Chapter 9 Introduction to the t-statistic PSY295 Spring 2003 Summerfelt

Overview CLT or Central Limit Theorem z-score Standard error t-score Degrees of freedom

Learning Objectives Know when to use the t statistic for hypothesis testing Understand the relationship between z and t Understand the concept of degrees of freedom and the t distribution Perform calculations necessary to compute t statistic Sample mean & variance estimated standard error for X-bar

Central limit theorem Based on probability theory Two steps Take a given population and draw random samples again and again Plot the means from the results of Step 1 and it will be a normal curve where the center of the curve is the mean and the variation represents the standard error Even if the population distribution is skewed, the distribution from Step 2 will be normal!

Z-score Review A sample mean (X-bar) approximates a population mean (μ) The standard error provides a measure of how well a sample mean approximates the population mean determines how much difference between X-bar and μ is reasonable to expect just by chance The z-score is a statistic used to quantify this inference obtained difference between data and hypothesis/standard distance expected by chance

What’s the problem with z? Need to know the population mean and variance!!! Not always available.

What is the t statistic? “Cousin” of the z statistic that does not require the population mean (μ) or variance (σ2)to be known Can be used to test hypotheses about a completely unknown population (when the only information about the population comes from the sample) Required: a sample and a reasonable hypothesis about the population mean (μ) Can be used with one sample or to compare two samples

When to use the t statistic? For single samples/groups, Whether a treatment causes a change in the population mean Sample mean consistent with hypothesized population mean For two samples, Coming later!

Difference between X-bar and μ Whenever you draw a sample and observe there is a discrepancy or “error” between the population mean and the sample mean difference between sample mean and population Called “Sampling Error” or “Standard error of the mean” Goal for hypothesis testing is to evaluate the significance of discrepancy between X-bar & μ

Hypothesis Testing Two Alternatives Is the discrepancy simply due to chance? X-bar = μ Sample mean approximates the population mean Is the discrepancy more than would be expected by chance? X-bar ≠ μ The sample mean is different the population mean

Standard error of the mean In Chapter 8, we calculated the standard error precisely because we had the population parameters. For the t statistic, We use sample data to compute an “Estimated Standard Error of the Mean” Uses the exact same formula but substitutes the sample variance for the unknown population variance Or you can use standard deviation

Estimated standard error of mean

Common confusion to avoid Formula for sample variance and for estimated standard error (is the denominator n or n-1?) Sample variance and standard deviation are descriptive statistics Describes how scatted the scores are around the mean Divide by n-1 or df Estimated standard error is a inferential statistic measures how accurately the sample mean describes the population mean Divide by n

The t statistic The t statistic is used to test hypotheses about an unknown population mean (μ) in situations where the value of (σ2) is unknown. T=obtained difference/standard error What’s the difference between the t formula and the z-score formula?

t and z Think of t as an estimated z score Estimation is due to the unknown population variance (σ2) With large samples, the estimation is good and the t statistic is very close to z In smaller samples, the estimation is poorer Why? Degrees of freedom is used to describe how well t represents z

Degrees of freedom df = n – 1 Value of df will determine how well the distribution of t approximates a normal one With larger df’s, the distribution of the t statistic will approximate the normal curve With smaller df’s, the distribution of t will be flatter and more spread out t table uses critical values and incorporates df

Four step procedure for Hypothesis Testing Same procedure used with z scores State hypotheses and select a value for α Null hypothesis always state a specific value for μ Locate a critical region Find value for df and use the t distribution table Calculate the test statistic Make sure that you are using the correct table Make a decision Reject or “fail to reject” null hypothesis

Example GNC is selling a memory booster, should you use it? Construct a sample (n=25) & take it for 4 weeks Give sample a memory test where μ is known to be 56 Sample produced a mean of 59 with SS of 2400 Use α=0.05 What statistic will you use? Why?

Steps State Hypotheses and alpha level Locate critical region (need to know n, df, & α) Obtain the data and compute test statistic Make decision