91Ó°ÊÓ

The U.S. Department of Labor collects data on unemployment insurance payments made to unemployed people in different states. Suppose that during 2011 a random sample of 1000 unemployed people in Florida received an average weekly unemployment benefit of \(\$ 219.65\), while a random sample of 900 unemployed people in Mississippi received an average weekly unemployment benefit of \(\$ 191.47\). Assume that the population standard deviations of 2011 weekly unemployment benefits paid to all unemployed workers in Florida and Mississippi were \(\$ 35.15\) and \(\$ 28.22\), respectively. (Note: A 2011 study by DailyFinance.com (http://www.dailyfinance.com/2011/05/12/unemployment- benefits-best-worst-states/) rated Mississippi and Florida as the two worst states for unemployment benefits.) a. Let \(\mu_{1}\) and \(\mu_{2}\) be the means of weekly unemployment benefits paid to all unemployed workers during 2011 in Florida and Mississippi, respectively. What is the point estimate of \(\mu_{1}-\mu_{2} ?\) b. Construct a \(96 \%\) confidence interval for \(\mu_{1}-\mu_{2}\). c. Using a \(2 \%\) significance level, can you conclude that the means of all weekly unemployment benefits paid to all unemployed workers during 2011 in Florida and Mississippi are different? IIse both the \(p\) -value and the critical-value annroaches to make this.te-

Step by step solution

01

Compute Point Estimate

The point estimate of \(\mu_{1}-\mu_{2}\) would be calculated as the difference between the sample means. Thus, the point estimate = \(\bar{x_{1}} - \bar{x_{2}}\) where \(\bar{x_{1}} = $219.65\) and \(\bar{x_{2}} = $191.47\).

02

Construct Confidence Interval

To construct a 96% confidence interval for \(\mu_{1}-\mu_{2}\), the Z statistic value corresponding to this confidence level (approximately 2.05) and the standard errors should be used. The confidence interval can be expressed as \((\bar{x_{1}} - \bar{x_{2}}) \pm Z_{\alpha/2} * \sqrt{\frac{S_{1}^{2}}{n_{1}} + \frac{S_{2}^{2}}{n_{2}}}\) where \(S_{1} = $35.15, S_{2} = $28.22, n_{1} = 1000\) and \(n_{2} = 900\).

03

Conduct a Significance Test

To interpret if the means are significantly different, a hypothesis test should be conducted. You can set the null hypothesis as: \(H_{0}: \mu_{1} = \mu_{2}\), and the alternative hypothesis as: \(H_{a}: \mu_{1} \neq \mu_{2}\). Using both p-value and critical-value approaches, you can then make a decision. For the p-value approach, calculate the p-value using the test statistic and if it's less than the significance level (0.02), reject the null hypothesis. Same applies for critical-value approach, you compute the critical value using the significance level and if the test statistic exceeds the critical value, you can reject the null hypothesis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Confidence Intervals

Confidence intervals provide a range of values that likely contain the true difference between two population means. In this exercise, we want to find a 96% confidence interval for the difference in weekly unemployment benefits between Florida and Mississippi. This means we calculate a range where we expect the true difference in mean weekly benefits for all unemployed individuals in these states to fall, with 96% certainty.

To construct this interval, we take the difference between the sample means (Florida's and Mississippi's average benefits), then adjust it by a margin of error. The margin of error accounts for variability and is influenced by the sample size and standard deviations of both states' benefits.

In our exercise, the confidence interval is calculated using the Z statistic for 96% confidence, which is approximately 2.05. Then, the formula includes the variability measured by the population standard deviations and sample sizes. A confidence interval helps in assessing if there's a statistically significant difference between the states, which is crucial for hypotheses testing.

Exploring Point Estimates

The point estimate is like a best guess for the true difference in what unemployed individuals receive weekly in the two states based on our sample data. It's simply the difference between the two sample means. Here, we calculate it by subtracting Mississippi's average unemployment benefit from Florida's average.

In this problem, the point estimate is calculated as \[ \bar{x_1} - \bar{x_2} = 219.65 - 191.47 = 28.18 \]

This result, $28.18, suggests that, in sample terms, Florida's unemployment benefits were on average this much higher than Mississippi's in 2011. It's important to consider that this single number does not convey the entire picture since it provides no information about the range of possible true differences or the uncertainty inherent in sample-based observations.

Significance Level in Hypothesis Testing

The significance level, often denoted by \( \alpha \), is a threshold used in hypothesis testing to determine whether to reject the null hypothesis. In this example, the significance level is set at 2% (0.02), indicating a high standard required to claim a significant difference in the unemployment benefits between the two states.

This level reflects the probability of rejecting the null hypothesis when it is actually true (Type I error). If our calculated p-value falls below this threshold, it suggests that the observed data is unlikely under the assumption that there's no difference in means, leading us to consider the alternative hypothesis that the means are different. Similarly, in the critical-value approach, we compare the test statistic to a pre-determined critical value, deciding to reject the null hypothesis if our test statistic exceeds this critical value.

A lower significance level implies more stringent criteria for determining a statistically significant result, important for drawing robust conclusions in analysis.

Unpacking Unemployment Benefits Statistics

The statistics surrounding unemployment benefits provide insight into the financial support offered to unemployed individuals within specific regions. The exercise focuses on two states, Florida and Mississippi, highlighting differences in average benefits during 2011.

Statistics such as the average benefit amount, sample size, and standard deviation provide a framework to evaluate variations between states. By analyzing these figures, researchers can gauge how state policies may affect unemployment benefits. The figures, e.g., Florida averaging $219.65 compared to Mississippi's $191.47, already indicate apparent differences, further emphasized through confidence intervals and hypothesis tests.

These statistics are crucial for policymakers in evaluating the effectiveness of unemployment benefits and making informed decisions aimed at improving financial assistance programs for unemployed individuals across different regions.