Chapter 7 and 10 Hypothesis Testing: One-Sample Case

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Questions and Answers

What is the difference between a one-sample z-test and a one-sample t-test?

A one-sample z-test is used when the population standard deviation is known, while a one-sample t-test is used when the population standard deviation is unknown.

What is the difference between a one-tailed test and a two-tailed test?

A one-tailed test is used when the researcher is interested in a specific direction of the difference between the sample and population means, while a two-tailed test is used when the researcher is interested in any difference between the sample and population means, regardless of direction.

What is the critical region?

The critical region is the portion of the sampling distribution that contains unlikely sample outcomes, and the size of the critical region is determined by the alpha level.

What is the difference between statistical significance and effect size?

<p>Statistical significance tells us how confident we are that a relationship exists, while effect size tells us how strong the relationship is.</p> Signup and view all the answers

What are the criteria for a large sample size in hypothesis testing?

<p>For a large sample size, both nP and n(1 - P) must be greater than or equal to 15, where P refers to the value in the null hypothesis.</p> Signup and view all the answers

What are the five steps of hypothesis testing?

<p>The five steps of hypothesis testing are stating the null hypothesis, stating the research hypothesis, selecting the sampling distribution and establishing the critical region, calculating the test statistic, and deciding whether to reject or fail to reject the null hypothesis.</p> Signup and view all the answers

What is a pooled estimate of the standard deviation?

<p>The pooled estimate of the standard deviation is a weighted average of the standard deviations of two samples, used to estimate the standard deviation of the population.</p> Signup and view all the answers

What the three assumptions needed for hypothesis testing?

<p>The three assumptions needed for hypothesis testing are random sampling, normal distribution, and either the population standard deviation is known or the sample size is large.</p> Signup and view all the answers

What is the purpose of a post hoc test?

<p>A post hoc test is used to determine which specific group means are significantly different from each other when the null hypothesis is rejected in a one-way ANOVA.</p> Signup and view all the answers

What is eta-squared (η^2)?

<p>Eta-squared is a measure of the strength of association in one-way ANOVA. It represents the proportion of the total variance in the dependent variable that is explained by the independent variable.</p> Signup and view all the answers

A sample of size 20 is considered a large sample size for testing proportions.

<p>False (B)</p> Signup and view all the answers

What is regression?

<p>Regression is a statistical technique used to predict the value of a dependent variable based on the value of an independent variable.</p> Signup and view all the answers

What three questions are addressed in regression analysis?

<p>The three questions addressed in regression analysis are: Is there a relationship between the variables? How strong is the relationship? What is the direction of the relationship?</p> Signup and view all the answers

What is the least-squares regression line?

<p>The least-squares regression line is the line that best fits the data by minimizing the sum of squared distances between the observed data points and the line. It is also known as the best-fitting straight line.</p> Signup and view all the answers

What is the slope of the regression line?

<p>The slope of the regression line (represented by b) is the amount of change produced in the dependent variable for each unit change in the independent variable.</p> Signup and view all the answers

What is the Y intercept of the regression line?

<p>The Y intercept of the regression line (represented by a) is the value of the dependent variable when the value of the independent variable is 0. It is the point where the regression line crosses the Y-axis.</p> Signup and view all the answers

What is Pearson’s r?

<p>Pearson’s r is a statistical measure that describes the strength and direction of the linear relationship between two interval-ratio level variables. Values of Pearson’s r range from -1.00 to +1.00, where 0.00 indicates no relationship, +1.00 indicates a perfect positive relationship, and -1.00 indicates a perfect negative relationship.</p> Signup and view all the answers

What is the coefficient of determination (r^2)?

<p>The coefficient of determination (represented by r^2) indicates the proportion of variance in the dependent variable that is explained by the independent variable. It is calculated by squaring Pearson’s r.</p> Signup and view all the answers

Why are dummy variables useful in regression?

<p>Dummy variables are useful in regression because they allow us to include nominal-level independent variables in the analysis, treating them as interval-ratio variables.</p> Signup and view all the answers

What is the ecological fallacy?

<p>The ecological fallacy occurs when we incorrectly infer a relationship between variables at the individual level based on observations at the aggregate level. The correlation between variables at the level of groups or populations may not reflect the same relationship between variables at the level of individuals.</p> Signup and view all the answers

What are the techniques used to analyze a relationship between two variables when both variables are measured at the interval-ratio level?

<p>Hypothesis tests and measures of association or correlation.</p> Signup and view all the answers

When analyzing a relationship between two variables, what is the usual first step?

<p>To construct and examine a scatterplot.</p> Signup and view all the answers

What does the regression line represent in a relationship between two variables?

<p>The line that best fits the pattern of the data points on the scatterplot. (B)</p> Signup and view all the answers

What is the Y intercept in a least-squares regression line?

<p>The point at which the regression line crosses the Y axis.</p> Signup and view all the answers

What does the slope in a least-squares regression line measure?

<p>The amount of change produced in the dependent variable (Y) by a unit change in the independent variable (X).</p> Signup and view all the answers

What is the purpose of the correlation coefficient (Pearson's r)?

<p>It measures the strength of the linear association between two interval-ratio variables.</p> Signup and view all the answers

Pearson's r can be used to predict scores on Y from scores on X.

<p>False (B)</p> Signup and view all the answers

What is the purpose of hypothesis testing for Pearson's r?

<p>To determine if the relationship between the variables can be assumed to exist in the population from which the sample was drawn</p> Signup and view all the answers

What are the assumptions that must be met to test the significance of Pearson’s r?

<p>Both variables must have bivariate normal distributions, the relationship between the variables should be linear in form, and the relationship should be homoscedastic.</p> Signup and view all the answers

What does homoscedasticity refer to?

<p>The variance of the Y scores is uniformed for all values of X.</p> Signup and view all the answers

What is the purpose of ANOVA?

<p>To test for the significance of the difference between the means of three or more categories or populations.</p> Signup and view all the answers

What is the assumption made in ANOVA?

<p>The populations from which the samples are drawn are equal.</p> Signup and view all the answers

Identify the components of the total variation of scores (SST).

<p>Sum of Squares Between (SSB) and Sum of Squares Within (SSW) (A)</p> Signup and view all the answers

What are the degrees of freedom for the Sum of Squares Within (SSW) and Sum of Squares Between (SSB)?

<p>The degrees of freedom for SSW is n-k, where n is the number of cases and k is the number of categories. The degrees of freedom for SSB is k-1.</p> Signup and view all the answers

How are the mean square estimates for the population variance calculated?

<p>By dividing the sum of squares for SSW and SSB by their respective degrees of freedom.</p> Signup and view all the answers

Explain what the F ratio represents in ANOVA.

<p>The ratio of the amount of variation between the categories to the amount of variation within the categories.</p> Signup and view all the answers

When is it appropriate to use post hoc tests?

<p>After rejecting the null hypothesis in ANOVA</p> Signup and view all the answers

What is the purpose of dummy variables in regression?

<p>To include independent variables measured at the nominal or ordinal level in regression analyses.</p> Signup and view all the answers

What is the key assumption when using dummy variables in regression?

<p>The dummy independent variablename a linear relationship with the dependent variable.</p> Signup and view all the answers

A large sample size is always necessary to conduct ANOVA or a hypothesis test for Pearson's r.

<p>False (B)</p> Signup and view all the answers

What is the purpose of a measure of association?

<p>To quantify the strength of the relationship between variables.</p> Signup and view all the answers

Flashcards

Hypothesis testing (one-sample)

A statistical method used to determine if a sample represents a population with a specific characteristic.

One-sample mean test

A hypothesis test used to determine if a single sample mean significantly differs from a known population mean.

One-tailed test

A hypothesis test where the critical region is in one tail of the sampling distribution, used when predicting the direction of the difference.

Two-tailed test

A hypothesis test where the critical region is split into two tails of the sampling distribution, used when the direction of the difference is unknown.

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Null hypothesis (H0)

The hypothesis of no difference between the sample and the population or that there is no effect.

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Research hypothesis (H1)

The hypothesis that proposes a difference between the sample and the population or an effect

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Sampling distribution

The probability distribution of all possible sample statistics calculated from samples of a given size.

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Standard Error

The standard deviation of the sampling distribution of a statistic

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Z-test

Used to test hypothesis with known population standard deviation

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t-test

Used to test hypothesis with unknown population standard deviation

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Degrees of freedom (df)

A parameter related to sample size that shapes the t-distribution

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Critical region

The range of values of a test statistic that leads to a rejection of the null hypothesis.

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Alpha (α)

Probability of rejecting the null hypothesis when it is actually true (significance level).

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Test statistic (z or t)

Computed from sample data to determine if it falls in the critical region.

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Confidence interval

A range of values likely to contain the population parameter.

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Type I error

Rejecting a true null hypothesis.

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Type II error

Failing to reject a false null hypothesis.

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Sample Proportion

Proportion of a sample that share a characteristic.

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Population Proportion

Proportion of the total population that shares a characteristic.

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One-sample case

A hypothesis test where you compare a single sample to a population.

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Hypothesis testing logic

Testing if a sample truly represents a population with a specific characteristic by assuming the opposite (null hypothesis) and checking for unlikely outcomes (critical region).

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One-Sample Proportion Test

Used to see if the proportion of a sample is significantly different from a known population proportion (assuming a large sample).

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What does a statistically significant outcome mean?

The difference or effect observed is unlikely to have occurred purely by random chance.

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How do you choose between a one-tailed and two-tailed test?

Use a one-tailed test if you know the direction of the difference, and a two-tailed test if you don't know the direction.

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What is the relationship between confidence interval and hypothesis testing?

If a confidence interval does not contain the value specified in the null hypothesis, then the null hypothesis is rejected at that alpha level.

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Why is the t-distribution used instead of the Z-distribution?

The t-distribution is used when the population standard deviation is unknown, while the Z-distribution is used when it is known.

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What are the steps involved in conducting a one-sample test?

  1. State the hypothesis (H0 and H1). 2. Choose alpha level. 3. Decide if one-tailed or two-tailed. 4. Choose the correct test (Z or t). 5. Calculate the test statistic. 6. Find the critical region. 7. Compare the test statistic to the critical region. 8. Make a decision and interpret.
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How does the size of the sample affect the test results?

A larger sample size generally leads to a more powerful test, making it easier to detect a significant difference or effect.

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Study Notes

Hypothesis Testing with Means and Proportions: The One-Sample Case

  • This chapter covers hypothesis testing for one sample mean and proportion.
  • A flowchart is provided to guide decisions on the appropriate test, depending on whether the population standard deviation (σ) is known or not.
  • The five-step model is used for hypothesis testing. The steps include: making assumptions, stating the null hypothesis, selecting the sampling distribution, computing the test statistic, and making a decision/interpreting results.
  • The appropriate sampling distribution (z or t) is selected according to whether the population standard deviation (σ) is known or unknown, and the sample size (n).
  • The logic of hypothesis testing as applied to the one-sample case is explained.
  • The significance of single-sample means and proportions are tested using the five-step model. The results are interpreted correctly.
  • The difference between one- and two-tailed tests is explained, and the appropriate test is specified.
  • A single-sample hypothesis test is conducted using a confidence interval.

Hypothesis Testing with Means and Proportions: The Two-Sample Case

  • This chapter covers hypothesis testing for two sample means and proportions.
  • A flowchart is provided to guide decisions on the appropriate test for two sample means.
  • The five-step model is used for hypothesis testing. The steps include: making assumptions, stating the null hypothesis, selecting the sampling distribution, computing the test statistic, and making a decision/interpreting results.
  • The appropriate sampling distribution (z or t) is selected according to whether the population standard deviation (σ) is known or unknown, and the combined sample size(n1+n2).
  • The logic of hypothesis testing as applied to the two-sample case is explained.
  • What an independent random sample is, is explained.
  • Tests of hypotheses for two sample means or two sample proportions are performed using the five-step model. The results are interpreted correctly.
  • The difference between statistical significance and the effect size of relationships between variables is explained.

Hypothesis Testing with More than Two Means: One-Way Analysis of Variance (ANOVA)

  • This chapter covers hypothesis testing for more than two sample means.
  • A flowchart is provided to guide decisions about the appropriate ANOVA test. The five-step model is used for the hypothesis test. The steps include making assumptions, stating the null hypothesis, selecting the sampling distribution, computing the test statistic, and making a decision/interpreting results.
  • The concepts of population variance, total sum of squares, sum of squares between, sum of squares within, and mean square estimates is described
  • Identifying significant differences between sample means using a post hoc test

Interpreting Statistics: Does Personal Well-Being Vary by Marital Status?

  • This chapter analyzes whether marital status has a relationship with personal well-being using a one-way ANOVA.
  • Describes different levels of satisfaction in happiness and life satisfaction
  • Explains the use of the five-step model in this context
  • Interprets the results statistically

Regression and Prediction

  • Explains the use of scatterplots to depict the relationship between variables
  • The least-squares regression line is described to aid in accurate predictions.
  • The relationship/direction and strength is examined through the slope and intercept
  • Demonstrates how to calculate the values using a computational tool

The Correlation Coefficient (Pearson's r)

  • It is a measure of association between two interval-ratio variables.
  • Describes how to calculate the correlation coefficient, r, r², and how to interpret them.
  • Explains situations where the variables are not linearly related, and possible solutions to these cases.

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Statistics: Hypothesis Testing with Means and Proportions (PDF)

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