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Presentation 8. First Part. Introduction to Inference: Confidence Intervals and Hypothesis Testing. What is inference?. Inference is when we use a sample to make conclusions about a population. . 2. Describe the SAMPLE. 1. Draw a Representative SAMPLE from the POPULATION.
Inference is when we use a sample to make conclusions about a population.
2. Describe the SAMPLE
1. Draw a Representative SAMPLE from the POPULATION
3. Use Rules of Probability and Statistics to make Conclusions about the POPULATION from the SAMPLE.
Example: What is the mean age of trees in the forest?
Estimate the proportion of US adults who would vote for candidate A.
2. Hypothesis Testing: (Ch. 11 & 13)
Example: Is the proportion of US adult women who would vote for candidate A >50%?
Valid Hypothesis: Is the mean age of trees in the forest > 50 years?
Invalid Hypothesis: Is the mean age of trees in the forest equal to 50 years?
For Hypothesis Tests and C.I.’s:
For Hypothesis Tests only:
Solution- CI for one population mean.
Solution- Test of one proportion
Ho: p ≤ .25
Ha: p > .25
Solution- CI for difference in 2-proportions.
Given the confidence level, β= 90%, 95%, 99%, etc conclude that with β % confidence the population parameter is within the confidence interval.
Example: Suppose the 90% CI for age of trees in the forest is (32,45) years. Then, we are 90% confident that the true mean age of trees in the forest is between 32 and 45 years.
Use the p-value to determine whether we can reject the null hypothesis.
We do not need to know the exact definition now, or how to calculate the p-value, but generally the p-value is a measure of how consistent the data is with the null hypothesis. A small p-value (<.05) indicates the data we obtained was UNLIKELY under the null hypothesis.
If the p-value is <.05 we REJECT the null hypothesis, and accept the alternative. We have a statistically significant result!
If the p-value is >.05 then we say that we do NOT have enough evidence to reject the null hypothesis.
Confidence Intervalsfor 1-Proportion
and the 95% C.I. for p is
Note that we are using instead of p for the condition!
So the 95% CI for p is:
Conclusion: We are 95% confident that the true population proportion of those who support candidate A is between 45.5% and 51.2%.Example 1: Obtaining a 95% C.I. for p.
Margin of Error=z* times the std. error
300 high-risk patients received an experimental AIDS vaccine. The patients were followed for a period of 5 years and ultimately 53 came down with the virus. Assuming all patients were exposed to the virus, construct a 99% CI for the proportion of individuals protected.
We have that the 99% CI for p is:
where z*= 2.58. (Can you see why using the Normal table?)
So the 99% CI for p = .823 ± 2.58(.0220) = (.767,.880)
We are 99% confident that the true proportion of those protected by the vaccine is between 76.7% and 88.0%.
n as the sample size increases the standard error of decreases and the confidence interval gets smaller. So a larger sample size gives us a more precise estimate of p.
z* as the confidence level increases (β%), the multiplier z* increases, leading to a wider CI.
So, if we want to control the length of the C.I. we can either adjust the confidence level or the sample size...
Question: What is an appropriate size in order to obtain a C.I. of a 95% confidence level that is not very large (i.e. with small Margin of Error)?