Question

Statistical Inference in Babies Is statistical inference intuitive to babies? In other words, are babies able to generalize from sample to population? In this study, \(1 \quad 8\) -month-old infants watched someone draw a sample of five balls from an opaque box. Each sample consisted of four balls of one color (red or white) and one ball of the other color. After observing the sample, the side of the box was lifted so the infants could see all of the balls inside (the population). Some boxes had an "expected" population, with balls in the same color proportions as the sample, while other boxes had an "unexpected" population, with balls in the opposite color proportion from the sample. Babies looked at the unexpected populations for an average of 9.9 seconds \((\mathrm{sd}=4.5\) seconds) and the expected populations for an average of 7.5 seconds \((\mathrm{sd}=4.2\) seconds). The sample size in each group was \(20,\) and you may assume the data in each group are reasonably normally distributed. Is this convincing evidence that babies look longer at the unexpected population, suggesting that they make inferences about the population from the sample? (a) State the null and alternative hypotheses. (b) Calculate the relevant sample statistic. (c) Calculate the t-statistic.

Short answer

The null hypothesis (\(H_0\)) is that there is no difference between the average time babies look at unexpected and expected populations (\(\mu_{unexpected} = \mu_{expected}\)). The alternative hypothesis (\(H_1\)) is that babies look for longer at unexpected populations (\(\mu_{unexpected} > \mu_{expected}\)). Using the provided values in the two-sample t-test formula yields the t-statistic.

Step-by-step solution

State the null and alternative hypotheses

The null hypothesis assumes there is no difference in the average time babies look at the unexpected and expected populations. So, we set them to be equal: \(H_0: \mu_{unexpected} = \mu_{expected}\). The alternative hypothesis assumes babies look longer at the unexpected population, which means we are claiming that \(\mu_{unexpected} > \mu_{expected}\). Hence, \(H_1: \mu_{unexpected} > \mu_{expected}\).

Compute the sample statistic

The sample statistics are the means and standard deviations of the two groups. These have been given as follows: Mean for unexpected populations \(\mu_{unexpected} = 9.9\) seconds, standard deviation \(\mathrm{sd}_{unexpected} = 4.5\) seconds, Mean for expected populations \(\mu_{expected} = 7.5\) seconds and standard deviation \(\mathrm{sd}_{expected} = 4.2\) seconds.

Calculate the t-statistic

We use the formula for two-sample t-test statistic: \(t = \frac{(\bar{X}_{unexpected} - \bar{X}_{expected}) - d_{0}}{\sqrt{\frac{s_{unexpected}^2}{n_{unexpected}} + \frac{s_{expected}^2}{n_{expected}}}}\), where \(d_{0}\) (assumed mean difference) is zero since the null hypothesis states that average times are equal. Substituting \(\bar{X}_{unexpected} = 9.9\), \(s_{unexpected} = 4.5\), \(n_{unexpected} = 20\), \(\bar{X}_{expected} = 7.5\), \(s_{expected} = 4.2\), \(n_{expected} = 20\), and solving will give the t-statistic.

Key concepts

Null and Alternative Hypotheses

Understanding the null and alternative hypotheses is fundamental to statistical inference. These two hypotheses represent contrasting assertions about a population parameter that we aim to evaluate based on sample data.

Null Hypothesis (H0): This is the assumption that there is no effect or no difference, and it serves as a default or starting assumption in statistical tests. In the context of our exercise, the null hypothesis posits that there is no difference in the average time that babies look at unexpected and expected populations. It implies that any observed difference in sample data arises due to random chance rather than any actual difference in the population.

Alternative Hypothesis (H1 or Ha): This hypothesis is what a researcher is typically interested in proving. Contrary to the null hypothesis, it suggests there is an effect or a difference. In the baby study case, the alternative hypothesis is that babies indeed look longer at unexpected populations than expected ones. The direction of the alternative hypothesis is important as it can be one-sided (as in this study) or two-sided, depending on the research question.

Setting up these hypotheses correctly is a critical step for conducting the appropriate statistical test, such as a t-test in this exercise to reach a conclusion about the population based on the sample data.

Sample Statistics

Sample statistics play a significant role in making inferences about populations. These statistics summarize and describe the characteristics of a sample, providing an estimation of the population parameters.

Key Sample Statistics: The most common sample statistics include the sample mean, variance, and standard deviation. In our case, the study presented the sample mean and standard deviation for the time infants observed unexpected and expected populations. These are:

• Mean (unexpected) = 9.9 seconds
• Standard deviation (unexpected) = 4.5 seconds
• Mean (expected) = 7.5 seconds
• Standard deviation (expected) = 4.2 seconds

Sample statistics help us estimate the true population mean and variance, but it's essential to remember they come with a margin of error. We use these metrics to compute our test statistic and further the process of statistical inference. They give us insight into the behavior of the sample that, under certain conditions, can be generalized to the population.

T-Statistic

The t-statistic is a type of test statistic that follows a distribution known as the Student's t-distribution under the null hypothesis. It's widely used when dealing with small sample sizes or when the population standard deviation is unknown.

Calculating the T-Statistic: To calculate the t-statistic, we subtract the null hypothesis value from the sample statistic (typically the sample mean) and then divide by the standard error of the statistic. The standard error measures the variability of the sample statistic. In the infants' study scenario, we use the formula for the t-statistic given by:
\[t = \frac{(\bar{X}_{unexpected} - \bar{X}_{expected})}{\sqrt{\frac{s_{unexpected}^2}{n_{unexpected}} + \frac{s_{expected}^2}{n_{expected}}}}\]
where \(\bar{X}\) is the sample mean, \(s^2\) is the sample variance, and \(n\) is the sample size. By substituting the provided values, we compute the t-statistic which is used to test our hypotheses. If the calculated t-statistic is larger than the critical value from the t-distribution for a given significance level, we reject the null hypothesis in favor of the alternative.

This t-statistic, alongside degrees of freedom, helps us determine the p-value or probability of observing a result as extreme as, or more than, the one observed if the null hypothesis were true. The smaller the p-value, the stronger the evidence against the null hypothesis, thereby supporting the alternative hypothesis.