1. Inference about a Mean Part II
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2. Today’s Plan Confidence Intervals Hypothesis testing Small samples
Large samples
Types of errors
Quick review of what we’ve learned so far
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3. What we’ve seen so far We’ve worked with univariate populations
Recall that we have the standardized normal Z distributed Z ~ N(0,1):
Ask question E(Y)? What is the probability that someone selected at random will have earnings of $300?
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4. What we’ve seen so far
Before when we were considering the distribution around my we were considering the distribution of Y
Now we are considering as a point estimator for my
The difference is that the distribution for has a variance of s2/n where as Y has a variance of s2
Having obtained an estimate of a parameter (my), and considered the properties of the estimator (BLUE), we need to find out how ‘good’ the estimate is. Estimation is the first side of statistical inference.
The other side of statistical inference: hypothesis testing
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5. Confidence Intervals
Recall our picture showing the distributions of Y and
You repeatedly take samples from the population and get different estimates of
The sampling distribution is the probability distribution for the values that takes on in the different samples
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6. Confidence Intervals (2)
How do we assign probability bounds on our estimate?
We don’t know what µy is, but we know the sample size and the sample estimates of
We can estimate µy give or take some amount of error
We know that is distributed
We use s2 as an estimate of 2
Our distribution of :
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7. Confidence Intervals (3)
Remember:
Large samples: use the Z distribution
Small samples: use the t distribution
We’ll use the Z distribution for this example
Our expression for the Z statistic is
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8. Confidence Intervals (4)
We have the standard normal distribution around µy -Z +Z
We want to describe how much area is between -Z and +Z
We can create a 95% confidence interval around Z
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9. Confidence Intervals (5)
We can write the confidence interval as
Where did we get the values and +1.96?
Look at the standard normal table
We see that 47.5% of the area under the curve can be found between 0 and Or 95% between +/- 1.96
-1.96
+1.96
47.5%
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10. Confidence Intervals (6)
So we can rewrite
This is the confidence interval estimate for µy at a 95% level of confidence
You can choose other levels
As you increase the confidence level you increase the number of possible values µy can take
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11. Using the t distribution
If we have a small sample, we should use the t distribution
Our t statistic looks like
What will our confidence interval look like?
We substitute t for Z
We don’t know the underlying population distribution
But we can use the central limit theorem to assume that the sample distribution is approximately normal
We can use the t distribution to approximate the sample distribution
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12. Using the t distribution (2)
We have to choose the confidence interval (1- ) that requires a choice of a
The area between the two t values is the confidence interval -t +t
Confidence interval
The usual accepted confidence level is 95% (a = 0.05)
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13. Using the t distribution (3)
If (1- ) is the area between the two t values, then () is the sum of the area under the two tails
if =0.95, (1- )=0.05
0.05/2 = .025
So for a 95% confidence level, of the area of the curve is found in each tail of the distribution
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14. The t table
In the first row, there is an upper number and a lower number
The upper gives you the area in one tail given a two tail test
The lower number gives the area in one tail or in two tails combined
At an infinite number of observations, 2.5% of the area under the curve is found in each of the tails when our t statistic is it approximates the normal
If our sample size is 10, 95% of the area under the t distribution is between and
Note: the t has fatter tails than the standard normal
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15. The t Table
For a small sample size, the t values corresponding to a 95% confidence interval are larger in absolute value than the Z values for the same interval
Depending on 3 things we get a very different approximation of the confidence interval
Sample size
Whether or not we use the population estimate for s
These determine the type of distribution we use
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16. Hypothesis Testing We want to ask:
What is the probability that µy is equal to some value?
Using hypothesis testing we can determine whether or not it’s plausible that µy equals a certain value
We have two types of samples
Large: n > 30
Small: 30  n
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17. Large Samples Large samples
Doesn’t matter if the population distribution is skewed or normal
Doesn’t matter if the population variance is known or unknown
Use the Z table
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18. Small Samples Small samples
If the population is normally distributed and the population variance is known, use either the Z or t table
If the population is normally distributed but the population variance is unknown use the t distribution with n-1 degrees of freedom (calculate the sample variance as an estimate of the population).
If the population is non-normally distributed, use neither the t nor the Z (I will never give you a case like this)
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19. Setting Up Hypotheses
In hypothesis testing you set up a null hypothesis H0
Under the null hypothesis µy will take a particular value
Example: we can create a null such that
H° : µy = 300
Once we have a null hypothesis we can set up an alternative hypothesis H1
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20. One and Two - Tailed Tests
We can represent this in the following graph:
One-tail tests
We calculate the area in the right-hand tail if H1 : µy > 300
We calculate the area in the left-hand tail if H1 : µy <300
Two tail test:
Find the area under both tails if H1 : µy  300
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21. Intervals and Regions
We also need to assign a significance level (or confidence interval)
For a two-tailed test we are looking to see if a value of 300 lies within the confidence interval
With hypothesis tests we are creating an acceptance region bound by critical values
Critical values are taken off the Z and t tables
The regions in the tails are the critical regions
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22. Intervals and Regions (2)
Critical region
/2
1 - 
Acceptance Region
Critical
value
 is the significance level
If you fail to reject the null, the Z or t statistic must fall in the acceptance region
If you reject the null, the Z or t must fall in one of the critical regions
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23. Types of Errors Type I errors
Rejecting a hypothesis when it is in fact true
Example:
In the confidence interval example we constructed the confidence interval (254  y  380). If the true pop. mean is 400 we can make H0 : y = In this case we’d falsely reject the null hypothesis!
Type II errors
Accepting a false hypothesis
Example: if the true mean is 400 but we accept H0 : y =300 we would be accepting a false hypothesis
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24. Types of Errors (2) Statisticians worry about Type I errors
They choose a significance level  that minimizes Type I errors
To minimize Type I errors choose a small , where  is the total area in both tails
Thus the area in each tail is /2
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25. Types of Errors (3)
As  decreases, the likelihood of rejecting a true null hypothesis also decreases
Most of the time  = 5% is used, and /2 = 2.5%
We can say that we accept or reject the null, but we can’t say that we accept the alternative!
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26. Hypothesis Testing in General
Null (H0):
If you are using the t instead, replace the Z’s with t’s
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27. Where are we now?
So far we have learned about inference and testing hypotheses using assumptions about distributions
Distributions
We had samples and populations and used weights to make inferences about the population using sample statistics
We assumed distributional forms such as the Z or t distributions
Sampling distribution of the mean
You should know the difference between E(Y) and
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28. Where are we now? (2)
BLUE: we’ll return to this in the next few lectures
Estimation and hypothesis testing
We now look to return to the regression line and consider the estimators for a and b from:
Have to consider the properties of the OLS estimator (BLUE), and how do we construct hypothesis tests on the estimates of the parameters a and b?
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