91Ó°ÊÓ

According to a Bureau of Labor Statistics release of March 25, 2015, financial analysts earned an average of \(\$ 110,510\) in 2014 . Suppose that the 2014 earnings of all financial analysts had a mean of \(\$ 110,510\). A recent sample of 400 financial analysts showed that they earn an average of \(\$ 114,630\) a year. Assume that the standard deviation of the annual earnings of all financial analysts is \(\$ 30,570\). a. Using the critical-value approach, can you conclude that the current average annual earnings of financial analysts is higher than \(\$ 110,510 ?\) Use \(\alpha=.01\). b. What is the Type I error in part a? Explain. What is the probability of making this error in part a? c. Will your conclusion of part a change if the probability of making a Type I error is zero? d. Calculate the \(p\) -value for the test of part a. What is your conclusion if \(\alpha=.01 ?\)

Step by step solution

01

Formulate the Hypothesis

Null hypothesis (\(H_0\)): The average annual earnings of financial analysts is \$110,510. It is given by \(μ = \$110,510\). Alternative hypothesis (\(H_1\)): The average annual earnings of financial analysts is higher than \$110,510, which is given by \(μ > \$110,510\).

02

Compute the Z Test Statistic

The Z test statistic is calculated as \(Z = \frac{\bar{x} - μ}{σ/√n}\) where \(\bar{x}\) is the sample mean = \$114,630, μ is the population mean = \$110,510, σ is the standard deviation = \$30,570, and n is the sample size = 400. Substituting the given numbers, \(Z = \frac{\$114,630 - \$110,510}{\$30,570/√400} ≈ 4.2368\).

03

Comparison and Conclusion

For the significance level \(\α = 0.01\), the z critical value (z*) from the standard normal distribution table is approximately 2.3263. Since the calculated z-value (4.2368) is greater than the critical z-value (2.3263), the decision is to reject the null hypothesis in favor of the alternative hypothesis. Therefore, it can be concluded that the current average annual earnings of financial analysts is significantly higher than \$110,510.

04

Interpret Type I Error

A type I error is when one wrongly rejects a true null hypothesis. Here, a type I error would be concluding that the current average annual earnings of financial analysts is greater than \$110,510 when it is actually not. The probability of making a Type I error in this case is the significance level \(\α = 0.01\) or 1%.

05

Answer to Part c

If the probability of making a Type I error is zero, that would mean absolutely certainty that the null hypothesis is false before rejecting it. In reality, this is almost impossible in statistics, and would not change the conclusion. The result would still be that earnings are above \$110,510.

06

Calculate P-Value

The p-value can be obtained from the z-table or any statistical software. For z=4.2368, the p-value is practically zero. If the p-value is less than or equal to the significance level (\(\α = 0.01\)), the decision is to reject the null hypothesis. So, the conclusion is still to reject the null hypothesis, i.e., the current earnings are higher than \$110,510.

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Key Concepts

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

Understanding the Z-Test

The z-test is a statistical method used to determine if there is a significant difference between the sample mean and a known population mean. It is most effective when dealing with large sample sizes and when the population standard deviation is known. In this scenario, we want to check if the average earnings of financial analysts have increased from the known average of \\(110,510 to a sample average of \\)114,630.
The formula for the z-test statistic is:

• \( Z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}} \)
• \( \bar{x} \) is the sample mean
• \( \mu \) is the population mean
• \( \sigma \) is the standard deviation
• \( n \) is the sample size

Using these values, we find the z-value and compare it to a critical value to make our statistical decision.

Exploring Type I Error

A Type I error occurs when we incorrectly reject a true null hypothesis. In simple terms, it's a false positive. Here, it means we might conclude that financial analysts' earnings have increased when they actually haven't.
The probability of making a Type I error is denoted by the significance level, \( \alpha \), which is set at 0.01 in this example. This translates to a 1% chance of mistakenly claiming that the earnings have increased when they haven't.
Being aware of Type I error helps researchers gauge the risk of making incorrect decisions based on their statistical tests.

Unveiling the P-Value

The p-value is a crucial concept in hypothesis testing. It represents the probability of observing the sample results, or something more extreme, assuming the null hypothesis is true.
For a calculated z-value of 4.2368, the p-value is found using statistical tables or software. In this test, the p-value is essentially zero, indicating it's highly unlikely to observe such a sample result by random chance if the null hypothesis is true.
Since the p-value is less than the significance level of 0.01, we reject the null hypothesis, supporting the claim that earnings have indeed increased.

Significance Level Explained

A significance level, often denoted as \( \alpha \), is a threshold set by the researcher to determine when to reject the null hypothesis. Commonly set at 0.05, it's lower here at 0.01, implying stricter criteria for statistical significance.
This level indicates the risk of making a Type I error. A smaller \( \alpha \) reduces this risk but also makes it harder to detect a true effect.

• In this context, the 0.01 level means we want to be 99% confident in our conclusion: that financial analysts' earnings are genuinely higher than \$110,510.

Choosing the right significance level is crucial, balancing sensitivity and the risk of error.