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Suppose that special federal funds are available for your state when the mean period of unemployment exceeds 40 weeks. As an economist with your state’s department of labor, you want to test whether the mean period of unemployment is more than 40 weeks. A random sample of 60 unemployed persons reveals the following. Sample Size = 60, Sample Mean = 41.7 weeks, Sample Standard Deviation = 6.1 weeks. Does the sample data provide sufficient evidence to conclude that the population mean period of unemployment is greater than 40 weeks (using α = 0.05)?

Paper For Above instruction

The increasing duration of unemployment poses significant economic and social challenges, prompting policymakers and economists to analyze whether recent trends reflect a true rise in unemployment durations. In this context, a hypothesis test can be used to evaluate whether the mean period of unemployment has exceeded a critical threshold—specifically, 40 weeks—using a sample drawn from the unemployed population. This paper systematically approaches such a hypothesis test, interpreting the implications for policy and economic conditions if the mean unemployment duration is found to be larger than 40 weeks.

Formulating the Hypotheses

The initial step involves establishing null and alternative hypotheses. The null hypothesis (H₀) assumes that the population mean unemployment duration is 40 weeks:

H₀: μ = 40

The alternative hypothesis (H₁), reflecting the concern that the duration has increased, states:

H₁: μ > 40

This sets up a right-tailed test, focusing on whether the mean exceeds 40 weeks.

Significance Level

The level of significance (α) is set at 0.05, indicating a 5% risk of rejecting the null hypothesis when it is actually true. This threshold balances the risk of Type I error and is standard in hypothesis testing.

Critical Value and Rejection Region

Given the sample size (n = 60) and the sample standard deviation, the test statistic follows a t-distribution with n - 1 = 59 degrees of freedom. Using a t-table or statistical software, the critical t-value for α = 0.05 in a one-tailed test with 59 df is approximately 1.671. The rejection region consists of values greater than this critical t-value, i.e., t > 1.671.

Calculating the Test Statistic

The test statistic (t) is calculated using the formula:

\[

t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}

\]

Where:

- \(\bar{x} = 41.7\) (sample mean)

- \(\mu_0 = 40\) (hypothesized population mean)

- \(s = 6.1\) (sample standard deviation)

- \(n = 60\)

Plugging in the values:

\[

t = \frac{41.7 - 40}{6.1 / \sqrt{60}} = \frac{1.7}{6.1 / 7.746} \approx \frac{1.7}{0.787} \approx 2.159

\]

The calculated t-value is approximately 2.159.

Decision Regarding the Null Hypothesis

Since 2.159 > 1.671, the test statistic falls into the rejection region. Consequently, we reject the null hypothesis at the 0.05 significance level.

Interpretation of the Results

Rejecting H₀ suggests that there is statistically significant evidence to support the claim that the population mean duration of unemployment exceeds 40 weeks. This indicates that unemployment durations are indeed longer than the critical threshold, underscoring potential issues in the labor market, such as skill mismatches or economic downturns prolonging unemployment spells.

Observed p-value and Its Interpretation

The p-value associated with the test statistic can be obtained using t-distribution tables or statistical software. For t = 2.159 and df = 59, the p-value is approximately 0.017. Since p

Conclusion and Policy Implications

Based on the hypothesis testing conducted, there is sufficient evidence at the 5% significance level to conclude that the mean unemployment duration exceeds 40 weeks. This finding has important policy implications; extended unemployment spells can signify structural issues within the labor market, such as mismatched skills, inadequate employment services, or economic slowdowns. Policymakers might consider targeted interventions, such as retraining programs or incentives for hiring, to address prolonged unemployment. Moreover, continuous monitoring of unemployment duration trends will be vital for making timely policy adjustments and for understanding the evolving nature of the labor market challenges.

Significance of Findings for Economic Policy

The results align with broader economic concerns about labor market health. If unemployment durations are consistently exceeding conventional thresholds, it suggests that workers face persistent barriers to re-employment, which can depress consumption, increase social welfare costs, and hinder economic growth. Recognizing these trends early enables policymakers to implement strategic measures to facilitate faster re-employment and economic stability.

Limitations and Future Research

While the statistical evidence indicates that the mean unemployment duration exceeds 40 weeks, the analysis is based on a sample that may not capture the entire population's heterogeneity. Further research could involve larger, more diverse samples, longitudinal studies, and analysis of additional factors influencing unemployment durations such as industry-specific trends, education levels, or regional economic conditions. Such comprehensive analysis would enable more nuanced policy responses tailored to the root causes of prolonged unemployment.

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