Chapter 9 Introduction to the t-statistic

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Chapter 9Introduction to the t-statistic

• PSY295 Spring 2003
• Summerfelt

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Overview

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

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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

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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!

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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

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Whats the problem with z?

• Need to know the population mean and variance!!!
  Not always available.

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What is the t statistic?

• Cousin of the z statistic that does not require
  the population mean (µ) or variance (s2)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

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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!

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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

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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

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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

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Estimated standard error of mean
Or
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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

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The t statistic

• The t statistic is used to test hypotheses about
  an unknown population mean (µ) in situations
  where the value of (s2) is unknown.
• Tobtained difference/standard error
• Whats the difference between the t formula and
  the z-score formula?

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t and z

• Think of t as an estimated z score
• Estimation is due to the unknown population
  variance (s2)
• 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

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Degrees of freedom

• df n 1
• Value of df will determine how well the
  distribution of t approximates a normal one
• With larger dfs, the distribution of the t
  statistic will approximate the normal curve
• With smaller dfs, the distribution of t will be
  flatter and more spread out
• t table uses critical values and incorporates df

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Four step procedure for Hypothesis Testing

• Same procedure used with z scores
• State hypotheses and select a value for a
• 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

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Example

• GNC is selling a memory booster, should you use
  it?
• Construct a sample (n25) 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 a0.05
• What statistic will you use? Why?

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Steps

1. State Hypotheses and alpha level
2. Locate critical region (need to know n, df, a)
3. Obtain the data and compute test statistic
4. Make decision