Question

The Autistic Brain Autistic children often have a small head circumference at birth, followed by a sudden and excessive increase in head circumference during the first year of life. A recent study \(^{29}\) examined the brain tissue in autopsies of seven autistic male children between the ages of 2 and 16. The mean number of neurons in the prefrontal cortex in non-autistic male children of the same age is about 1.15 billion. The prefrontal cortex is the part of the brain most disrupted in autism, as it deals with language and social communication. In the sample of seven autistic children, the mean number of neurons in the prefrontal cortex was 1.94 billion with a standard deviation of 0.50 billion. The values in the sample are not heavily skewed. Use the t-distribution to test whether this sample provides evidence that autistic male children have more neurons (on average) in the prefrontal cortex than non-autistic children. (This study indicates that the causes of autism may be present before birth.)

Short answer

The answer would be dependent on the calculated t-value and the critical t-value for \( df=6 \) degrees of freedom. If the calculated t-value exceeds the critical t value, we would reject the null hypothesis and conclude that there is evidence that the average number of neurons in the prefrontal cortex of autistic children is greater than that of non-autistic children.

Step-by-step solution

State the null and alternative hypotheses

Our null hypothesis is that the mean number of neurons for autistic children is equal to that of non-autistic children (\( \mu = 1.15 \) billion) and the alternative hypothesis is that the mean for autistic children is greater than that of non-autistic children (\( \mu > 1.15 \) billion).

Choose hypothesis test

We are going to carry out a one-sample t-test since we are comparing the sample mean to a known value.

Calculate the test statistic

The formula for t-value is \( t = \frac{\overline{x} - \mu}{s/\sqrt{n}} \), where \( \overline{x} \) is the sample mean, \( \mu \) is the assumed population mean, \( s \) is the standard deviation and \( n \) is the number of values in the sample. So, \( t = \frac{1.94 - 1.15}{0.50/\sqrt{7}} \).

Find the p-value

We will compare our calculated t-value with t-distribution table for \( df = n-1 = 6 \) degrees of freedom and our \( \alpha = 0.05 \). If our t-value is greater than critical t-value, we will reject the null hypothesis in favor of the alternative.

Interpret the results

If we reject the null hypothesis, this supports the claim that autistic children have a higher mean number of neurons in the prefrontal cortex than non-autistic children.

Key concepts

Autism and Brain Development

Autism, a neurodevelopmental disorder, presents challenges in social interaction, communication, and restricted interests or repetitive behaviors. Research suggests that differences in brain development, detectable as early as the first year of life, could be linked to its manifestation.

For instance, studies have noted atypical patterns of growth in the heads of autistic children, showing a smaller head circumference at birth that expands rapidly within the first year. This could correlate with changes in the number of neurons in various parts of the brain, especially the prefrontal cortex, which is vital for complex functions like language and social interaction. The prefrontal cortex's alteration in autistic individuals might contribute to the difficulties experienced in these areas.

An increase in the number of neurons in the prefrontal cortex, as indicated by autopsy studies in autistic children, hints at a neurobiological foundation for the disorder. These findings support the hypothesis that autism stems from developmental differences that begin before birth, adding another layer to our understanding of the condition's complexity.

One-sample t-test

The one-sample t-test is a statistical method used to determine whether the mean of a single sample differs significantly from a known or hypothesized population mean. It's particularly useful when the population standard deviation is unknown and the sample size is small, with the t-distribution providing a more accurate estimation of the standard error.

In our autism study, a one-sample t-test compares the average number of neurons in the prefrontal cortex between autistic and non-autistic male children. We calculate the test statistic by comparing the sample mean to the assumed population mean, considering the sample standard deviation and size. The result helps determine if the observed difference is due to natural variation or if it's statistically significant, suggesting a true difference in neuron count.

Null and Alternative Hypotheses

Central to hypothesis testing is defining the null hypothesis (H0) and the alternative hypothesis (Ha). The null hypothesis represents a default position that there is no effect or no difference, and it is what we attempt to reject through our test. In the context of our study, H0 posits that the mean number of neurons in the prefrontal cortex of autistic children is equal to that of non-autistic children.

The alternative hypothesis, on the other hand, proposes what we suspect might be true instead. For the autism study, Ha contends that autistic children have, on average, a greater number of neurons in the prefrontal cortex than non-autistic children. By conducting the t-test, we aim to gather evidence to support or refute Ha and hence determine whether H0 can be rejected.

Statistical Significance

Statistical significance is a vital concept in hypothesis testing that helps researchers determine if their findings are likely the result of a real effect rather than random chance. A result is considered statistically significant if the observed p-value is less than the predefined threshold, often set at 0.05 (5% significance level).

In our exercise, statistical significance comes into play when we compare the calculated t-value against the critical value from the t-distribution table. If the t-value exceeds the critical value, the p-value will be lower than our threshold, leading us to reject the null hypothesis. Rejecting H0 indicates a statistically significant difference in neuron count between autistic and non-autistic children's prefrontal cortex, offering evidence that developmental brain differences in autism are present from an early stage.