p-value approach
The p-value approach is a method used in hypothesis testing to determine the significance of results. Essentially, it tells you how probable it is to observe your data if the null hypothesis is true. To use this approach, first, calculate a test statistic, such as a z-score, for your sample. Then, refer to statistical tables or software to find the p-value. If the p-value is smaller than the predetermined significance level (often written as \(\alpha\)), it suggests strong evidence against the null hypothesis. In our example, we used \(\alpha = 0.01\). This means that we desire a high degree of certainty (99%) before rejecting the null hypothesis.
classical approach
The classical approach involves comparing a calculated test statistic to a critical value determined from statistical tables. Unlike the p-value approach, which looks at probabilities, the classical approach uses critical values to decide whether to reject the null hypothesis. The critical value acts as a threshold; if the test statistic falls beyond this point, the null hypothesis is rejected. In hypothesis testing with the classical approach, it's important to know whether your test is one-tailed or two-tailed, as this affects the critical value.
z-score calculation
The z-score is a statistic that measures the number of standard deviations a data point is from the mean. It's critical in hypothesis testing, particularly for comparing a sample mean to a population mean. In our example, the z-score is computed using the formula: \[Z = \frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\]where \(\bar{x}\) is the sample mean, \(\mu\) is the population mean, \(\sigma\) is the standard deviation, and \(n\) is the sample size. By calculating this value, you can decide how many standard deviations the sample mean is away from the population mean.
standard deviation
Standard deviation is a measure of the dispersion or variability within a set of data. It shows how much individual data points deviate from the mean of the data set. In hypothesis testing, knowing the standard deviation helps in calculating the z-score, which allows you to judge how significant a sample mean is with respect to a known population mean. In this exercise, the standard deviation is known to be 10.7 minutes, which quantifies the variability in commute time across the sample.
sample mean
The sample mean is the average of all the data points in a sample, calculated as the sum of the data points divided by the number of data points. In hypothesis testing, the sample mean is compared against the population mean to assess claims about the population. In the exercise, the sample mean of 21.7 minutes is compared with the nationwide average of 24.3 minutes to determine if there is a significant difference.
null hypothesis
The null hypothesis (symbolized as \(H_0\)) is a statement that there is no effect or no difference and acts as a default or companion to the alternative hypothesis. In hypothesis testing, the aim is to either reject or fail to reject the null hypothesis based on statistical evidence. For this exercise, the null hypothesis states that the average commute time of the sample is equal to the national average, which is formalized as \(H_0: \mu = 24.3\) minutes.
alternative hypothesis
The alternative hypothesis (denoted as \(H_a\)) is what you want to prove or argue is true. It suggests that there is a statistically significant effect or difference. In this exercise, the alternative hypothesis suggests that the workers' average commute time is less than the national average. It's put forward as \(H_a: \mu < 24.3\) minutes. By testing the alternative hypothesis, researchers aim to find evidence to support this claim, against what is stated in the null hypothesis.
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