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The Z-test is a type of hypothesis test used when the population standard deviation is known and the sample size is large. It is commonly used to determine if there is a significant difference between the population mean and a sample mean. The Z-test helps to assess whether any observed differences are due to random chance or if they are statistically significant. In the context of this exercise, a Z-test enables us to compare the average amount of money lost in flexible spending accounts in 2009 to that in 2007.

• Standard deviation is known
• Sample size is large (usually n > 30)
• Assumes normal distribution

The formula for the Z-test is: \[ Z = \frac{(X - \mu_0)}{(\sigma/\sqrt{n})} \]where \(X\) is the sample mean, \(\mu_0\) is the population mean under the null hypothesis, \(\sigma\) is the standard deviation of the population, and \(n\) is the sample size. This formula shows how many standard deviations the sample mean is from the population mean. A higher Z-score indicates a higher deviation from what is expected, leading to a potential rejection of the null hypothesis.

The null hypothesis, often represented as \(H_0\), is a statement that there is no effect or no difference. It serves as the default or starting assumption in hypothesis testing. In the exercise, the null hypothesis is that there is no difference in the average amount forfeited from flexible spending accounts between 2007 and 2009. Mathematically, this is expressed as: \[ H_0: \mu = \$723 \]where \(\mu\) represents the population mean in 2009. By testing the null hypothesis, researchers can determine if any observed differences in the data are statistically significant or if they might have occurred by random chance. Rejecting the null hypothesis suggests that there is evidence against it, supporting the idea that the population mean might be different from 723.

The alternative hypothesis, denoted as \(H_1\) or \(H_a\), is the statement that opposes the null hypothesis. It represents what the researcher aims to demonstrate. In this exercise, the alternative hypothesis suggests that the average amount forfeited in 2009 is different from the amount in 2007. Therefore, it is expressed as:\[ H_1: \mu eq \$723 \]This two-tailed alternative hypothesis is used when we are interested in detecting any difference from the hypothesized mean, whether it is higher or lower. If the hypothesis test leads to the rejection of the null hypothesis, it provides support for the alternative hypothesis, indicating a significant change in the average amounts forfeited from the flexible spending accounts between the two years.

The population mean, denoted as \(\mu\), is the average of a set of values for a whole population. In many real-world scenarios, including this exercise, the population mean can be a theoretical value that we can only estimate through sampling due to the large size or inaccessibility of the population. Here, the parameter of interest is the population mean of money forfeited from flexible spending accounts.

• A theoretical average of the entire data set
• Often estimated through sample data
• Different from the sample mean, which is specific to one sample

The task within this hypothesis test is to determine if \(\mu\) in 2009 could be the same as \$723, the population mean in 2007, or if there has been a significant change. This value forms the basis of the null hypothesis, serving as a point of comparison for the sample mean obtained from the 2009 data.

Standard deviation, represented by the symbol \(\sigma\), is a measure of the dispersion or spread of a set of values around the mean. A smaller standard deviation indicates that the values tend to be close to the mean, while a larger standard deviation shows more variability. In hypothesis testing, knowing the standard deviation of the population is critical for calculating the Z-test statistic.

• Measure of spread in data
• Population standard deviation is used in Z-test

In this particular exercise, the population standard deviation of the amount forfeited is \$307. This value helps determine how much variation there is in the money forfeited by employees. The standard deviation is used in the Z-test formula to assess how far the sample mean is from the hypothesized population mean, helping us decide whether to accept or reject the null hypothesis. Understanding this metric is crucial for interpreting the results of the hypothesis test accurately.