An economist is evaluating how frequently the U.S. inflation rate exceeds the Federal Reserve's long-term target of 2% per $yr\approx 0.17\%$ per month. The economist finds that in 34 of the 48 sampled months, the monthly inflation rate did exceed $0.17\%$.
Make a $95\%$ confidence interval for the true proportion of months in which the inflation rate exceeds the target.
We are ___$\%$ confident that the inflation rate exceeds the target in between (_––––,_––––) of months. 

• A. We are $95\%$ confident that the inflation rate exceeds the target in between $(0.61,0.75)$ of months. 
• B. We are $95\%$ confident that the inflation rate exceeds the target in between $(0.58,0.75)$ of months. 
• C. We are $95\%$ confident that the inflation rate exceeds the target in between $(0.61,0.84)$ of months. 
• D. We are $95\%$ confident that the inflation rate exceeds the target in between $(0.58,0.84)$ of months. 

Answer: (D)

Verified step by step guidance

1. Step 1: Identify the sample proportion (p̂) by dividing the number of months where the inflation rate exceeded the target (34) by the total number of sampled months (48). The formula is p̂ = x / n, where x is the number of successes and n is the sample size.
2. Step 2: Calculate the standard error (SE) for the sample proportion using the formula SE = sqrt((p̂ * (1 - p̂)) / n), where p̂ is the sample proportion and n is the sample size.
3. Step 3: Determine the z-value for a 95% confidence level. For a 95% confidence interval, the z-value is approximately 1.96 (this value corresponds to the critical value for a two-tailed test).
4. Step 4: Calculate the margin of error (ME) using the formula ME = z * SE, where z is the z-value for the desired confidence level and SE is the standard error.
5. Step 5: Construct the confidence interval by adding and subtracting the margin of error from the sample proportion. The confidence interval is given by (p̂ - ME, p̂ + ME).