On Friday, Wall Street traders were anxiously awaiting the...

On Friday, Wall Street traders were anxiously awaiting the federal govemment's release of numbers on the January increase in nonfarm payrolls. The esrly consensus estimate among economists was for a growth of 250,000 new jobs (CNBC. February 3, 2006). However, a sample of 20 economists taken Thursday afternoon provided a sample mean of 266,000 with a sample standard deviation of 24,000 . Financial analysts often call such a sample mean, based on late-breaking news, the whisper number. Treat the "consensus estimate" as the population mean. Conduct a hypothesis test to determine whether the whisper number justifies a conclusion of a statistically significant increase in the consensus estimate of economists. Use $a=.01$ as the level of significance.

Key Concepts

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis represents the status quo or the stated value of a parameter, while the alternative hypothesis represents a change or difference that you aim to detect. The hypothesis test then evaluates whether the observed sample data is consistent with the null hypothesis or provides sufficient evidence in favor of the alternative hypothesis.

Hypothesis Testing

This is a statistical procedure used to decide whether there is enough evidence in a sample of data to infer that a certain condition holds true for the entire population. It involves formulating a null hypothesis (a default assumption) and an alternative hypothesis (what you want to test), then using sample data to determine which hypothesis is more likely to be true.

One-Tailed Test

When the alternative hypothesis is directional—indicating that the parameter is either greater than or less than the hypothesized value—a one?tailed test is appropriate. This type of test concentrates all of the significance level into one tail of the sampling distribution, making it more sensitive to differences in the specified direction.

Test Statistic

This is a standardized value computed from sample data during a hypothesis test. It measures the distance between the observed sample statistic and the value stated in the null hypothesis, relative to the standard error of the statistic. Common test statistics include the z-score and the t-score, depending on the sample size and whether the population standard deviation is known.

t-Distribution

When dealing with small sample sizes or when the population standard deviation is unknown, the t-distribution is used instead of the normal distribution. The t-distribution accounts for extra uncertainty in the estimation and is characterized by its degrees of freedom, which typically equal the sample size minus one.

Standard Error

This is the estimated standard deviation of the sampling distribution of a statistic, such as the sample mean. It is calculated by dividing the sample standard deviation by the square root of the sample size. The standard error quantifies the variability of the estimator and is crucial in determining the test statistic.

Significance Level

The significance level, often denoted as alpha (?), represents the threshold for rejecting the null hypothesis. It is the probability of making a Type I error, which occurs when a true null hypothesis is incorrectly rejected. A more stringent significance level (like 0.01) requires stronger evidence from the data to reject the null hypothesis.

p-Value

The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming that the null hypothesis is true. It provides a measure of the evidence against the null hypothesis: a smaller p-value indicates stronger evidence favoring the alternative hypothesis. If the p-value is lower than the set significance level, the null hypothesis is rejected.

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