Given the confidence interval [tex]\(-0.000482 < p_1 - p_2 < -0.000329\)[/tex]:
1. Understanding the Confidence Interval:
- The confidence interval provides a range of values within which the true difference in proportions [tex]\( p_1 - p_2 \)[/tex] is likely to fall.
- Here, [tex]\( p_1 \)[/tex] represents the proportion of children who got polio after receiving the vaccine, and [tex]\( p_2 \)[/tex] represents the proportion of children who got polio after receiving a placebo.
2. Analyzing the Claim:
- The claim suggests that the rate of polio is different between the group given the vaccine and the group given the placebo.
- We need to see if this interval includes zero to determine if there is a significant difference between the rates.
3. Interpreting the Results:
- The confidence interval does not include zero (0). This implies there's no overlap including zero between the lower and upper bounds of the interval.
- Because the confidence interval does not contain zero, it suggests there is a significant difference between the two proportions.
4. Direction of the Difference:
- The entire confidence interval consists of values less than zero. This indicates that [tex]\( p_1 - p_2 \)[/tex] is consistently negative within the interval.
- Since [tex]\( p_1 < p_2 \)[/tex] within this range, it suggests that the proportion of children who got polio is less in those given the vaccine compared to those given the placebo.
To summarize:
- Because the confidence interval does not contain zero, there does appear to be a significant difference between the two proportions.
- Because the confidence interval consists only of negative values, it appears that the first proportion ([tex]\( p_1 \)[/tex]) is less than the second proportion ([tex]\( p_2 \)[/tex]).
- There is evidence to support the claim that the rate of polio is less for children given the vaccine than it is for children given a placebo.
