1.5 EVPI.

5.2 Portray a decision analysis problem.

5.2.1 Construct and interpret a decision tree.
5.2.2 Construct and interpret a payoff table.
5.2.3 Construct and interpret an opportunity loss table.

6.0 ADDITIONAL NONPARAMETRIC STATISTICS
The instructional goal is to explore distribution-free techniques for a test of central tendency, a test of the difference between two sampled populations, a test of the association between two variables, and a test for comparing more than two populations.  

6.1 Conduct and interpret the Sign test for a population median.
6.2 Conduct and interpret the Wilcoxon rank test for comparing two independent samples.
6.3 Conduct and interpret the Wilcoxon rank test for comparing two paired samples.
6.4 Calculate the Spearman rank correlation coefficient between two sets of ranked data.
6.5 Conduct and interpret the Kruskal-Wallis rank test for one-factor ANOVA.
6.6 Conduct and interpret the Friedman rank test for two-factor ANOVA.
6.7 Using technology, input sample data and execute the commands to perform an appropriate nonparametric test; interpret the output.

Department Notes

This is the second term of a two-term sequence (MTH 243 and MTH 244). This course is intended to provide an introduction to statistics in a data-based setting.