Final answer:
A paired sample t-test should be used to analyze the data to determine if there is a statistical difference in CARS scores before and after the GFCF diet. With a mean difference of 1.0 and a standard deviation of 3, the test will compare to an alpha of 0.05 to assess significance. If the p-value is below alpha, the null hypothesis is rejected, indicating a potential effect.
Step-by-step explanation:
To evaluate if there is a difference after the GFCF (gluten-free, casein-free) diet on the severity of autistic symptoms as measured by the Childhood Autism Rating Scale (CARS), one would typically use a paired sample t-test. This statistical test is appropriate because you are comparing the means of two related groups (the same participants' CARS scores before and after the diet). The conditions that allow for a paired t-test, such as the differences having a normal distribution, are assumed to be met.
The hypothesis being tested is whether the mean difference in CARS scores is significantly different from zero. Given the provided mean difference of 1.0, a standard deviation of 3, and a significance level of alpha = 0.05, a paired sample t-test will determine if the observed mean difference is statistically significant.
If the p-value calculated from the t-test is less than the significance level (0.05), you would reject the null hypothesis. Since a hypothetical p-value given is 0.0004, which is indeed less than 0.05, the decision will be to reject the null hypothesis, suggesting that the diet may have an effect on the severity of autistic symptoms as measured by CARS.