the single-sample t test chapter 9. t distributions >sometimes, we do not have the population...
TRANSCRIPT
![Page 1: The Single-Sample t Test Chapter 9. t distributions >Sometimes, we do not have the population standard deviation. (that’s actually really common). >So](https://reader035.vdocuments.us/reader035/viewer/2022081511/5697bf7c1a28abf838c83db3/html5/thumbnails/1.jpg)
The Single-Sample t Test
Chapter 9
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t distributions
> Sometimes, we do not have the population standard deviation.• (that’s actually really common).
> So what can we do?
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t distributions
> The t distribution is used when we do not know the population information.• So we use the sample to estimate the
population information.• Because we are using the sample, the t
distribution changes based on that sample.
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t distributions
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> When sample size increases, s (the spread of t) approaches σ and t and z become more equal
> The t distributions• Distributions of differences between means
The t Statistic
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Wider and Flatter t Distributions
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Check Your Learning
> When would you use a z test?> When would you use a t test?
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Types of t
> Single sample t• One sample (group of people), population
mean to compare against> Dependent sample t
• One sample tested twice to compare those two scores
> Independent sample t• Two samples to compare those two groups
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t distributions
N
MXSD
2)(
)1(
)( 2
N
MXs
Sample Standard Deviation
Population Standard Deviation
What we did before…Biased estimate
New formula…Unbiased estimate
Based on some error
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Calculating the Estimated Population SD
> Step 1: Calculate the sample mean
> Step 2: Use the sample mean in the corrected standard deviation formula
)1(
)( 2
N
MXs
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= 8.8 = 2.97
Steps to calculating s:
)1(
)( 2
N
MXs )15(
2.35
(8 12 16 12 14)12.4
5M
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SPSS Steps
> Remember you can get the SD from SPSS!• (chapter 4)
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> Using the standard error
> The t statistic
Calculating Standard Error for the t Statistic
N
sSM
M
M
S
Mt
)(
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= 2.97
Steps to calculating t statistic using standard error:
)1(
)( 2
N
MXs
> From previous example:
> Assume population mean is 11:
2.971.33
5M
sS
N
( ) (12.4 11)1.05
1.33M
M
Mt
S
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t distributions
> SPSS will also calculate these values for you!• There are several types of t tests (covered
in the next chapter).• Let’s go over single sample t.
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SPSS
> Analyze > compare means > one-sample t
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SPSS
> Move variable over to the right.> Be sure to change the test value.
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SPSS
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SPSS
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Hypothesis Tests: The Single Sample t Test
> The single sample t test • When we know the population mean, but
not the standard deviation• Degrees of freedom
df = N - 1 where N is sample size
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Stop and think. Which is more conservative: one-tailed or two-tailed tests? Why?
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> The t test• The six steps of hypothesis testing
> 1. Identify population, distributions, assumptions> 2. State the hypotheses> 3. Characteristics of the comparison distribution> 4. Identify critical values
df =N-1
> 5. Calculate> 6. Decide
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> Draw a picture of the distribution> Indicate the bounds> Look up the t statistic> Convert the t value into a raw mean
Calculating Confidence Intervals
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Example Confidence Interval
STEP 1: Draw a picture of a t distribution that includes the confidence interval
STEP 2: Indicate the bounds of the confidence interval on the drawing
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Confidence Interval Continued
STEP 3: Look up the t statistics that fall at each line marking the middle 95%
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STEP 4: Convert the t statistics back into raw means.
Confidence Interval Example
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Confidence Interval Completed
STEP 5: Check that the confidence interval makes sense
The sample mean should fall exactly in the middle of the two ends of the interval:
4.71-7.8 = -3.09 and 10.89 - 7.8 = 3.09
The confidence interval ranges from 3.09 below the sample mean to 3.09 above the sample mean.
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Interpretation of Confidence Interval
If we were to sample five students from the same population over and over, the 95% confidence interval would include the population mean 95% of the time.
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Calculating Effect size
s
Md
)(
For the counseling center data:
(M ) (7.8 4.6)d 1.29
s 2.490