PSCI lecture 6

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Questions and Answers

What is the primary objective of inferential statistics?

To describe population values without estimation

To estimate sample statistics

To infer from sample to population (correct)

To collect sample data for analysis

Which of the following describes a point estimate?

A range of values that estimate a population value

A specific sample statistic used to estimate a population value (correct)

A statistical technique to infer population values

A method for collecting population data

What is a confidence interval?

A range of values used to estimate a population parameter (correct)

A summary of all sample statistics

A single point statistic representing the entire population

A technique used to eliminate sampling error

Which statement about estimators is true?

An estimator should be unbiased and efficient. Signup and view all the answers

What characterizes an unbiased estimator?

The mean of its sampling distribution equals the population value. Signup and view all the answers

What is the Z score used for when constructing a confidence interval?

It indicates the level of certainty in the estimate. Signup and view all the answers

If the sample mean is 105 and the calculated confidence interval range is from 102.92 to 107.08, what is the value of the Z score used?

2.08 Signup and view all the answers

According to the Central Limit Theorem, what must be true about the sample size to ensure normality in the sampling distribution of sample means?

The sample size must be larger than 100. Signup and view all the answers

What adjustment do we make if we do not know the population standard deviation when constructing a confidence interval?

Apply a different formula for the confidence interval. Signup and view all the answers

If a community has an average IQ of 105 and the population standard deviation is 15, how does this affect the construction of the confidence interval?

It simplifies the estimation process since the population standard deviation is known. Signup and view all the answers

What is the main implication of the Central Limit Theorem regarding non-normally distributed traits?

Repeated large samples can create a normal distribution. Signup and view all the answers

What statistical measure is represented by the term 'standard error of the mean'?

It estimates the precision of the sample mean estimate. Signup and view all the answers

In the calculation of a confidence interval with an alpha level of 0.05, what level of confidence does this correspond to?

95% confidence Signup and view all the answers

What does an unbiased sampling distribution mean regarding the mean or proportion?

It is equal to the population parameter. Signup and view all the answers

How does the standard error of the sampling distribution relate to the sample size?

It decreases as the sample size increases. Signup and view all the answers

What is the term that describes the extent to which the sampling distribution is clustered around the mean?

Efficiency Signup and view all the answers

In the context of point estimates, what is the significance of having a larger sample size?

It allows for more precise estimations. Signup and view all the answers

Given two unbiased samples with the same mean, how does one sample's standard deviation affect the standard error when compared to the other?

Higher standard deviation increases standard error. Signup and view all the answers

What is the relationship between the standard deviation of a sampling distribution and the mean?

They are inversely related through standard error. Signup and view all the answers

What is indicated when a sample is reported as being 3 standard deviations from the mean?

It is a rare occurrence. Signup and view all the answers

What effect does a smaller standard error have on the sampling distribution?

It results in more clustering around the mean. Signup and view all the answers

What happens to the width of a confidence interval as the confidence level increases?

It becomes wider. Signup and view all the answers

Which of the following values of alpha corresponds to a 90% confidence interval?

0.10 Signup and view all the answers

If the sample size is increased, what effect does it have on the confidence interval?

It decreases. Signup and view all the answers

For a 95% confidence interval with a lower limit of $114,035.78 and an upper limit of $115,964.22, what is the estimated mean value?

$115,000 Signup and view all the answers

What is the minimum sample size required to achieve a margin of error of ±4%?

625 Signup and view all the answers

What does a confidence interval of $115,000 ±$1,269.23 indicate?

The estimated mean is $115,000. Signup and view all the answers

Which sampling error would require at least a sample size of 2,500 to achieve?

±2% Signup and view all the answers

In the context of confidence intervals, what is generally affected by the level of alpha?

Both level of confidence and interval width. Signup and view all the answers

What is the significance level commonly used in research?

0.05 Signup and view all the answers

What does a point estimate represent?

A single value without accounting for uncertainty Signup and view all the answers

What Z score corresponds to a 95% confidence level?

1.96 Signup and view all the answers

How is the probability divided for a significance level of 0.05?

0.025 in both tails Signup and view all the answers

What represents a greater level of confidence in an estimate?

99% confidence intervals Signup and view all the answers

What is the typical range for constructing confidence intervals in social sciences?

90% to 99% Signup and view all the answers

What does the confidence interval account for?

Sampling variability and uncertainty Signup and view all the answers

Why is a larger sample beneficial for confidence in estimations?

It increases the precision of the estimators Signup and view all the answers

Which Z score corresponds to a 90% confidence level?

1.65 Signup and view all the answers

What would indicate that only a small chance exists for the population parameter to fall outside the confidence interval?

A low significance level Signup and view all the answers

Study Notes

Estimation Procedures

Estimation procedures estimate population values using sample statistics.
The goal of inferential statistics is to infer from sample data to the population.
There are two main types of estimation procedures:
- Point Estimate: A single value used to estimate a population value.
- Confidence Interval: A range of values used to estimate a population value.
Estimators are statistics calculated from a sample, used to estimate a population parameter.
Estimators must be unbiased and efficient.

Bias

An estimator is unbiased if the mean of its sampling distribution equals the population value of interest.
If an estimator is unbiased, the sampling distribution mean will equal the population parameter if the sample size is large enough.

Efficiency

Efficiency measures how closely the sampling distribution is clustered around the mean.
A smaller standard error means the sample means are more clustered around the population mean, improving the estimator's efficiency.
Standard error decreases as the sample size increases, leading to higher efficiency.

Point Estimates

Point estimates are calculated by drawing an EPSEM sample and calculating the sample mean or proportion.
Larger samples lead to more confidence in the point estimate because they are more efficient.

Confidence Intervals

Confidence intervals are safer than point estimates because they provide a range of values rather than a single value.
They increase the likelihood of including the true population parameter.
Alpha (α) represents the significance level and the risk a researcher is willing to take of being wrong.
Common alpha levels are 0.05 (95% confidence level) and 0.10 (90% confidence level).

Constructing Confidence Intervals

Step 1: Set alpha (α).
Step 2: Find the corresponding Z-score for α. Divide the probability into the upper and lower tails of the sampling distribution.
Step 3: Construct the confidence interval based on the information available (population standard deviation known vs unknown, sample size, normality).

Confidence Interval Procedures

When population standard deviation is known:
- Confidence interval formula: c.i. = ± Z * σ / √n
  - c.i.: Confidence interval
  - Z: Z-score
  - σ: Population standard deviation
  - n: Sample size
When population standard deviation is unknown, the population is normally distributed, and the sample size is 100 or greater:
- Confidence interval formula: c.i. = ± t * s / √n
  - c.i.: Confidence interval
  - t: t-score (calculated using the t-distribution)
  - s: Sample standard deviation
  - n: Sample size

Controlling Confidence Interval Width

Factors influencing confidence interval width:
- Alpha level (α): Higher confidence levels yield wider intervals.
- Sample size: Larger sample sizes result in narrower intervals.

Margin of Error

Margin of error is tied to sample size.
Larger sample sizes have smaller margins of error, resulting in more precise estimates.
For example, a sample size of 1000 would have a margin of error of ±3%, while a sample size of 400 would have a margin of error of ±5%.

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Description

This quiz explores estimation procedures in inferential statistics, focusing on point estimates and confidence intervals. Learn about the importance of unbiased and efficient estimators in estimating population values from sample statistics. Test your understanding of bias and efficiency in sampling distributions.