1. Intro to Confidence Intervals Introduction to Inference
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Chapter 14
Introduction to Inference
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2. Statistical Inference
Two forms of statistical inference:
Confidence intervals
Significance tests of hypotheses
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3. Population Mean μ
We wish to infer population mean μ using sample mean “x-bar”. The following conditions prevail:
Data acquired by Simple Random Sample
Population distribution is Normal
The value of σ is known
The value of μ is NOT known
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4. Example
Statement of problem: Young people have a better chance of good jobs and wages if they are good with numbers. We want to know the average NAEP math score of a population.
Response variable ≡ NAEP math scores
Range from 0 to 500
Have a Normal distribution
Population standard deviation σ = 60
Population mean μ not known
A simple random sample of n = 840 individuals derives sample mean (“x-bar”) = 272
We want to estimate population mean µ
Reference: Rivera-Batiz, F. L. (1992). Quantitative literacy and the likelihood of employment among young adults. Journal of Human Resources, 27,
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5. The Sampling Distribution of the Mean and its Standard Deviation
Sample means would vary from sample to sample, forming a sampling distribution
This distribution will be Normal with mean μ
The standard deviation of the distribution of means will equal population standard deviation σ divided by the square root of the sample size:
This relationship is known as the square root law
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6. Sampling distribution of the mean
For our example, σ = 60 and n = 840; thus, the sampling distribution of the mean will be Normal with
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7. Margin of Error for 95% Confidence
The rule says 95% of x-bars will fall in the interval μ ± 2σxbar
More accurately, 95% will fall in μ ± 1.96∙σxbar
1.96∙σxbar is the margin of error m for 95% confidence
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8. In repeated independent samples:
We call these intervals Confidence Intervals (CIs)
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9. How Confidence Intervals Behave
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10. Other Levels of Confidence
Confidence intervals can be calculated at various levels of confidence by altering critical value z* :
Common levels of confidence
Confidence level C
90%
95%
99%
Critical value z*
1.645
1.960
2.576
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11. CI for μ when σ known To estimate μ with confidence level C, use
Use Table C to look up z* values for various levels of confidence
Confidence level C
90%
95%
99%
Critical value z*
1.645
1.960
2.576
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12. In this example: x-bar = 272 and σxbar = 2.1
Conf Interval for μ
In this example: x-bar = 272 and σxbar = 2.1
For 99% confidence; use z* = 2.576
For 90% confidence, use z* = 1.645
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13. Tests of Significance Recall: two forms of statistical inference
Confidence intervals
Hypothesis tests of significance
The objective of confidence intervals is to estimate a population parameter
The objective of a test of significance is to test a statistical hypothesis
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14. Tests of Significance: Reasoning
Like with confidence intervals, we ask what would happen if we repeated the sample or experiment many times
We assume the same conditions as before: (SRS, Normal population, population standard deviation σ known)
We recognize that sample means will vary from sample to sample
The text explains the reasoning behind tests of significance on pp. 369 – 376
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15. Tests of Statistical Significance: Procedure
A. The claim is stated as a null hypothesis H0 and alternative hypotheses Ha
Calculate a test statistic
The test statistic is converted to a probability statement called a P-value
The P-value is interpreted in terms of the weight of evidence against H0
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16. Example
Example: We want to test whether data in a sample provides reliable evidence that a population has gained weight.
Based on prior research, we know that weight gain in the population is Normal with σ = 1 lb
We select n = 10 individuals
The sample mean weight gain x-bar = 1.02 lbs.
Is this level of evidence strong enough to conclude a population weight gain?
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17. Intro to Confidence Intervals
HS 67
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The Null Hypothesis
The null hypothesis H0 is a statement of “no difference” in the form: H0: μ = μ0 where μ0 represents the value of the population if the null hypothesis is true
In our example, μ0 = 0 and the null hypothesis is H0: μ = 0
Does the sample mean of 1.02 based on n = 10 provide strong evidence that the null hypothesis is untrue?
The first step in the procedure is to state the hypotheses null and alternative forms. The null hypothesis (abbreviate “H naught”) is a statement of no difference. The alternative hypothesis (“H sub a”) is a statement of difference. Seek evidence against the claim of H0 as a way of bolstering Ha.
The next slide offers an illustrative example on setting up the hypotheses.
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9: Basics of Hypothesis Testing 17
The Basics of Significance Testing
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18. The Alternative Hypothesis
The alternative hypothesis Ha contradicts the null hypothesis
The alternative hypothesis can be stated in one of two ways
The one-sided alternative specifies the direction of the difference
For the current example, Ha: μ > 0, indicating a positive weight change in the population
The two-sided alternative does not specific the direction of the difference
For the current example Ha: μ ≠ 0, indicating a “weight change in the population”
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19. Sampling Distribution of the Mean
This is the sampling distribution of the mean if the null hypothesis is true (i.e., μ = 0) is shown here
Note that the standard deviation of the mean is
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20. Test Statistic for H0: μ = μ0 When population σ is known
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21. Test Statistic, Example
For data example, x-bar = 1.02, n = 10, and σ = 1
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22. Basics of Significance Testing
P-Value from z table
Convert z statistics to a P-value:
For Ha: μ > μ0 P-value = Pr(Z > zstat) = right-tail beyond zstat
For Ha: μ < μ0 P-value = Pr(Z < zstat) = left tail beyond zstat
For Ha: μ ¹ μ0 P-value = 2 × one-tailed P-value
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23. Basics of Significance Testing
P-value: Example
For current example, zstat = 3.23 and the one-sided P-value = Pr(Z > 3.23) = 1 − = .0006
Two-sided P-value = 2 × one-sided P = 2 × = .0012
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24. P-value: Interpretation
P-value (definition) ≡ the probability the sample mean would take a value as extreme or more extreme than observed test statistic when H0 is true
P-value (interpretation) Smaller-and-smaller P-values → stronger-and-stronger evidence against H0
P-value (conventions)
.10 < P < 1.0  evidence against H0 is not significant
.05 < P ≤ .10  evidence against H0 is marginally signif.
.01 < P ≤ .05  evidence against H0 is significant
0 < P ≤ .01  evidence against H0 is highly significant
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25. Basics of Significance Testing
P-value: Example
Let us interpret the current two-sided P-value of .0012
This provides strong evidence against H0
Thus, we reject H0 and say there is highly significant evidence of a weight gain in the population
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26. Basics of Significance Testing
Significance Level
α ≡ threshold for “significance”
If we choose α = 0.05, we require evidence so strong that it would occur no more than 5% of the time when H0 is true
Decision rule
P-value ≤ α  evidence is significant
P-value > α  evidence not significant
For example, let α = The two-sided P-value = is less than .01, so data are significant at the α = .01 level
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27. Summary one sample z test for a population mean
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