the single-sample t test chapter 9. the t distributions >distributions of means when the...
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The Single-Sample t Test
Chapter 9
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The t Distributions
> Distributions of Means When the Parameters Are Not Known
> Using t distributions • Estimating a population standard deviation
from a sample
N
MXSD
2)(
)1(
)( 2
N
MXs
Sample Standard Deviation
Population Standard Deviation
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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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> 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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> When sample size increases, s 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? Give an example.
> When would you use a t test? Give an example.
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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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STEP 1: Identify population, distribution, assumptions
Population 1: All clients at this counseling center who sign a contract to attend at least 10 sessionPopulation 2: All clients at this counseling center who do not sign a contract to attend at least 10 sessions
• The comparison distribution will be a distribution of means
• Use a single-sample t test because there is one sample and we know the population mean but not the population standard deviation
• Assumptions?
Example: Single Sample t Test
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Calculating the Single Sample t Test
STEP 2: State the hypotheses
0 1 2
1 1 2
H : =
H :
STEP 3: Determine the characteristicsof the comparison distribution.
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t Test Calculation Continued
STEP 4: Determine the critical values, or cutoffs
df = N -1 = 5 -1 = 4
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STEP 5: Calculate the test statistic
STEP 6: Make a decision
t Test Calculation Completed
( ) (7.8 4.6)2.873
1.114M
M
Mt
S
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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
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Dot Plots
> The dot plot is a graph that displays all the data points in a sample, with the range of scores along the x-axis and a dot for each data point above the appropriate value.
> Dot plots serve a similar function to stem-and-leaf plots.
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> The three steps to creating a dot plot
STEP 1: We determine the lowest score and highest score of the sample
STEP 2: We draw an x-axis and label it, including the values from the lowest through highest scores
STEP 3: We place a dot above the appropriate value for every score.
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Example Dot Plot
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> When would you use a z test over a t test?
> When would you use an independent sample t test? Think of a specific study.
Stop and Think