student’s t-distributions. student’s t-model: family of distributions similar to the normal...
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Student’s t-Model:
Family of distributions similar to the Normal model but changes based on degrees-of-freedom.
Degrees-of-freedom = sample size (n) – 1
Differs because we need to estimate the mean and standard deviation of a population using our sample data
The graph of Student’s t is generally wider than the normal model.
The larger the sample size the closer Student’s t-model approach the Normal model.
http://www.stat.tamu.edu/~jhardin/applets/signed/T.html
One Sample t-intervalConditions:Randomization10% ConditionNearly Normal – Two possible graphs to check:
1. Histogram: unimodal and symmetric, with no outliers
2. Normal Probability Plot: a straight line, no curve
**It can be skewed if the sample size is large.
Generally n > 30. CLT kicks in at that point.
All conditions have been met to use Student’s t-model for a one sample t-interval
Normal Probability PlotFinds the z-score of each value compared to the
sample mean and standard deviation. Plots these against the values themselves.
No curve in the plot shows a unimodal and symmetric distribution
Mechanics:Calculate sample mean, , and sample standard
deviation, sFind the degrees of freedom, df = n – 1Calculate critical value using given confidence level
and degrees of freedom,
One Sample t-interval
x
dft
CI : df
sx t
n
Example: Coffee MachineA coffee vending machine dispenses coffee into a
paper cup. You’re supposed to get 10 ounces of coffee, but the amount varies slightly from cup to cup. Below are the amounts measured in a random sample of 20 cups. Is there evidence that the machine is shortchanging customers? Construct a 95% confidence interval.
9.9 9.7 10.0 10.1 9.99.6 9.8 9.8 10.0 9.59.7 10.1 9.9 9.6 10.29.8 10.0 9.9 9.5 9.9
Conditions:Randomization: Stated as a random sample10% Condition: 20 cups is less than 10% of all cups
dispensed by the coffee machineNearly Normal Condition: The histogram is unimodal
and approximately symmetric. Or
The normal probability plot shows an
approximately linear pattern.All conditions have been met to use
Student’s t-model for a one sample
t-interval.
Mechanics:
(9.752, 9.938)
We are 95% confident that the true mean amount of coffee from the coffee machine is between 9.752 to 9.938 ounces.
9.845x 0.199s 20n
. . 20 1 19d f
19 2.093t 0.199
CI : 9.845 2.09320
df
sx t
n
1 Sample t-TestInference about the mean of a population, µ.
Hypotheses:
H0: µ = µ0 The population mean is _____
HA: µ <;>;≠ µ0
Conditions:
1.Randomization
2.10% Condition
3.Nearly Normal (draw a picture)
Mechanics:Calculate sample mean, , and sample standard
deviation, sFind the degrees of freedom, df = n – 1State alpha valueCalculate test statistic, Find P-Value
Conclusion:Compare P-Value to alpha.State conclusion in context
x
0. .d f
xt
s
n
Hypothesis:
H0: µ = 10; The mean amount of coffee dispensed by the coffee machine is 10 ounces.
HA: µ < 10; The mean amount of coffee dispensed by the coffee machine is less than 10 ounces.
Conditions:
1.Randomization: Random sample
2.10% Condition: 20 cups is less than 10% of all cups dispensed.
3.Nearly Normal: The histogram is
unimodal and approximately symmetric
with no outliers.
Example: Coffee Machine
Mechanics:
Conclusion:
Since the P-Value is less than alpha (0.0012 < 0.05) we reject the null hypothesis. There is statistically significant evidence that the mean amount of coffee from the coffee machine is less than 10 ounces.
9.845x 0.199s
20n . . 20 1 19d f
0.05
019
xt
s
n
9.845 100.199
20
3.49
19( 3.49) 0.0012P Value P t
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