A group of 477 healthy children, aged 5-11 years, were randomly divided into two groups. Children in the "sugar-free" group were given one can (250 mL) of an artificially sweetened beverage each day for 18 months. Children in the "sugar-containing" group were given one can each day for 18 months of a beverage sweetened with sugar. The researchers developed the two drinks specifically for this experiment and made them taste and look as similar as possible.
The researchers recorded body mass index (BMI, which is weight [kg] divided by the square of height [m].) In order to control for age and sex effects, the response variable was change in BMI z score; that is, the number of SDs that a BMI differed from the national average for the child's age and sex. The table below shows the means and standard deviations of change in BMI z score for the two groups. Consider a t test with a directional alternative to investigate the research hypothesis that change in BMI is lower for the sugar-free condition.

(a) Write the null hypothesis and the alternative hypothesis in symbols.
(b) The P-value for the t test is 0.0003. Using α = 0.01, state your conclusion regarding the hypotheses in the context of this problem.

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Sugar-free Sugar-containing 0.02 0.41 0.15 0.42 Mean SD 225 252