An economist is evaluating how frequently the U.S. inflation rate exceeds the Federal Reserve's long-term target of 2% per $yr\thickapprox0.17\%$ per month. The economist finds that in 34 of the 48 sampled months, the monthly inflation rate did exceed $0.17\%$.
Under stable conditions the inflation rate should not exceed the target more than $20\%$ of the time. Can the economist conclude that inflation has exceeded the target more than $20\%$?

• A. No since $0.2$ is in $\left(0.61,0.75\right)$. 
• B. Yes since $0.2$ is not in $\left(0.58,0.84\right)$.
• C. No since $0.2$ is not in $\left(0.58,0.84\right)$.
• D. Yes since $0.2$ is in $\left(0.61,0.75\right)$. 

Answer: (B)

Verified step by step guidance

1. Step 1: Define the null hypothesis (H0) and the alternative hypothesis (H1). The null hypothesis (H0) states that the proportion of months where inflation exceeds the target is 20% (p = 0.20). The alternative hypothesis (H1) states that the proportion of months where inflation exceeds the target is greater than 20% (p > 0.20).
2. Step 2: Calculate the sample proportion (p̂). The sample proportion is the number of months where inflation exceeded the target divided by the total number of months sampled. Use the formula: p̂ = x / n, where x is the number of successes (34 months) and n is the total sample size (48 months).
3. Step 3: Compute the standard error (SE) of the sample proportion. The standard error is calculated using the formula: SE = sqrt((p0 * (1 - p0)) / n), where p0 is the hypothesized proportion (0.20) and n is the sample size (48).
4. Step 4: Calculate the z-test statistic. The z-test statistic is computed using the formula: z = (p̂ - p0) / SE, where p̂ is the sample proportion, p0 is the hypothesized proportion, and SE is the standard error calculated in Step 3.
5. Step 5: Compare the z-test statistic to the critical value or p-value. For a one-tailed test at a significance level of 0.05, the critical z-value is approximately 1.645. If the z-test statistic is greater than 1.645, reject the null hypothesis (H0) and conclude that inflation has exceeded the target more than 20% of the time. Otherwise, fail to reject the null hypothesis.