Paired Sample Statistics and Hypothesis Testing in Infant Behavioral Research

Research Context: Social Functions of Music in Infancy

• Core Research Inquiries:

  • Investigation into the underlying psychological and social functions that music serves for infant development.
  • Assessment of whether infants exhibit social bonding preferences: specifically, if parents sing a particular song to their infant, determining whether the infant is subsequently more likely to bond with an unfamiliar individual singing that identical song.

• Primary Empirical Reference:

  • Study: Mehr, S. A., Song, L. A., & Spelke, E. S. (2016). For 5-month-old infants, melodies are social. Psychological Science, 27, 486–501.
  • Data source file: lullabyexperiment.xlsx (retrieved via ICON).

• Experimental Design and Methodology:

  • Sample Size: $n = 32$ infants and their parents.
  • Familiarization Phase: Parents are taught a specific lullaby and instructed to sing it to their baby daily across a duration of $1$ to $2$ weeks.
  • Baseline Phase: The infant is presented with silent video recordings of two novel, unfamiliar people; researchers record and compute the baseline proportion of visual attention the baby devotes to each person.
  • Test Phase: The infant is presented with video recordings of the same two novel individuals singing; one sings the familiar lullaby previously taught by the parents, while the other sings an unfamiliar, novel lullaby. Researchers measure the proportion of visual attention the infant directs toward the person singing the familiar song.

Analytical Framework and Descriptive Analysis

• Core Analytical Objective:

  • Determine whether a statistically significant difference exists in the duration and proportion of attention an infant directs toward a novel individual when silent versus when singing the familiarized lullaby.

• Descriptive Statistical Procedures:

  • Calculate intra-subject difference scores between the Baseline condition and the Test condition for each infant ($d_i = \text{Test}_i - \text{Baseline}_i$).
  • Compute the sample mean difference ($\bar{d}$) across all $32$ subjects.
  • Construct a clustered column chart visually plotting individual Baseline and Test attentional proportions side by side for each infant in the dataset.

• Inferential Statistical Procedures:

  • Construct a paired-sample confidence interval around the mean difference ($\bar{d}$).
  • Conduct a formal paired-sample $t$-test to evaluate hypotheses regarding population differences.

Paired Sample Confidence Intervals

• Mathematical Formulation and Derivation Steps:

  • Compute individual subject differences: $d_i = x_{1,i} - x_{2,i}$.
  • Compute the sample mean of differences ($\bar{d}$):     $\bar{d} = \frac{1}{n} \sum_{i=1}^n d_i$
  • Calculate the sample standard deviation of differences ($s_d$):     $s_d = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (d_i - \bar{d})^2}$
  • Determine degrees of freedom ($df$):     $df = n - 1$
  • Determine the critical value ($t_{\alpha/2}$) using the two-tailed inverse Student's $t$-distribution function in Excel:     $= \text{T.INV.2T}(\alpha, df)$
  • Compute the standard error of the mean difference ($SE_{\bar{d}}$):     $SE_{\bar{d}} = \frac{s_d}{\sqrt{n}}$
  • Calculate the Margin of Error ($ME$):     $ME = t_{\alpha/2} \times \frac{s_d}{\sqrt{n}}$(Alternatively calculated directly in Excel via = CONFIDENCE.T(alpha, standard_dev, size)).
  • Establish the confidence interval limits:     $\text{Confidence Interval} = \bar{d} \pm t_{\alpha/2} \frac{s_d}{\sqrt{n}}$

• Empirical Findings and Interpretation:

  • The $95\%$ confidence interval for the true mean difference is determined to lie between $-13\%$ ($-0.13$) and $-1\%$ ($-0.01$).
  • Substantive meaning: Infants pay approximately $1\%$ to $13\%$ more focused attention to a novel individual when that individual sings a familiar lullaby compared to when that same individual is silent.
  • Inferential conclusion: Because the hypothesized mean difference of zero ($0\%$, representing null difference) falls entirely outside the $95\%$ confidence interval ($[-13\%, -1\%]$), the difference between the silent condition and the singing condition is statistically significant at the $\alpha = 0.05$ level.

Paired Two-Sample t-Test for Means

• Hypothesis Testing Setup:

  • Null Hypothesis ($H_0$): The true population mean difference between conditions is zero.     $H_0: \mu_d = 0$
  • Alternative Hypothesis ($H_A$): The true population mean difference between conditions is not zero (two-tailed test).     $H_A: \mu_d \neq 0$
  • Significance Threshold: $\alpha = 0.05$.

• Manual Calculation Procedure ("The Hard Way"):

  • Calculate the mean difference ($\bar{d}$) and standard deviation of differences ($s_d$) across all $n = 32$ participants.
  • Compute the test statistic ($t$):     $t = \frac{\bar{d} - \mu_0}{\frac{s_d}{\sqrt{n}}}$     where $\mu_0 = 0$ represents the hypothesized null population difference.
  • Calculate degrees of freedom:     $df = n - 1 = 32 - 1 = 31$
  • Derive the exact $p$-value in Excel utilizing the distribution function:     $= \text{T.DIST}(t, df, \text{cumulative})$ or = T.DIST.2T(ABS(t), df) for a two-tailed probability.

• Automated Implementation via Excel Analysis Toolpak ("The Easy Way"):

  • Open the Analysis Toolpak menu and select: t-Test: Paired Two Sample for Means.
  • Variable 1 Range: Reference the Baseline data column (cells $G$1:$G$33, including variable header label).
  • Variable 2 Range: Reference the Test data column (cells $H$1:$H$33, including variable header label).
  • Hypothesized Mean Difference: Enter 0.
  • Labels: Ensure the checkbox is checked to designate that row 1 contains variable names.
  • Alpha: Set to 0.05.
  • Output Options: Select New Worksheet Ply.

• Results and Empirical Conclusions:
  • The two-tailed $p$-value produced by the paired sample $t$-test equals $0.02$.
  • Comparative evaluation: Because $p = 0.02 < \alpha = 0.05$, the decision is to reject the null hypothesis ($H_0$).
  • Substantive conclusion: Infants exhibit a statistically significant difference in attention toward unfamiliar individuals who sing a parental lullaby versus unfamiliar individuals who remain silent, demonstrating empirical support for the hypothesis that infant social engagement and bonding are systematically enhanced by exposure to familiar melodies.

Independent Two-Sample t-Test (Assuming Unequal Variances)

• Sex-Based Comparative Inquiries:

  • Investigation into whether infant baseline attention toward silent strangers varies systematically by biological sex.
  • Research expectation: Male infants exhibit a greater baseline visual attention toward silent individuals than female infants.

• Hypothesis Formulation (One-Tailed Test):

  • Let index $1$ denote male infants and index $2$ denote female infants.
  • Null Hypothesis ($H_0$): The population mean attention for males is less than or equal to that of females.     $H_0: \mu_1 \le \mu_2$
  • Alternative Hypothesis ($H_A$): The population mean attention for males is strictly greater than that of females.     $H_A: \mu_1 > \mu_2$
  • Significance Level: $\alpha = 0.05$.

• Manual Calculation Procedure ("The Hard Way"):

  • Calculate sample metrics for each independent group:
  • Sample means: $\bar{x}_1$ and $\bar{x}_2$
  • Sample standard deviations: $s_1$ and $s_2$
  • Sample group sizes: $n_1$ and $n_2$
  • Compute the Welch's $t$-test statistic assuming unequal variances:     $t = \frac{(\bar{x}_1 - \bar{x}_2) - \Delta}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$     where $\Delta = 0$ represents the hypothesized difference under the null hypothesis.
  • Compute Welch–Satterthwaite degrees of freedom ($df$):     $df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{1}{n_1 - 1}\left(\frac{s_1^2}{n_1}\right)^2 + \frac{1}{n_2 - 1}\left(\frac{s_2^2}{n_2}\right)^2}$
  • Compute the corresponding $p$-value via = T.DIST(t, df, cumulative) or = T.DIST.RT(t, df) for an upper-tail test.

• Automated Implementation via Excel Analysis Toolpak ("The Easy Way"):

  • Open the Analysis Toolpak menu and select: t-Test: Two-Sample Assuming Unequal Variances.
  • Variable 1 Range: Input range for Group 1 (e.g., Male observations: $A$1:$A$16, containing $n_1 = 15$ observations plus header label).
  • Variable 2 Range: Input range for Group 2 (e.g., Female observations: $B$1:$B$18, containing $n_2 = 17$ observations plus header label).
  • Hypothesized Mean Difference: Enter 0.
  • Labels: Ensure the checkbox is checked.
  • Alpha: Set to 0.05.
  • Output Options: Select New Worksheet Ply.

Academic Support and Supplemental Instruction (SI)

• Supplemental Instruction (SI) Framework:

  • Structured academic support program led by an undergraduate SI Leader (Ava) who previously achieved high academic standing in the course.
  • Total frequency: Three weekly sessions.

• Schedule and Location Details:

  • Mondays: 10:30 am
  • Wednesdays: 4:30 pm
  • Thursdays: 2:00 pm
  • Location: Room C410, Pappajohn Business Building (PBB)

• Session Scope and Structure:

  • Delivers interactive, collaborative learning activities focused on statistical problem-solving, cognitive study strategies, and quiz preparation.
  • Policy Boundary: SI sessions do not constitute homework completion sessions or individual assignment assistance; specific homework and graded assignment questions must be directed to faculty or teaching assistant office hours.