91Ó°ÊÓ

The U.S. Department of Labor collects data on unemployment insurance payments. Suppose that during 2009 a random sample of 70 unemployed people in Alabama received an average weekly benefit of \(\$ 199.65\), whereas a random sample of 65 unemployed people in Mississippi received an average weekly benefit of \(\$ 187.93\). Assume that the population standard deviations of all weekly unemployment benefits in Alabama and Mississippi are \(\$ 32.48\) and \(\$ 26.15\), respectively. a. Let \(\mu_{1}\) and \(\mu_{2}\) be the means of all weekly unemployment benefits in Alabama and Mississippi paid during 2009 , respectively. What is the point estimate of \(\mu_{1}-\mu_{2}\) ? b. Construct a \(96 \%\) confidence interval for \(\mu_{1}-\mu_{2}\). c. Using the \(4 \%\) significance level, can you conclude that the means of all weekly unemployment benefits in Alabama and Mississippi paid during 2009 are different? Use both approaches to make this test.

Step by step solution

01

Compute the Point Estimate of \(\mu_{1}-\mu_{2}\)

The point estimate of the difference in means, \(\mu_{1}-\mu_{2}\), is simply given by the difference in sample means. Using the given data, \( \bar{x}_{1} = \$199.65, \bar{x}_{2} = \$187.93 \), the point estimate of \(\mu_{1}-\mu_{2}\) is \(\bar{x}_{1} - \bar{x}_{2} = \$199.65 - \$187.93 = \$11.72\).

02

Construct a 96% Confidence Interval for \(\mu_{1}-\mu_{2}\)

A 96% confidence interval for (\(\mu_{1}-\mu_{2}\)) can be obtained using the formula: \[\bar{x}_{1} - \bar{x}_{2} \pm Z_{\alpha/2}\sqrt{\frac{\sigma_{1}^2}{n_{1}} + \frac{\sigma_{2}^2}{n_{2}}}\] Where \(\sigma_{1}\), \(\sigma_{2}\) are the standard deviations of the first and second populations, and \(n_{1}\), \(n_{2}\) are the sample sizes of the first and second samples. \(Z_{\alpha/2}\) can be obtained from the standard normal distribution table, which gives us the value of 2.05. As per the values available in the exercise: \[CI = \$11.72 \pm 2.05 \sqrt{\frac{(\$32.48)^2}{70} + \frac{(\$26.15)^2}{65}}\] Upon calculation, this will give us the required confidence interval.

03

Test if the means are statistically different

To test if \(\mu_{1}\) and \(\mu_{2}\) are different, we do a two-tailed hypothesis test. For the 4% level of significance, the null hypothesis, \( H_{0}: \mu_{1} = \mu_{2} \) signifies that the means are equal, while the alternative, \( H_{1}: \mu_{1} \neq \mu_{2} \) is that they are unequal. The test statistic is \[Z = \frac{(\bar{x}_{1}-\bar{x}_{2})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^2}{n_{1}} + \frac{\sigma_{2}^2}{n_{2}}}}\] We can use this test statistic and compare it to our critical value from the standard normal table. If the computed statistic is beyond the critical value, we reject the null hypothesis and conclude that the means are statistically different.

04

P-value approach

This approach consists of calculating a p-value, which is the smallest level of significance at which we would reject the null hypothesis, given the observed sample. If the obtained p-value is lower than the significance level of 4%, we reject the null hypothesis and consider the means as different.

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Key Concepts

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

Hypothesis Testing

Hypothesis testing is a vital process in statistics that allows us to make judgements about a population based on sample data. In the context of the exercise given, hypothesis testing helps determine whether the average weekly unemployment benefits in Alabama and Mississippi are different.

The process begins with formulating two hypotheses:

• **Null Hypothesis ( \( H_0 \)):** This usually states that there is no effect or no difference. In our exercise, it assumes that the means of the benefits are equal, written as \( \mu_1 = \mu_2 \).
• **Alternative Hypothesis ( \( H_1 \)):** Contrary to the null, it suggests that there is an effect or a difference. Here, it posits that the means are different, represented by \( \mu_1 eq \mu_2 \).

The objective is to analyze sample data to decide which hypothesis is supported. If the sample provides strong evidence against the null hypothesis, it is rejected in favor of the alternative.

Point Estimate

The concept of a point estimate is crucial in summarizing data into a single value. This is a simple, yet powerful statistical method. In the provided exercise, the point estimate represents the estimated difference between the average weekly benefits received by individuals in two states, namely Alabama and Mississippi.

Calculated as the difference between the sample means, the point estimate gives us an idea of what the difference in population means might be. Using values \( \bar{x}_1 = \\(199.65 \) and \( \bar{x}_2 = \\)187.93 \), the calculation becomes:

• Point Estimate: \( \bar{x}_1 - \bar{x}_2 = \\(199.65 - \\)187.93 = \\(11.72 \)

This suggests that, on average, the unemployment benefit in Alabama is approximately \)11.72 higher than in Mississippi based on our sample, however, this is an estimate, not a conclusion.

Standard Error

The standard error provides insight into the level of accuracy of a sample mean estimate. It's a measure of the variability within a sample statistic over numerous samples from the same population. In this lesson, the standard error helps estimate how much the point estimate of the mean difference might fluctuate due to random sampling.

The formula for calculating the standard error of the difference between two means in our exercise is:

• \[ SE = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}} \]

Where \(\sigma_1\) and \(\sigma_2\) are the population standard deviations ( \(32.48 and \)26.15), and \(n_1\) and \(n_2\) are the sample sizes (70 and 65 respectively). A smaller standard error indicates a more precise point estimate, whereas a larger standard error suggests less reliability. This information is pivotal in constructing confidence intervals and hypothesis tests.

Significance Level

In hypothesis testing, the significance level ( \( \alpha \)) plays a critical role. It defines the probability of rejecting the null hypothesis when it is actually true (Type I error). For our exercise, a 4% significance level is utilized.

Commonly chosen significance levels are 1%, 5%, and 10%, reflecting a willingness to accept some risk of making a wrong decision. In this case, using a 4% significance level implies that there is allowable room for this error at 4%. It helps establish a threshold, known as the critical value, beyond which the null hypothesis is deemed unlikely and thus rejected in favor of the alternative.

Moreover, the value plays a pivotal role when comparing it to the p-value in hypothesis testing. If the p-value is less than or equal to the significance level, the null hypothesis is rejected, indicating the evidence is strong enough to support the alternative hypothesis.