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What is the primary objective of inferential statistics?
What is the primary objective of inferential statistics?
To infer from a sample to a population.
What are the two types of estimation techniques used in inferential statistics?
What are the two types of estimation techniques used in inferential statistics?
A point estimate is a range of values used to estimate a population value.
A point estimate is a range of values used to estimate a population value.
False
A confidence interval consists of a range of values used to estimate a population value.
A confidence interval consists of a range of values used to estimate a population value.
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What are estimators?
What are estimators?
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What are the two key characteristics that estimators must possess?
What are the two key characteristics that estimators must possess?
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An estimator is considered unbiased if the mean of its sampling distribution is equal to the population value of interest.
An estimator is considered unbiased if the mean of its sampling distribution is equal to the population value of interest.
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A large sample size ensures that an estimator will always be unbiased.
A large sample size ensures that an estimator will always be unbiased.
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The smaller the standard error, the more clustered the sampling distribution is around the mean.
The smaller the standard error, the more clustered the sampling distribution is around the mean.
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The standard error of a sampling distribution increases as the sample size increases.
The standard error of a sampling distribution increases as the sample size increases.
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A point estimate provides a range of values that is likely to contain the true population parameter.
A point estimate provides a range of values that is likely to contain the true population parameter.
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What is the purpose of constructing a confidence interval?
What is the purpose of constructing a confidence interval?
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A confidence interval is always constructed using the same formula, regardless of the information available.
A confidence interval is always constructed using the same formula, regardless of the information available.
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What is the significance level (α) and what does it represent in confidence interval construction?
What is the significance level (α) and what does it represent in confidence interval construction?
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What does the Central Limit Theorem state?
What does the Central Limit Theorem state?
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The t-distribution is used when the sample size is smaller than 100 and the population standard deviation is unknown.
The t-distribution is used when the sample size is smaller than 100 and the population standard deviation is unknown.
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As the degrees of freedom increase in the t-distribution, it becomes more similar to the Standard Normal Distribution.
As the degrees of freedom increase in the t-distribution, it becomes more similar to the Standard Normal Distribution.
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Study Notes
Quantitative Research Methods in Political Science
- Lecture 6 covered estimation procedures in political science.
- Estimation procedures are techniques used to estimate population values from sample statistics.
- Inferential statistics are used to infer from sample data to the population.
- Two types of estimation procedures exist:
- Point Estimate: a sample statistic used to estimate the population value.
- Confidence Interval: a range of values used to estimate the population value.
Estimators
- Estimators are sample statistics that estimate a fact about the population.
- Examples:
- A sample mean can be used to estimate the population mean.
- A sample standard deviation can be used to estimate the population standard deviation.
- Estimators must be unbiased and efficient.
Bias
- An unbiased estimator has a sampling distribution mean equal to the population value of interest.
- Based on the sampling distribution, we can know if an estimator is unbiased when the sample size is large enough.
- If an estimator is unbiased, the sampling distribution mean/proportion is equal to the population parameter.
Efficiency
- Efficiency refers to the extent to which the sampling distribution is clustered around the mean.
- A smaller standard error indicates a more clustered sampling distribution around the mean.
- The standard error of a sampling distribution decreases as sample size increases.
- Efficiency increases as the standard error decreases.
Point Estimates
- A point estimate is a single value used as an estimate for a population value.
- A sample mean is an example of a point estimate.
- In practical use, a point estimate doesn't fully account for the uncertainty of sampling.
- A larger sample size will improve the confidence in the estimator.
Interval Estimates / Confidence Intervals
- More complex method than point estimates used to estimate population parameter values.
- Confidence intervals specify a range of values that are more likely to contain the true population value.
Step 1 - Select Alpha
- Alpha represents the significance level (risk of being wrong) a researcher is willing to accept.
- The most common value for alpha is 0.05 (95% confidence level)
Step 2 - Find Corresponding Z Score for Alpha
- Divide the probability into lower and upper tails of the sampling distribution.
- Locate the Z-score corresponding to the given alpha level by using the "Area Beyond Z" from the textbook or lecture notes.
- The Z-score for alpha/2 is useful in this calculation.
Step 2 - Find Corresponding Z Score for Alpha Cont'd
- The corresponding Z-score for a 95% confidence interval is 1.96.
Step 3 - Construct Confidence Interval
- Different formulas are used depending on the information provided about the population.
Interval Estimation Procedures for Samples (known)
- The formula for a confidence interval when the population standard deviation is known is c.i. = x ± Z (σ /√n).
Example for Constructing Confidence Interval (known)
- A formula is used with an example of calculating a confidence interval given the standard deviation of IQs for a given population and sample.
A Note on Sample Size
- Central Limit Theorem applies when sample size is 100 or larger.
- Regardless of whether the population distribution is normal or not, with large enough samples, the sample means will be normally distributed.
Interval Estimation Procedures for Samples (unknown)
- The confidence interval is calculated by c.i. = x ± t (s /√n) using the sample standard deviation instead of the population standard deviation.
Student's t Distribution
- Used in situations with sample standard deviation.
- Sample sizes smaller than 100 may also require a t-distribution.
- The t-distribution depends on the degrees of freedom.
Student's t Distribution Cont'd
- As sample sizes increase, the t-distribution starts to resemble the Z-distribution.
Using the t Distribution
- Example illustrating how to use the t-distribution table to find the t-score for a given confidence level and degrees of freedom.
Controlling Width of Confidence Intervals
- Two main factors affect the width of confidence intervals—alpha level and sample size.
- Higher alpha levels mean wider intervals.
- Larger sample sizes mean narrower intervals.
Interval Width and Alpha
- Comparing interval widths for different confidence levels (90%, 95%, 99%) to show the relation.
Interval Width and Sample Size
- Illustrates how larger samples result in narrower confidence intervals for the same confidence level.
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Description
This quiz covers Lecture 6 on estimation procedures in political science, focusing on point estimates and confidence intervals. Understand how estimators function, their bias, and their efficiency in estimating population values based on sample statistics.
