PSCI lecture 5

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Questions and Answers

What is a primary purpose of inferential statistics?

To generalize findings from a sample to a population (correct)

To calculate measures of central tendency

To describe the relationship between multiple variables

To summarize data from a single variable

Which of the following is NOT considered a component of descriptive statistics?

Sample selection process (correct)

Measures of central tendency

Measures of dispersion

Proportions and percentages

When using inferential statistics, what type of data is analyzed?

Data from a carefully chosen sample (correct)

Population data exclusively

All available data in the dataset

Only categorical data

Which of the following describes a univariate statistic?

Statistical analysis focusing on one variable only Signup and view all the answers

What distinguishes bivariate statistics from univariate statistics?

Bivariate statistics assess the relationship between two variables Signup and view all the answers

What shape will the sampling distribution of sample means take if the population trait is normally distributed?

Normal Signup and view all the answers

Which formula represents the calculation for the Standard Error (SE)?

SE = $σ/√n$ Signup and view all the answers

According to the Central Limit Theorem, what happens to the sampling distribution of sample means as the sample size (n) becomes large?

It approaches normality. Signup and view all the answers

What is the minimum conservative sample size suggested for achieving normality in the sampling distribution?

100 Signup and view all the answers

What can be estimated about a population using sample statistics, according to the theorems provided?

Population mean and standard deviation Signup and view all the answers

Which of the following statements is true regarding the normality of the population distribution according to Theorem #2?

As long as the sample size is large enough, we can still achieve normality in the sampling distribution. Signup and view all the answers

What factors may influence the sufficient sample size needed for normality?

Distribution of the population and size of the population Signup and view all the answers

What does the standard deviation of the sampling distribution signify?

It measures the variability of the sampling distribution. Signup and view all the answers

What is the primary characteristic of a quota sample?

It targets individuals based on accessibility. Signup and view all the answers

Which of the following best explains the snowball sampling method?

It involves reaching out to a source who refers the researcher to others. Signup and view all the answers

What distinguishes probability sampling from other sampling methods?

It involves random selection to ensure representativeness. Signup and view all the answers

What is a potential issue with using convenience samples?

They can lead to biased results that are not generalizable. Signup and view all the answers

How does a quota sample attempt to represent a population?

By focusing on achieving a representation from specific categories. Signup and view all the answers

What would be an inappropriate method of sampling if the goal is to understand the broader Canadian population?

Surveying shoppers at a local grocery store. Signup and view all the answers

What is a key reason why inferential statistics require a representative sample?

To ensure findings can be generalized to the wider population. Signup and view all the answers

What is implied by the term 'strata' in the context of quota sampling?

It identifies specific categories within the population. Signup and view all the answers

What occurs to the shape of the sampling distribution as the sample size increases?

It approaches a normal distribution. Signup and view all the answers

How does the sampling distribution change if the population is normally distributed?

It becomes taller and narrower. Signup and view all the answers

What is required for a sampling distribution to achieve normality when the population is highly asymmetrical?

Sample sizes of at least 100. Signup and view all the answers

If a sample of size 2 is taken from a population of 4 with the amounts $2, $4, $6, and $8, how many unique sample means can be produced?

16 Signup and view all the answers

What describes the relationship between the standard error and the sample representation of the population?

A smaller standard error means more representative samples. Signup and view all the answers

What does the Central Limit Theorem state regarding larger samples?

Larger samples tend to reflect the population's distribution more closely. Signup and view all the answers

In constructing the sampling distribution from a population of four individuals with known amounts, how frequently does the mean of $5 occur?

Four times Signup and view all the answers

What dictates the shape, central tendency, and dispersion of the sampling distribution?

Theorems related to inferential statistics. Signup and view all the answers

What is the formula for calculating the standard error of the sampling distribution?

$rac{ ext{population standard deviation}}{ ext{square root of sample size}}$ Signup and view all the answers

Which statement accurately describes the law of large numbers?

Larger samples lead the mean to approach that of the population. Signup and view all the answers

What is the minimum sample size required for the sampling distribution to be normal in shape?

100 Signup and view all the answers

Which statement about the sampling distribution is correct?

The standard deviation of the sampling distribution is called the Standard Error. Signup and view all the answers

What percentage of sample means falls within 1 standard error of the mean?

68.26% Signup and view all the answers

What does a very small percentage (0.0026%) of sample means represent in the context of standard errors?

Means that fall beyond 3 standard errors from the mean Signup and view all the answers

What is the relationship between the sampling distribution and the population mean?

They always have the same mean. Signup and view all the answers

Which is a correct statement regarding the computation of sample means?

It is impractical to calculate means for every possible sample. Signup and view all the answers

What can be inferred about sampling means within 2 standard errors from the population mean?

95.44% of sample means will be within this range. Signup and view all the answers

What is the purpose of the theorems related to sampling distribution?

To link the sample to the population. Signup and view all the answers

Study Notes

Descriptive vs. Inferential Statistics

Descriptive statistics summarize data and describe the distribution of a single variable or the relationship between two or more variables.
Inferential statistics use information from a sample to generalize findings to the entire population.

Types of Non-probability Sampling

Convenience Samples: Targets individuals who are easily accessible to the researcher.
Snowball Samples: Useful for populations that are hard to reach, like those in conflict zones or underground groups. The sample expands as new participants are identified through existing participants.
Quota Samples: Non-random counterpart to stratified sampling. The researcher seeks to collect a sample that reflects the population in categories of interest (e.g., specific income bracket).

Representativeness and Probability Sampling

A representative sample accurately reflects the characteristics of the population.
Probability sampling, also known as random sampling, uses techniques that ensure each member of the population has a known chance of being selected.

Probability Sampling Cont’d

A bad example of sampling: Surveying the first 1000 people who exit a grocery store. This sample would not be representative as it would only reflect people who shop at that specific store on that particular day.

Theorem #1: Sampling Distribution of Sample Means

If a trait is normally distributed in the population, the sampling distribution of sample means will also be normally distributed.
The mean of the sampling distribution will be the same as the population mean.
The standard deviation of the sampling distribution (standard error) is calculated by dividing the population standard deviation by the square root of the sample size (n).

Theorem #2: The Central Limit Theorem

Even if a trait is not normally distributed in the population, as the sample size (n) increases, the sampling distribution of sample means will approach a normal distribution.
This holds true regardless of the shape of the original population distribution.
This is essential because it allows us to apply the principles of normal distribution to larger samples, even if the population is not normally distributed.

Demonstrating the Central Limit Theorem

A conservative estimate for a sufficient sample size to achieve normality is 100.
However, the required sample size can vary depending on the distribution of the population and its size.

Constructing a Sampling Distribution Example

Example: Imagine a population of four individuals with varying amounts of money: 2,2, 2,4, 6,and6, and 6,and8.
The population mean is 5andthepopulationstandarddeviationis5 and the population standard deviation is 5andthepopulationstandarddeviationis2.45.
If we take samples of size 2 with replacement (meaning we can select the same person twice), we generate 16 possible samples.
Each sample has a mean, and the distribution of these sample means forms a sampling distribution.

The Standard Error

The smaller the standard error, the more representative the sample is of the population.
This is because a larger sample size results in a smaller standard error, indicating the sample mean is more likely to be close to the population mean.
The law of large numbers states that as the sample size increases, the sample mean will approach the population mean.

Linking the Population, Sampling Distribution, and Population Review

Inferential statistics allow us to link information about a sample to the larger population.
The theorems help us understand the statistical characteristics of the sampling distribution (shape, central tendency, and dispersion), enabling us to make inferences about the population based on sample data.
In practice, we usually only need to collect one sample to learn about the population.

Final Points

While we can't realistically calculate the mean of every possible sample, the theorems provide the theoretical foundation for inferential statistics.
The sampling distribution is a theoretical concept, but its properties allow us to make inferences about the population based on a single sample.
In future lectures, we will delve deeper into how to use this knowledge to perform statistical analysis and draw conclusions about the population.

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Description

Explore the key concepts of descriptive and inferential statistics in this quiz. Test your understanding of various sampling methods, including non-probability sampling techniques. Gain insights into how to accurately represent populations in statistical studies.