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Introduction to Inference: Confidence Intervals and Hypothesis Testing

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## Introduction to Inference: Confidence Intervals and Hypothesis Testing

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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

3. Use Rules of Probability and Statistics to make Conclusions about the POPULATION from the SAMPLE.

Population Parameters

- p = population proportion
- µ = population mean
- σ = population standard deviation
- β1= population slope (we will see this in Ch. 14)

Sample Statistics

- = sample proportion
- = sample mean
- s = sample standard deviation
- b1 = sample slope (we will see this in Ch. 14)

Two Types of Inference

- Confidence Intervals: (Ch. 10 & 12)
- Confidence Intervals give us a range in which the population parameter is likely to fall.
- We use confidence intervals whenever the research question calls for an estimation of a population parameter.

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)

- Hypothesis tests are tests of population parameters.

Example: Is the proportion of US adult women who would vote for candidate A >50%?

- We can only prove that a population parameter is ‘different’ than our null value. We cannot prove that a population parameter is equal to some value.

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?

Types of CIs and Hypothesis Tests

For Hypothesis Tests and C.I.’s:

- 1-proportion (1-categorical variable)
- 1-mean (1-quantitative variable)
- Difference in 2 proportions (2-categorical variables, both with 2 levels)
- Difference in 2 means (1-quantitative and 1-categorical variable, or 2-quantitative variables, independent samples)
- Regression, Slope (2-quantitative variables)

For Hypothesis Tests only:

- Chi-Square Test (2-categorical variables, at least one with 3 or more levels!)

Some Examples…

- Mike wants to estimate the mean high-school GPA of incoming freshman at Penn State.

Solution- CI for one population mean.

- George wants to know if the proportion of students who engage in under age drinking is greater than 25%.

Solution- Test of one proportion

Ho: p ≤ .25

Ha: p > .25

- Doug wants to estimate the difference in the proportion of men and women who smoke.

Solution- CI for difference in 2-proportions.

Interpreting CI and Hypothesis Testing

- Confidence Intervals:

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.

- Hypothesis Testing:

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.

Decision Rule:

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

Review of Ch.9: Sample Proportion

- If np and n(1-p) are greater or equal to 10, the sampling distribution of is approximately normal with mean p and standard deviation .

From Sampling Distributions to Confidence Intervals…

- The sample proportion will fall close to the true proportion.
- Thus the true proportion is likely to be close to the observed sample proportion. How close?
- 95% of the would be expected to fall within ± 2 standard deviations of the true proportion p.
- So if we were to construct intervals around ‘s with a width of ± 2 standard deviations these intervals would contain the TRUE population proportion 95% of the times!

Margin of Error & C.I.

- is an estimator of p but it is not exactly equal to p.
- How far is from p?
- Margin of Error is a measure of accuracy providing a likely upper limit for the difference between and p.
- This difference is almost always less that the Margin of Error.
- The almost always is translated with large probability. Usually we are talking about 90%, 95% or 99% probability.
- This probability is the confidence level. For example, if the confidence level is 95%, it means that 95% of the times the difference between and p is less than the Margin of Error. (i.e. we expect 38 out of 40 samples to give a such that its difference with p is less than the Margin of Error.)
- Example: Based on a sample of 1000 voters, the proportion of voters who favor candidate A are 34% with a 3% Margin of Error based on a 95% confidence level. What does this tell us?

95% C.I. for 1-proportion (Derivation)

- If np and n(1-p) are ≥ 10, the sampling distribution of is approximately normal with mean p and standard deviation
- From the empirical rule we have that for about 95% of the samples, is going to fall within from p, i.e. with 95% probability we have

- There is a problem here! Since p is the unknown parameter of interest, is also unknown. Thus, we substitute with the . Doing so we have that if are both ≥10, then with 95% probability we have

95% Margin of Error and C.I. for p

- Thus, if the 95% Margin of Error is

and the 95% C.I. for p is

Note that we are using instead of p for the condition!

A sample of 1200 people is polled to determine the percentage that are in favor of candidate A. Suppose 580 say they are in favor. Construct a 95% CI for the true population proportion.

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.Any C.I. for 1-proprtion

- Conditions: We need to have
- β% CI for p :
- z* multiplier depends on the desired confidence level, β%.
- z* is such that P(-z*<Z<z*)= β%. The most common multipliers are
- Interpretation: We are β% confident that the true population proportion, p, is contained within the confidence interval. Another interpretation is that for about β% samples from the population, the CI captures p.

Margin of Error=z* times the std. error

Example 2: Obtaining a 99% C.I. for p.

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%.

The Width of a Confidence Interval is affected by:

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)?

- The Margin of Error for 95% CI is equal to 2 x s.e( ).
- Before collecting the sample, is unknown, thus we cannot calculate the exact Margin of Error.
- A conservative Margin of Error is equal to
- This implies that differs from p at most ___________ .
- Using the conservative Margin of Error, the length of the C.I. is equal to _____________.
- How large should n beto get a 95% CI of some length L?

n=___________.

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