Statistical Inference: Population, Sampling, and Confidence Intervals

Review: Key components of a study

• A research question or hypothesis

• Independent and dependent variables

• Measurement for the variables

• Data collection

• For what? Statistical inference (to answer the research question)

• On whom? (next lecture) = Sampling

Preview

• Population vs. Sample

• Statistical Inference

• Sampling

• Confidence Intervals

Population vs. Sample

• Population

  • The entire pool from which a sample is drawn

  • The individuals or group the sample is supposed to represent

  • In general, large numbers of people

• Sample

  • Usually who researchers actually deal with

  • A portion of the population

  • In general, a much smaller number of individuals

Population vs. Sample (Parameters vs. Statistics)

• Parameter

  • A characteristic of the population

  • Ex: population mean

  • Example: Average height for an adult male in the U.S.

• Sample statistic

  • A characteristic of a sample calculated from data in the sample

  • Ex: sample mean

  • Example: Average height for an adult male in the sample of 100 U.S. male participants

• Statistical inference

  • Generalizing from a sample to a population with calculated degree of certainty

Population vs. Sample (Symbols)

• Population → Parameter(s): N, μ, σ

• Sample → Statistic(s): n, x̄, s

• Note: Parameter represents the true population value; statistics are derived from the sample data

Statistical Inference

• Determining a specific population value to better understand the population

• Determining if there is a relationship between variables in a population

• A population value = a population parameter = a "true" value

Examples of Statistical Inference

• "The average Netflix subscriber spends 2 hours a day on the service."

• "On average, people live 78.93 years."

• "The average income of former UH Manoa students who are working and no longer in school is $49,300."

Statistical Inference and Sampling

• Researchers use sophisticated sampling procedures and collect data from the sample

• Researchers infer, with a relatively high degree of accuracy (95%), the characteristics of a given population from the characteristics of a representative sample

• Concept: Statistical inference

Example: Netflix hours (how many hours, on average, do Netflix subscribers spend a day on the service?)

• Population: Netflix subscribers (182.8 million)

• A population value or parameter: 2 hours (only Netflix knows this)

• How can we find out for ourselves?

Statistical Inference and Sampling (continued)

• Our sample should be representative of the population of interest to enable generalization

• In reality, there is no such thing as a perfectly representative sample

• There will ALWAYS be a degree of difference between a sample statistic and a “true” value that would have been derived from the entire population

Statistical Inference and Sampling (error and solutions)

• Example context: Average hours spent on Netflix with a sample size of 400 participants

• Potential sources of error:

  • Measurement error

  • Respondent error

  • Researcher error

• How to reduce error: draw multiple smaller samples and calculate the means multiple times and take the average

• By taking multiple samples, the error is divided by the number of samples

Example: four samples of 100 each

• Take a 100-sample and calculate the mean

  • Sample A: 115 minutes

• Take another 100-sample, calculate the mean

  • Sample B: 128 minutes

• Take another 100-sample, calculate the mean

  • Sample C: 100 minutes

• Take another 100-sample, calculate the mean

  • Sample D: 145 minutes

• Incorporate results from these four samples

• Take the average of the 4 means

• Average:

  • {\left(115+128+100+145\right)}{4}=122

In Real Research

• The greater the number of samples (and participants) we use, the closer the sample mean will be to the true population mean

• In real research, we can’t randomly sample as many times as we want

• This leads to a certain degree of error

• Statisticians found some laws:

  • Representative sample

  • Large sample size

  • If these conditions are met, the mean of the sample means ≈ the population mean

In Real Research (continued)

• There is almost always a degree of difference between a sample statistic and a true population value

• Researchers DO NOT deterministically report findings from a sample

• Instead, they report findings with a certain degree of confidence

• Concept: Confidence interval

Confidence Intervals

• A range computed using sample statistics to estimate an unknown population parameter with a stated level of confidence

• A range within which the population parameter (the true value) is likely to fall, with a certain probability (usually 95%)

• Lower end =x̄-1.96\times𝑆𝐸

• Upper End=x̄+1.96\times𝑆𝐸

Confidence Interval Interpretation

• "We are 95% confident that the confidence interval contains the population parameter (the true value)."

• "95 out of 100 samples (95%) from the same population will produce confidence intervals that contain the population parameter (the true value)."

Example 1 (UH height)

• Researchers collected data from a random sample of 525 UH students

• Mean height = 67 inches

• 95% CI: [66.62, 67.38]

• Population: All UH students

• Sample: 525 UH students (random sample)

• Interpretation: "We are 95% confident that the mean height of all UH students (population parameter, the true value) is between 66.62 inches and 67.38 inches."

Example 2 (MSU IQ)

• Random sample of 300 students at MSU

• Mean IQ = 102.2

• 95% CI: [98.72, 108.38]

• Population: All MSU students

• Sample: 300 MSU students (random sample)

• Interpretation:

  • "We are 95% confident that the mean IQ score of all students at Michigan State University is between 98.72 and 108.38."

A Relationship between Variables

• A relationship between variables is studied for correlation purposes

• Examples:

  • Aggression among high school students is highly associated with playing violent video games

  • Anti-smoking ads depicting serious health consequences significantly decreased smoking cessation intention

• Often used for correlation studies

Example: Relationship between social media use and loneliness (UH Manoa)

• Sample: 325 UH students surveyed in Spring 2021

• Found that social media use and loneliness are highly correlated with Pearson’s correlation coefficient r = 0.67

• 95% confidence interval for the correlation:

  • [0.54, 0.71]

• Interpretation guidelines:

  • r ≥ 0.5 = Moderate correlation

  • r ≥ 0.7 = Strong correlation

Interpretation of the UH Manoa example

• We are 95% confident that the correlation between social media use and loneliness in the population of all college students is between 0.54 and 0.71

• This provides evidence of a moderate to strong relationship between social media use and loneliness among college students

To Recap

• Sampling: Population → Sample

• Collect data → Calculate sample statistics (e.g., sample mean)

• Infer population parameters (e.g., population mean)

• Statistical inference: drawing conclusions about a population from sample data with a stated level of confidence

• Core concepts: population, sample, parameter, statistic, sampling error, confidence intervals, and relationships between variables (correlation)