Wall Street securities firms paid out record year-end...

Wall Street securities firms paid out record year-end bonuses of $$\$ 125,500$$ per employee for 2005 (Fortune, February 6, 2006). Suppose we would like to take a sample of employees at the Jones \& Ryan securities firm to see whether the mean year-end bonus is different from the reported mean of $$\$ 125,500$$ for the population. a. State the null and alternative hypotheses you would use to test whether the year-end bonuses paid by Jones \& Ryan were different from the population mean. b. Suppose a sample of 40 Jones & Ryan employees showed a sample mean year-end bonus of $$\$ 118,000$$. Assume a population standard deviation of $$\sigma=\$ 30,000$$ and compute the $p$-value. c. With $a-05$ as the level of significance, what is your conclusion? d. Repeat the preceding bypohesis test using the critical value approach.

Wall Street securities firms paid out record year-end bonuses of $$\$ 125,500$$ per employee for 2005 (Fortune, February 6, 2006). Suppose we would like to take a sample of employees at the Jones \& Ryan securities firm to see whether the mean year-end bonus is different from the reported mean of $$\$ 125,500$$ for the population.
a. State the null and alternative hypotheses you would use to test whether the year-end bonuses paid by Jones \& Ryan were different from the population mean.
b. Suppose a sample of 40 Jones & Ryan employees showed a sample mean year-end bonus of $$\$ 118,000$$. Assume a population standard deviation of $$\sigma=\$ 30,000$$ and compute the $p$-value.
c. With $a-05$ as the level of significance, what is your conclusion?
d. Repeat the preceding bypohesis test using the critical value approach.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions or inferences about populations based on sample data. It involves formulating two mutually exclusive statements – the null hypothesis, which represents a status quo or a baseline assumption, and the alternative hypothesis, which represents a claim of difference or effect – and then using a test statistic to assess the evidence provided by the sample data to decide whether to reject the null hypothesis.

Null Hypothesis and Alternative Hypothesis

The null hypothesis (H0) is a statement asserting that there is no significant difference between the sample parameter and the population parameter, while the alternative hypothesis (Ha) posits that there is a difference. Formulating these hypotheses is crucial as they define the framework of the statistical test, determine the direction of the test (one-tailed or two-tailed), and guide the interpretation of the results.

Z-Test

A Z-test is a hypothesis testing procedure used when the population standard deviation is known and the sample size is sufficiently large. This test involves standardizing the sample mean using the standard error to obtain a Z statistic, which can then be compared to critical values from the standard normal distribution. It is commonly used to determine whether a sample mean significantly differs from a known population mean.

p-value

The p-value is the probability, under the null hypothesis, of observing a test statistic as extreme as, or more extreme than, the value obtained from the sample data. It serves as a measure of the strength of evidence against the null hypothesis. A small p-value indicates that such an extreme result is unlikely under the null hypothesis, leading to potential rejection of the null.

Significance Level

The significance level, often denoted by alpha (?), is a threshold set by the researcher, which defines the probability of committing a Type I error (rejecting a true null hypothesis). Common values for ? are 0.05 or 0.01. If the p-value is less than or equal to ?, the null hypothesis is rejected, suggesting that the observed effect is statistically significant.

Critical Value Approach

The critical value approach involves determining a cutoff value or values from the test statistic’s probability distribution corresponding to the chosen significance level. The calculated test statistic is then compared to these critical values. If the test statistic falls in the rejection region beyond the critical value(s), the null hypothesis is rejected. This method provides an alternative way to decide on the hypothesis test outcome, relying on predetermined thresholds rather than a computed p-value.

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