Distribution of Means, Z Tests, and Confidence Intervals

Review of Populations, Samples, and Research Objectives

• Population vs. Sample:

  • A population is the entire group that a researcher is interested in (e.g., all depressed people).

  • A sample is a smaller subset of that population (e.g., a sample of $75$ depressed people).

  • Because it is usually not feasible to study an entire population, researchers u8ose samples to make inferences about the larger group.

• Statistics vs. Parameters:

  • Parameters are numerical values that describe a population.

  • Statistics are numerical values that describe a sample.

  • Researchers use statistics to estimate population parameters.

• Objective of Inferences:

  • The primary goal is for samples to represent the population accurately so that results can be claimed about the population as a whole.

Sampling Error and Estimation

• Defining Sampling Error:

  • This refers to the difference between the values actually existing in the population and the values observed in a sample.

  • It is the likelihood that sample values (such as the Mean, $M$, and Standard Deviation, $SD$) will differ from the actual population parameters.

  • It does not imply that mistakes were made in data collection or analysis; it is a natural byproduct of sampling.

• Managing Sampling Error:

  • In real-world scenarios, the exact amount of sampling error is unknown because the population parameters are rarely known.

  • Therefore, researchers estimate the average amount of sampling error present in the data.

• Illustration of Sampling Error:

  • Population (NY College Students):

    • Total population size ($N$) = $145,000$.

    • Population Parameters: Mean Age = $21.3$, Mean IQ = $112.5$, Gender = $65\%$ Female, $35\%$ Male.

  • Sample #1 (U. Albany Students, $n = 100$):

    • Sample Statistics: Mean Age = $20.4$, Mean IQ = $114.2$, Gender = $40\%$ Female, $60\%$ Male.

  • Sample #2 (Ithaca College Students, $n = 100$):

    • Sample Statistics: Mean Age = $23.3$, Mean IQ = $118.0$, Gender = $80\%$ Female, $20\%$ Male.

  • Sample #3 (NYU Students, $n = 100$):

    • Sample Statistics: Mean Age = $19.8$, Mean IQ = $104.6$, Gender = $60\%$ Female, $40\%$ Male.

The Distribution of Means

• Conceptual Overview:

  • When samples of more than one individual are taken, the comparison distribution for the null hypothesis shifts from a distribution of individual scores to a distribution of means.

  • Also referred to as the "Sampling Distribution of Means" or "Distribution of Sample Means."

  • These means tend to "pile up" around the actual population mean ($\mu$).

• Characteristics of the Distribution of Means:

  • Mean of the Distribution of Means: The mean of the sampling distribution is identical to the mean of the population of individuals ($\mu$).

  • Impact of Sample Size ($n$):

    • Larger sample sizes lead to sample means that are closer to the population mean on average.

    • Large samples are generally better representatives of the population than small samples.

  • Variance of the Distribution of Means ($\sigma_M^2$ or $\text{Var}_M$):

    • Calculated as the variance of the population divided by the number of individuals in each sample: $\text{Variance} = \frac{\sigma^2}{n}$.

  • Standard Deviation of the Distribution of Means:

    • This is the square root of the variance of the distribution of means.

The Standard Error of the Mean (SEM)

• Definition: The standard deviation of the sampling distribution of means is formally called the Standard Error of the Mean (SEM).

• Function:

  • It reflects the accuracy with which sample means estimate the population mean.

  • it represents the average deviation (sampling error) of sample means from the population mean ($\mu$).

• Mathematical Relationship and Sample Size:

  • As sample size ($n$) increases, the SEM decreases, meaning the sample mean becomes a better estimate of the population mean.

  • If the SEM is large, sample means will tend to differ significantly from one another, and many will not be accurate representations of $\mu$.

  • It answers the practical question: "If I had taken another sample, how different would my result be?"

Hypothesis Testing with Z Tests

• Applicability: Z tests are used when population parameters (population mean and standard deviation) are known.

• The Procedure:

  • A sample mean is compared to a known null population mean using the SEM (adjusted for sample size) as the standard of measure.

  • Z-Test Formulas:

    • Standardizing the mean: $Z = \frac{M - \mu}{SEM}$

    • Where $SEM = \frac{\sigma}{\sqrt{n}}$

• Z Test Example: College XYZ IQ Scores:

  • Known Population Parameters: Mean ($\mu$) = $100$, Standard Deviation ($\sigma$) = $15$.

  • Hypotheses:

    • Null Hypothesis ($H_0$): $\mu_1 = \mu_2$ (No difference).

    • Alternative Hypothesis ($H_1$): $\mu_1 \neq \mu_2$ (Difference exists).

• Scenario 1: Small Sample ($n = 10$):

  • Observed Mean ($M$) = $107$.

  • Alpha Level ($\alpha$) = $0.05$ (two-tailed).

  • Critical Value ($Z$) = $\pm 1.96$.

  • Calculation:

    • $SEM = \frac{15}{\sqrt{10}} = 4.7434... \approx 4.75$

    • $Z = \frac{107 - 100}{4.75} = 1.47$

  • Conclusion: 1.47 < 1.96. Since the observed $Z$ is smaller than the critical value, the null hypothesis is not rejected. The result is inconclusive.

• Scenario 2: Large Sample ($n = 50$):

  • Observed Mean ($M$) = $107$.

  • Calculation:

    • $SEM = \frac{15}{\sqrt{50}} = 2.1213... \approx 2.12$

    • $Z = \frac{107 - 100}{2.12} = 3.3$ (or $3.33$)

  • Conclusion: 3.33 > 1.96. Since the observed $Z$ is larger than the critical value, the null hypothesis is rejected in favor of the alternative. The class mean appears to come from a population with a mean higher than $100$.

Confidence Intervals (CI)

• Purpose: An alternative to the "all-or-none" decision of hypothesis testing. It establishes a range of values around a sample mean where the population mean is likely to reside.

• Calculation Components:

  • Uses the sample mean ($M$).

  • Uses the standard error (SEM).

  • Uses a specific level of confidence (typically $95\%$ or $99\%$).

• Confidence Levels and Z Scores:

  • For a $95\%$ CI, the Z score is $1.96$.

  • For a $99\%$ CI, the Z score is $2.57$.

• Formula:

  • $CI = M \pm (Z \times SEM)$

  • Lower Limit: $M - (Z \times SEM)$

  • Higher Limit: $M + (Z \times SEM)$

Confidence Interval Examples

• Example: IQ of 25 Children:

  • Given: $M = 107$, Variance of population = $324$, $n = 25$.

  • Step 1: Calculate SEM:

    • Variance of D of M = $\frac{324}{25} = 12.96$

    • $SEM = \sqrt{12.96} = 3.6$

  • $95\%$ Confidence Interval:

    • $107 \pm (1.96 \times 3.6)$

    • $107 \pm 7.056$

    • Interval: $99.944$ to $114.056$

  • $99\%$ Confidence Interval:

    • $107 \pm (2.57 \times 3.6)$

    • $107 \pm 9.252$

    • Interval: $97.748$ to $116.252$

• Observations on Intervals:

  • The $99\%$ CI is wider than the $95\%$ CI. A larger range is required to be more confident that the interval contains the population mean.

  • Effect of Sample Size:

    • As $n$ increases, SEM decreases, making the confidence interval smaller (more precise).

    • Example at $95\%$\ confidence: If $n = 25$, the range is $99.94$ to $114.06$. If $n = 100$, the range narrows to $103.65$ to $110.35$.

• Interpretation:

  • It is technically more accurate to say we are calculating a range of values that contains the true population mean based on the sample mean, rather than being "$95\%$\ confident this specific interval contains it."

SPSS Application and Data Visualization

• Error Bar Graphs:

  • Used to visualize confidence intervals for different categories.

  • Path: Graph -> Error Bar -> Simple -> Define.

• Interpreting Overlap:

  • Nonsignificant Results: Confidence intervals around the means will typically fall within each other's limits (they overlap significantly).

  • Significant Results: The confidence interval around one mean does not overlap with the confidence interval of the other mean.

• Variables Examined in Class:

  • Independent Variable (IV) vs. Dependent Variable (DV).

  • Examples: Family_Visit by Sex, Current_GPA by Sex.

Scholarly Research Examples

• Trypophobia and Comfort Levels (Pipitone & DiMattina, 2020):

  • Study involved $31$ trypophobic images with three manipulated versions (original, scrambled, phase).

  • Sample size: $146$ participants.

  • Results (Error bars representing $95\%$ CIs):

    • Phase explained the most variance in comfort ($24.9\%$).

    • Amplitude explained $9\%$\ variance.

    • Interaction of phase and amplitude explained $6.1\%$.

• Kinship and Fertility in Iceland (Helgason et al., 2008):

  • Analyzed all known Icelandic couples born between $1800$ and $1965$.

  • Found a significant positive association between kinship and fertility.

  • Greatest reproductive success observed for couples related at the level of third and fourth cousins.

  • Used $95\%$ confidence intervals across seven intervals of kinship level to measure:

    • Total number of children.

    • Number of children who reproduced.

    • Number of grandchildren.

    • Mean life expectancy of children.