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## PowerPoint Slideshow about 'Chapter 9 Introduction to the t-statistic' - hayes

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Presentation Transcript

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

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

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