HSE230 Chapter 9: Introduction to T-testing

t Distribution

- a family of distributions, one for each value of degrees of freedom -approximates the shape of normal distribution --flatter than normal distribution --more spread out than normal distribution --more variability (fatter tails) in t distribution -to find proportion for t-statistics, use Table B.2 in place of Unit Normal table

t Statistic

-Alternative to z -used to test hypotheses about an unknown population mean, μ, when the value of σ is unknown t=M-μ/sm

Learning Check

-When n is small (less than 30), the t distribution is flatter &more spread out than normal z-distribution

Factors that Influence a Hypothesis Test with the t Statistic

-an increase in the significance level (such as using a=.05 instead of a=.01) the t-statistic stays the same -a decrease in the obtained difference (M-μ) decreases the t-statistic -a switch from using a two-tailed test to a one-tailed test the t-statistic stays the same -an increase in the sample variance (s^2) decreases the t-statistic -an increase the sample standard deviation decreases the t-statistic -an increase in sample size increases the t-statistic

Degrees of freedom

-considers all variance of sample scores except one, the sample mean. --only n-1 scores in a sample are independent &free to vary --researchers call n-1 the degrees of freedom -noted as df -df=n-1 (#of scores/observations-1) -the greater the df for a sample, the better the s^2 represents the σ^2 & better the t statistic approximates the z score.

Percentage of variance explained

-determining the amount of variability in scores explained by the treatment effect is an alternative method for measuring effect size. r^2=variability accounted for/ total variabililty= t^2/t^2+df -r^2=0.01 small effect r^2=0.09 medium effect r^2=0.25 large effect

Hypothesis Tests with the t Statistic

-null hypothesis for the one-sample t test statistic --H0:M=μ t=sample mean- pop. mean/estimated standard error= M-μ/sm=0 -Alternative hypothesis for the one-sample t test statistic --H1: M doesn't equal μ

To be able to reject the Null Hypothesis

-obtained statistical value>critical value --ex. Zobt > Zcrit ---Zobt-> the obtained statistical value ---Zcrit->the critical boundary based upon the alpha level set by the researcher -p-value < alpha level --ex. p<a --- p->based upon obtained statistic, the probability of Type II error ---a->researcher's accepted probability of Type II error

Cohen's d for the t-test

-original equation included pop. parameters -estimated Cohen's d is computed using the sample standard deviation estimated d=mean difference/sample standard deviation= M-μ/s

Measuring Effect Size

-recall that effect size is the absolute magnitude of a treatment effect -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 -results from a hypothesis test should be accompanied by a measure of effect size

Single Sample t Statistic in Action Continued

-tobt=M-μ/sm = 79-70/4= 9/4=2.25 tobt=2.25 -use a=.05(2 tail0 &find the tcrit --df=24->tcrit=2.064 --reject the null as t(24)=2.25, p<.05 -but what is the size of effect? -Cohen's d= M-μ/s= 79-70/20=9/20= 0.45=medium effect using r^2, what is % of variance accounted for by the treatment? -r^2=t^2/t^2+df=2.25^2/2.25^2+24=5.0625/5.0625+24=.17=17%=medium effect Reporting the results: -reject the null as t(24)=2.25, p<.05,d=0.45 -reject the null as t(24)=2.25, p<.05, r^2=0.17

Z-score Statistic

-use z-score statistic to quantify inferences about the pop. Z=M-μ/σm=obtained difference between data &hypothesis/ standard distance between M &μ -use unit normal table to find critical region if z-scores form a normal distribution --when n>=30 OR when the original distribution is approximately normally distributed -requires more info than researchers typically have available -requires knowledge of pop standard deviation

Estimated Standard Error (sm) for t Tests

-uses sample variance (s^2) to estimate the population variance (σ^2) estimated standard error=sm=s/square root n or square root of s^2/n -estimate standard error is used as an estimate of the real standard error when the value of σm is unknown --provides an estimated distance between M and μ

Hypothesis Testing

-using sample data to evaluate a hypothesis about a population: --alpha level:boundaries that separate the unlikely from the likely; converted to proportion is the critical boundary --Critical Region:all the values that are unlikely to occur from chance alone;values that fall beyond the boundary defined by the alpha level

Assumptions of the t test

-values in the sample are independent observations -the population sample must be normal --with large samples, this assumption can be violated without affecting the validity of the hypothesis test

Four steps for Hypothesis Testing

1. State the null & alternative hypotheses Select an alpha level 2. Locate the critical region using the t distribution table and df 3. Calculate the t test statistic 4. Make a decision regarding H0

Finding Critical Region:

1. What is the critical value for a 2-tailed t test with n=12 using a=.01? Answer: tcrit=3.106 2.What is the critical value for a 1-tailed t test with n=25 using a=.05? Answer:tcrit=1.711 3. What is the critical calue for a 2-tailed t test with n=65 using a=.05? Answer: tcrit=2.00

Examples:

1. You are planning to evaluate the mean of a single continuous variable from a study with a sample of n=45 using the t statistic. What are degrees of freedom for the sample? Answer: 44 2. With another study study, where you also plan on evaluating a mean using the t statistic, you have a sample of n=41 that has an SS of 600. What is the variance for the sample? Answer: 15 3. For a sample of n=64 that has a sample variance of 1600, what is the estimated standard error for the sample? Answer: 5

Single Sample t Statistic in Action

A random sample is obtained from a pop. with a mean of μ=70. A treatment is administered n=25 individuals in the sample &after treatment, the sample mean is M=79 w/a standard deviation of s=20 -was there a difference in the sample treatment mean from the pop mean? Use a=.05 -use t equation to find tobt= M-μ/sm -first we need to find estimated standard error s/square root n -sm=s/square root n= 20/square root 25= 20/5= 4.00

Ex.

Critical region in the t distribution for a=.05 and df=8 -ALL the shaded portion in the tail(s) is the critical region -+/- 2.306 is the tcrit, also known as critical boundary

Using the Table for t Distribution

Let's consider the Output of SPSS for a One-Sample t Test: 1. Assume we are testing an IQ hypothesis with an a=0.05 (2 tail) 2. Find the df in the output and then in Table B.2. 3.Look up the tcrit for df=9(row) with .05 proportion in two tails combined (column) 4. You should have found tcrit=2.262

Example

μ=56 M=59 n= 35 SS=2400 d=.05 df=24 s^2= SS/n-1=2400/24 s^2=100 s=square root of s^2 s=10 sm=square root of s^2/n= square root of 100/25 =10/5 sm=2 tobt= M-μ/sm =59-56/2 tobt=1.50 fail to reject the null

Basic Equation for a Confidence Interval

μ=M+- t(sm)