Question

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 study1 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 autisticmale 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.)^1Adapted from Courchesne,E., et.al., "Neuron Number and Size in Prefrontal Cortex of Children with Autism," Journal of the American Medical Association, November 2011; 306(18): 2001-2010.

State the null and alternative hypotheses.
Give the test statistic and the P -value and state the conclusion of the test.
Round your answer for the test statistic to two decimal places and your answer for the P -value to three decimal places,
test statistic =
P -value =
Conclusion:
___ H_0.

Does the sample provide evidence that autistic male children have more neurons (on average) in the prefrontal cortex than non-autistic children?
O Yes
O No

Answer 1

The null hypothesis states that the mean number of neurons in the prefrontal cortex in autistic male children is the same as in non-autistic male children. The alternative hypothesis states that the mean number of neurons in the prefrontal cortex in autistic male children is greater than in non-autistic male children. A one-sample t-test is performed to test this hypothesis.

• The null hypothesis (H0) states that the mean number of neurons in the prefrontal cortex in autistic male children is the same as the mean number of neurons in non-autistic male children.
• The alternative hypothesis (H1) states that the mean number of neurons in the prefrontal cortex in autistic male children is greater than the mean number of neurons in non-autistic male children.
• To test this hypothesis, we can perform a one-sample t-test.
• The test statistic is calculated by subtracting the hypothesized mean (1-15 billion) from the sample mean (1.94 billion) and dividing it by the sample standard deviation (0.50 billion) divided by the square root of the sample size (7).
• The calculated test statistic is 5.80, and the p-value is less than 0.001.
• Since the p-value is less than the significance level (usually 0.05), we reject the null hypothesis and conclude that there is evidence to suggest that autistic male children have more neurons in the prefrontal cortex than non-autistic male children.
• Conclusion: Yes, the sample provides evidence that autistic male children have more neurons (on average) in the prefrontal cortex than non-autistic children.