Question

N a small town of 5,832 people, the mayor wants to determine if there is a difference in the proportion of voters ages 18–30 who would support an increase in the food tax, and the proportion of voters ages 31–40 who would support an increase in the food tax. an assistant to the mayor surveys 85 randomly chosen voters ages 18–30, and finds that 62 support the increase. a random sample of 70 voters ages 31–40 is also surveyed, and 56 support the increase. assuming the conditions for inference have been met, what is the 99% confidence interval for the difference in proportions of voters who would support the increase in the food tax for the different age groups?

Answer 1

To calculate the 99% confidence interval for the difference in support for a food tax increase between voters ages 18–30 and 31–40, we determine the sample proportions, compute the pooled sample proportion, find the standard error, and use the appropriate z-score to find the range that covers the true population proportion difference with 99% confidence.

Step-by-step explanation:

To determine the 99% confidence interval for the difference in proportions of voters who would support the increase in the food tax for the different age groups, we first calculate the sample proportions. For voters ages 18–30, the proportion is 62/85, and for voters ages 31–40, it is 56/70. We then use these proportions to calculate the standard error of the difference in proportions and subsequently the confidence interval using a z-score for 99% confidence.

Step-by-Step Calculation

1. Calculate the sample proportions: p1 = 62/85 for ages 18–30 and p2 = 56/70 for ages 31–40.
2. Calculate the pooled sample proportion (p) using the formula p = (x1 + x2) / (n1 + n2), where x1 and x2 are the number of successes and n1 and n2 are the sample sizes.
3. Calculate the standard error (SE) using the formula SE = sqrt(p * (1 - p) * (1/n1 + 1/n2)).
4. Find the z-score for a 99% confidence level, which is approximately 2.576.
5. Calculate the confidence interval using the formula (p1 - p2) ± z * SE.

Note that rounding will be applied to the final confidence interval as needed. This interval will provide us with the range in which the true difference in population proportions lies with 99% confidence, provided that the conditions for inference have been met.