Sample Size and Population Size

Example of a Confidence Interval

By definition, a confidence interval is a statistical range calculated around a sample statistic that is designed to contain which of the following a specified percentage of the time?

If a calculated confidence interval completely excludes a specific hypothetical population mean, a researcher can conclude that the sample mean is not statistically significantly different from that hypothetical value.

A cognitive psychologist is testing a new training method to determine its effect on reaction times. The average reaction time in the general population is known to be 400 milliseconds (ms). Match each of the following research results (presented as 95% Confidence Intervals) to the correct statistical conclusion regarding the training group's performance.

A researcher is analyzing the results of a psychological study and determines that the $95\%$ confidence interval is too wide to be useful. Arrange the following steps in the logical order that describes how the researcher's decision to increase the sample size leads to a more precise estimate.

A developmental psychologist is designing a mock-up of a research poster to teach students how to evaluate statistical findings. The goal is to create a hypothetical scenario where a new intervention's impact is judged to be 'statistically effective' compared to a baseline population mean of $100$, but where the specific estimate of that impact remains highly uncertain (represented by a wide range). Which of the following result summaries should the psychologist construct to fulfill these specific design goals?

The most commonly used confidence interval in psychological research is set to contain the true population parameter $95\%$ of the time.

A student in an introductory psychology research methods course is learning about interval estimation. Match each component or outcome of a confidence interval with its correct conceptual interpretation.

A clinical psychologist is evaluating a claim that a specific mindfulness technique has no impact on stress levels (where a difference of $0$ indicates no impact). The researcher's study results in a $95\%$ confidence interval for the mean reduction in stress of $[4.2, 10.8]$. Because the value representing 'no impact' falls entirely outside this interval, the researcher should judge the initial claim as _____.

A social psychologist surveys 200 college students about daily social media use (in minutes) and obtains a $95\%$ confidence interval of [85, 115]. After examining the interval, the researcher notes that a media organization's claimed population mean of 130 falls completely _____ the interval, while a second published estimate of 95 falls completely inside it — leading the researcher to conclude that only the first claim is statistically refuted at the .05 level.

An undergraduate student completing a research methods paper must decide whether to report statistical results using only a null hypothesis test (p-value) or a $95\%$ confidence interval. Arrange the following steps of the student's evaluative reasoning in the most logical order.

Define a confidence interval and specify what factors determine its width (specifically comparing the role of sample size and population size).

Based on this scenario and the relationship between confidence intervals and null hypothesis tests, explain how the psychologist should evaluate the skeptic's claim. What conclusion should be made regarding the statistical significance of the difference between the sample mean and the hypothesized population mean of $230$, and why?

A developmental psychologist surveys a sample of $1,000$ children and calculates a $95\%$ confidence interval of [47, 53] for the percentage of children who read daily, where the population size is one hundred thousand. If the psychologist conducts the same study with a sample of $1,000$ children from a population of one hundred million, what will the new $95\%$ confidence interval be, and why?