Ch15 data exploration (ii)
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This document discusses key concepts in inferential statistics, including sampling distributions, hypothesis testing, significance, effect size, power, chi-square tests, analysis of variance (ANOVA), and t-tests. It explains that inferential statistics are used to draw conclusions about populations based on samples, and covers topics like the standard error of the mean, properties of sampling distributions, null and alternative hypotheses, one-tailed and two-tailed tests, and how to interpret ANOVA tables.
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PAGE O&M Statistics – Inferential Statistics Hypothesis Test.docx
PAGE
O&M Statistics – Inferential Statistics: Hypothesis Testing
Inferential Statistics
Hypothesis testing
Introduction
In this week, we transition from confidence intervals and interval estimates to hypothesis testing, the basis for inferential statistics. Inferential statistics means using a sample to draw a conclusion about an entire population. A test of hypothesis is a procedure to determine whether sample data provide sufficient evidence to support a position about a population. This position or claim is called the alternative or research hypothesis.
“It is a procedure based on sample evidence and probability theory to determine whether the hypothesis is a reasonable statement” (Mason & Lind, pg. 336).
This Week in Relation to the Course
Hypothesis testing is at the heart of research. In this week, we examine and practice a procedure to perform tests of hypotheses comparing a sample mean to a population mean and a test of hypotheses comparing two sample means.
The Five-Step Procedure for Hypothesis Testing (you need to show all 5 steps – these contain the same information you would find in a research paper – allows others to see how you arrived at your conclusion and provides a basis for subsequent research).
Step 1
State the null hypothesis – equating the population parameter to a specification. The null hypothesis is always one of status quo or no difference. We call the null hypothesis H0 (H sub zero). It is the hypothesis that contains an equality.
State the alternate hypothesis – The alternate is represented as H1 or HA (H sub one or H sub A). The alternate hypothesis is the exact opposite of the null hypothesis and represents the conclusion supported if the null is rejected. The alternate will not contain an equal sign of the population parameter.
Most of the time, researchers construct tests of hypothesis with the anticipation that the null hypothesis will be rejected.
Step 2
Select a level of significance (α) which will be used when finding critical value(s).
The level you choose (alpha) indicates how confident we wish to be when making the decision.
For example, a .05 alpha level means that we are 95% sure of the reliability of our findings, but there is still a 5% chance of being wrong (what is called the likelihood of committing a Type 1 error).
The level of significance is set by the individual performing the test. Common significance levels are .01, .05, and .10. It is important to always state what the chosen level of significance is.
Step 3
Identify the test statistic – this is the formula you use given the data in the scenario. Simply put, the test statistic may be a Z statistic, a t statistic, or some other distribution. Selection of the correct test statistic will depend on the nature of the data being tested (sample size, whether the population standard deviation is known, whether the data is known to be normally distributed).
The sampling distribution of the test statistic is divided into t.
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1
To make tests of hypotheses about more than two population means, we use the:
t distribution
normal distribution
chi-square distribution
analysis of variance distribution
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- 1. Research Methods, 9th Edition Theresa L. White and Donald H. McBurney Chapter 15 Data Exploration Part 2: Inferential Statistics
- 2. 22 Some basic terms Empirical data Facts derived from experience Population All members of some group Sample A subset of a population Statistic A quantity computed from a sample Parameter A quantity computed from a population
- 3. 33 Statistics Descriptive Statistics Summarize a set of data Chapter 14 Inferential Statistics Assist in drawing conclusions about populations by examining a sample drawn from the population.
- 4. 44 Sampling Distributions Distribution of means of samples from a population Standard Error of the Mean Standard Deviation of a sampling distribution
- 5. 55 Sampling Distributions Three properties: The sampling distribution has the same mean as the original distribution. The sampling distribution has a smaller standard deviation than the population distribution. The larger the size of the samples that are drawn from the population, the smaller the standard deviation of the sample distribution.
- 6. 66 Sampling Distributions The standard error of the mean is the standard deviation of the population divided by the square root of the sample size The sample size becomes larger, the shape of the distribution approaches a normal distribution, regardless of the shape of the population from which the samples are drawn.
- 7. 77 Hypothesis Testing H1 (Alternative) vs H0 (Null)
- 8. 88 Hypothesis Testing H1 (Alternative) vs H0 (Null)
- 9. 99 Hypothesis Testing H1 (Alternative) vs H0 (Null)
- 10. 1010 Hypothesis Testing H1 (Alternative) vs H0 (Null)
- 11. 1111 Hypothesis Testing Directional hypothesis An alternative hypothesis that predicts that the results of one condition will be greater (or less) than another, rather than a prediction that they will simply differ.
- 15. 1515 Hypothesis Testing One-tailed hypothesis test Statistical test of a directional hypothesis Two-tailed hypothesis test Statistical test of a nondirectional hypothesis
- 18. 1818 Hypothesis Testing Statistical significance is the probability that a result happened by chance Alpha is the probability of deciding that the null hypothesis is false when it’s actually true. Probability of a Type I error
- 23. 2323 Significance Probability that an experimental result happened by chance Generally alpha less than .05 (statistical significance) Does not necessarily mean that the result was important or large
- 24. 2424 Significance The significance of significance Size of result is measured with Effect Size Effect size shows the strength of the relationship between the independent and dependent variables. Be measured by practice or by Cohen’s d (eta- squared)
- 26. 2626 Power of the Test of the Null Hypothesis Against the Alternative Power is the probability of rejecting the null hypothesis when it actually IS false
- 27. 2727 Power of the Test of the Null Hypothesis Against the Alternative Power Three things influence the power of a test The value of alpha The smaller your alpha level, the smaller your power. Experimental error Sample sizes
- 28. 2828 Chi Square Statistic Tests Frequency Data to determine whether two categorical variables are related. Expected Frequency vs Observed Frequency
- 29. 2929 Chi Square Statistic contingency table also referred to as cross tabulation or crosstab In statistics, a contingency table is a type of table in a matrix format that displays the (multivariate) frequency distribution of the variables.
- 31. 3131 ANOVA Compares more than two conditions For only two, use a t-test Tests the significance of a difference among several conditions in an experiment by making two different estimates of the variability that would be expected if the null hypothesis is true. Between variability Within variability
- 32. 3232 T test
- 33. 3333 ANOVA Ha: at least two means differ (Note the alternative hypothesis is sometimes stated as "at least one mean differs")
- 34. 3434 ANOVA
- 35. 3535 ANOVA
- 36. 3636 How to Read an ANOVA Table Between Subjects Single Factor Design
- 37. 3737 How to Read an ANOVA Table Within Subjects Single Factor Design
- 38. 3838 How to Read an ANOVA Table Between Subjects Factorial Design