Question

Do Babies Prefer Speech? Psychologists in Montreal and Toronto conducted a study to determine if babies show any preference for speech over general noise. \(^{61}\) Fifty infants between the ages of \(4-13\) months were exposed to both happy-sounding infant speech and a hummed lullaby by the same woman. Interest in each sound was measured by the amount of time the baby looked at the woman while she made noise. The mean difference in looking time was 27.79 more seconds when she was speaking, with a standard deviation of 63.18 seconds. Perform the appropriate test to determine if this is sufficient evidence to conclude that babies prefer actual speaking to humming.

Short answer

To check whether babies prefer speech to humming, run a one-sample t-test with the null hypothesis that the mean difference in looking time is zero. If the resulting p-value is less than your chosen significance level (such as 0.05), then we can conclude that babies do show a preference for speech over humming.

Step-by-step solution

State the Null and Alternative Hypothesis

Null hypothesis, \(H_0\): There is no preference- The mean difference is equal to 0.\nAlternative hypothesis, \(H_a\): There is a preference- The mean difference is not equal to 0.

Perform Hypothesis Test

Using a t-test for the sample mean, calculate the test statistic using the formula: \(t = \frac{x - \mu_{0}}{\sigma / \sqrt{n}}\). \nWhere: \(\mu_{0}\)= the hypothesized population mean from the null hypothesis, x = sample mean, \(\sigma\) = standard deviation of the population, n = sample size or the number of observations. Here, \(\mu_{0} = 0\), x = 27.79, \(\sigma = 63.18\), n = 50. Then find the tail-area probability or p-value associated with this t-statistic.

Conclusion Based on the Results

If the p-value is less than your significance level (usually 0.05), you reject the null hypothesis in favor of the alternative. This would suggest that there is a statistical difference in babies' preference for speech over humming.

Key concepts

Null and Alternative Hypothesis

In the study examining whether babies have a preference for speech over humming, the null hypothesis (H0) is a statement of no effect. It posits that there is no difference in the amount of time babies look at a woman while engaged in speaking versus humming. In this case, it suggests the mean difference in looking time is zero. Conversely, the alternative hypothesis (Ha) denotes that there is a difference – that the mean looking time is not equal to zero which would suggest a preference. These hypotheses are fundamental to hypothesis testing as they frame the question being investigated and determine the statistical tests to be used.

In constructing these hypotheses, clarity and specificity are key. They ensure the test is accurately addressing the research question. Always define the populations, behaviours, or phenomena that your null and alternative hypotheses pertain to, to avoid ambiguity and to provide clear direction for your analysis.

T-test for Sample Mean

When examining mean differences, a t-test is often used, particularly when the sample size is small or the population variance is unknown. The t-test for the sample mean evaluates whether the observed sample mean significantly differs from a hypothesized or known population mean. The formula \(t = \frac{{x - \mu_0}}{{\sigma / \sqrt{n}}}\) represents this test statistic, where \(x\) is the sample mean, \(\mu_0\) is the hypothesized population mean, \(\sigma\) is the standard deviation of the sample, and \(n\) is the sample size.

In performing a t-test, it's important that the data meet certain assumptions including independence of observations, the random sample from the population, and a normally distributed population. When these assumptions are met, the t-test can be a powerful tool for assessing the probability that the observed sample comes from a particular population defined by the null hypothesis.

Statistical Significance

The concept of statistical significance is pivotal in making a decision on whether to accept or reject the null hypothesis. In the context of our example, it addresses the question of whether the difference in mean looking times is due to real effects or simply due to random chance. A result is deemed statistically significant if the observed probability of occurrence under the null hypothesis is less than a pre-defined threshold level, commonly \(\alpha = 0.05\).

Statistical significance denotes reliability in the results – that, if the experiment was repeated multiple times, the outcome would replicate itself consistently. However, it's crucial to understand that 'significant' does not necessarily equate to 'important' or 'meaningful.' Therefore, when interpreting results, one should consider not just the statistical significance but also the practical significance of the findings.

Mean Difference in Statistics

When discussing the mean difference in statistics, we refer to the arithmetic difference between the means of two groups or conditions. In experiments, this is often calculated to determine whether an intervention has made a significant change. The difference provides a measure of effect size which complements the statistical significance, offering insight into the magnitude of the finding.

To improve our understanding, it's essential to report the mean difference with a measure of variability such as the standard deviation or confidence interval. These additional statistics contextualize the mean difference, indicating the precision and reliability of the estimated effect.

P-value in Hypothesis Testing

The p-value in hypothesis testing is a measure of the strength of evidence against the null hypothesis. It represents the probability that the observed data, or data more extreme, could occur if the null hypothesis were true. A low p-value (< \(\alpha\), often set at 0.05) indicates that the observed data are unlikely under the null hypothesis and thus lead to its rejection in favor of the alternative.

It's crucial for students to remember that the p-value does not indicate the probability that the null hypothesis is true or false. It simply assesses whether the observed evidence is unusual under the assumption that the null hypothesis is correct. Interpretation of p-values should always be done in the context of the pre-determined significance level and the study design.