Chapter9-Tagged.pdf

Chapter 9 Learning Outcomes

• Know when to use t statistic instead of z-score hypothesis test

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• Perform hypothesis test with t-statistics

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Tools You Will Need

• Sample standard deviation (Chapter 4)

• Standard error (Chapter 7)

• Hypothesis testing (Chapter 8)

9.1 The t statistic: An alternative to z

• Sample mean (M) estimates (& approximates) population mean (μ)

• Standard error describes how much difference is reasonable to expect between M and μ.

• either or

nM

 

nM

2 

• Use z-score statistic to quantify inferences about the population.

• Use unit normal table to find the critical region if z-scores form a normal distribution– When n ≥ 30 or

– When the original distribution is approximately normally distributed

Reviewing the z-Score Statistic





and Mbetween distance standard

hypothesis and databetween difference obtained



M

Mz

The Problem with z-Scores

• The z-score requires more information than researchers typically have available

• Requires knowledge of the population standard deviation σ

• Researchers usually have only the sample data available

Introducing the t Statistic

• t statistic is an alternative to z.

• t might be considered an “approximate” z.

• Estimated standard error (sM) is used as in

place of the real standard error when the value of σM is unknown.

Estimated standard error

• Use s2 to estimate σ2.

• Estimated standard error:

• Estimated standard error is used as estimate of the real standard error when the value of σM is

unknown.

n

sor error standard

2

n

ssestimated M 

The t-Statistic

• The t statistic uses the estimated standard error in place of σM.

• The t statistic is used to test hypotheses about an unknown population mean μ when the value of σ is also unknown.

Ms

Mt



Degrees of Freedom

• Computation of sample variance requires computation of the sample mean first– Only n-1 scores in a sample are independent

– Researchers call n-1 the degrees of freedom

• Degrees of freedom– Noted as df

– df = n-1

Figure 9.1 Distributions of the t Statistic

The t Distribution

• Family of distributions, one for each value of degrees of freedom

• Approximates the shape of the normal distribution– Flatter than the normal distribution

– More spread out than the normal distribution

– More variability (“fatter tails”) in t distribution

• Use Table of Values of t in place of the Unit Normal Table for hypothesis tests

Figure 9.2 The t Distribution for df=3

9.2 Hypothesis Tests with the t Statistic

• The one-sample t test statistic (assuming the null hypothesis is true)

0error standard estimated

mean population -mean sample



Ms

Mt



Figure 9.3 Basic Research Situation for t Statistic

Hypothesis Testing: Four Steps

• State the null and alternative hypotheses andselect an alpha level

• Locate the critical region using the t distribution table and df

• Calculate the t test statistic

• Make a decision regarding H0 (null hypothesis)

Figure 9.4 Critical Region in the t

Distribution for α = .05 and df = 8

Assumptions of the t Test

• Values in the sample are independent observations.

• The population sampled must be normal– With large samples, this assumption can be

violated without affecting the validity of the hypothesis test

Influence of Sample Size and Sample Variance

• The larger the sample, the smaller the error.

• The larger the variance, the larger the error.

Learning Check 1 (slide 1 of 4)

• When n is small (less than 30), the t distribution ______.

• is almost identical in shape to the normal z distributionAA

• is flatter and more spread out than the normal z distributionBB

• is taller and narrower than the normal z distributionCC

• cannot be specified, making hypothesis tests impossibleDD

Learning Check 1 – Answer (slide 2 of 4)

• When n is small (less than 30), the t distribution ______.

• is almost identical in shape to the normal z distributionAA

• is flatter and more spread out than the normal z distributionBB

• is taller and narrower than the normal z distributionCC

• cannot be specified, making hypothesis tests impossibleDD

Learning Check 1 (slide 3 of 4)

• Decide if each of the following statements is True or False.

• By chance, two samples selected from the same population have the same size (n = 36) and the same mean (M = 83). That means they will also have the same t statistic.

T/FT/F

• Compared to a z-score, a hypothesis test with a t statistic requires less information about the population.

T/FT/F

Learning Check 1 – Answers (slide 4 of 4)

• The two t values are unlikely to be the same; variance estimates (s2) differ between samples.

False

False

• The t statistic does not require the population standard deviation; the z-test does.

TrueTrue

9.3 Measuring Effect Size

• Hypothesis test determines whether the treatment effect is greater than chance– No measure of the size of the effect is included

– A very small treatment effect can be statistically significant

• Therefore, results from a hypothesis test should be accompanied by a measure of effect size.

Estimated Cohen’s d

• Original equation included population parameters

• Estimated Cohen’s d is computed using the sample standard deviation

s

M

deviation standardsample

difference meand estimated



Figure 9.5 Distribution for Examples 9.1 & 9.2

Percentage of Variance Explained

• Determining the amount of variability in scores explained by the treatment effect is an alternative method for measuring effect size.

• r2 = 0.01 small effect

• r2 = 0.09 medium effect

• r2 = 0.25 large effect

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t

yvariabilit total

for accountedy variabilitr

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22

Figure 9.6 Deviations with and without the Treatment Effect

Confidence Intervals for Estimating μ (slide 1 of 3)

• Alternative technique for describing effect size

• Estimates μ from the sample mean (M)

• Based on the reasonable assumption that M should be “near” μ

• The interval constructed defines “near” based on the estimated standard error of the mean (sM)

• Can confidently estimate that μ should be located in the interval

Confidence Intervals for Estimating μ (slide 2 of 3)

• Every sample mean has a corresponding t:

• Rearrange the equations solving for μ:

Ms

Mt



MtsM 

Confidence Intervals for Estimating μ (slide 3 of 3)

• In any t distribution, values pile up around t = 0.

• For any α we know that (1 – α ) proportion of t values fall between ± t for the appropriate df.

• E.g., with df = 9, 90% of t values fall between ±1.833 (from the t distribution table, α = .10).

• Therefore, we can be 90% confident that a sample mean corresponds to a t in this interval.

Figure 9.7t Distribution with df = 8

Confidence Intervals for Estimating μ (continued)

• For any sample mean M with sM

• Pick the appropriate degree of confidence (80%? 90%? 95%? 99%?) 90%

• Use the t distribution table to find the value of t (For df = 9 and α = .10, t = 1.833)

• Solve the rearranged equation

• μ = M ± 1.833(sM)

• Resulting interval is centered around M

• 90% confident that μ falls within this interval

Factors Affecting Width of Confidence Interval

• Confidence level desired

• More confidence desired increases interval width

• Less confidence acceptable decreases interval width

• Sample size• Larger sample smaller SE smaller interval

• Smaller sample larger SE larger interval

In the Literature

• Report whether (or not) the test was “significant”

• “Significant”  H0 rejected

• “Not significant”  failed to reject H0

• Report the t statistic value including df, e.g., t(12) = 3.65

• Report significance level, either:• p < alpha, e.g., p < .05 or

• Exact probability, e.g., p = .023

9.4 Directional Hypotheses and One-Tailed Tests

• Non-directional (two-tailed) test is most commonly used

• However, directional test may be used for particular research situations

• Four steps of hypothesis test are carried out– The critical region is defined in just one tail of the t

distribution.

Figure 9.8 Example 9.6One-Tailed Critical Region

Learning Check 2 (slide 1 of 4)

• The results of a hypothesis test are reported as follows: t(21) = 2.38, p < .05. What was the statistical decision and how big was the sample?

• The null hypothesis was rejected using a sample of n = 21AA

• The null hypothesis was rejected using a sample of n = 22BB

• The null hypothesis was not rejected using a sample of n = 21CC

• The null hypothesis was not rejected using a sample of n = 22DD

Learning Check 2 – Answer(slide 2 of 4)

• The results of a hypothesis test are reported as follows: t(21) = 2.38, p < .05. What was the statistical decision and how big was the sample?

• The null hypothesis was rejected using a sample of n = 21AA

• The null hypothesis was rejected using a sample of n = 22BB

• The null hypothesis was not rejected using a sample of n = 21CC

• The null hypothesis was not rejected using a sample of n = 22DD

Learning Check 2 (slide 3 of 4)

• Decide if each of the following statements is True or False

• Sample size has a great influence on measures of effect size.

T/FT/F

• When the value of the t statistic is near 0, the null hypothesis should be rejected.

T/FT/F

Learning Check 2 – Answers (slide 4 of 4)

• Measures of effect size are not influenced to any great extent by sample size.

False

False

• When the value of t is near 0, the difference between M and μ is also near 0.

False

False