A 2010 poll asked people in the United States whether they were satisfied with their financial situation. A total of 299 out of 1155 people said they were satisfied. The same question was asked in 2012, and 337 out of 830 people said they were satisfied.

Question

Construct a 99.5% confidence interval for the difference between the proportions of adults who said they were satisfied in 2012 and 2010. Let pv1 denote the proportion of individuals satisfied with their financial situation in 2012 and pv2 denote the proportion of individuals satisfied with their financial situation in 2010. Round the answers to three decimal places.

Answer 1

To construct a 99.5% confidence interval for the difference in proportions (\( p_1 - p_2 \)), we first need to calculate the sample proportions and then the standard error of the difference in proportions.

1. Calculate the sample proportions:

   • For 2010: \[ p_2 = \frac{299}{1155} \approx 0.258 \]
   • For 2012: \[ p_1 = \frac{337}{830} \approx 0.406 \]

2. Calculate the difference between the two sample proportions: \[ p_1 - p_2 \approx 0.406 - 0.258 = 0.148 \]

3. Calculate the standard error of the difference in proportions: The formula for the standard error (SE) of the difference between two proportions is: \[ SE = \sqrt{\frac{p_1(1 - p_1)}{n_1} + \frac{p_2(1 - p_2)}{n_2}} \] Where:

   • \( n_1 = 830 \) (sample size for 2012)
   • \( n_2 = 1155 \) (sample size for 2010)

   Calculating each part:

   • For \( p_1 \): \[ p_1(1 - p_1) = 0.406(1 - 0.406) = 0.406 \times 0.594 \approx 0.241 \]
   • For \( p_2 \): \[ p_2(1 - p_2) = 0.258(1 - 0.258) = 0.258 \times 0.742 \approx 0.191 \]

   Now we can calculate the standard error: \[ SE = \sqrt{\frac{0.241}{830} + \frac{0.191}{1155}} = \sqrt{\frac{0.241}{830} + \frac{0.191}{1155}} \approx \sqrt{0.000290 + 0.000165} \approx \sqrt{0.000455} \approx 0.0213 \]

4. Determine the critical z-value for a 99.5% confidence interval: For 99.5% confidence level, we need to find the z-value that corresponds to the upper tail probability of \( \frac{0.005}{2} = 0.0025 \) (since it's two-tailed). The z-value for 0.0025 can be found using standard z-tables or calculators: \[ z \approx 2.576 \]

5. Calculate the confidence interval: The confidence interval for the difference in proportions is given by: \[ (p_1 - p_2) \pm z \cdot SE \] Substituting the values: \[ 0.148 \pm 2.576 \cdot 0.0213 \] Calculate the margin of error: \[ 2.576 \cdot 0.0213 \approx 0.0549 \]

   Thus, the confidence interval is: \[ 0.148 \pm 0.0549 \]

   This yields: \[ (0.148 - 0.0549, 0.148 + 0.0549) \approx (0.0931, 0.2029) \]

6. Round to three decimal places: The final 99.5% confidence interval for the difference in proportions is: \[ \boxed{(0.093, 0.203)} \]