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A well-designed and safe workplace can contribute greatly to increasing productivity. It is especially important that workers not be asked to perform tasks, such as lifting, that exceed their capabilities. The following data on maximum weight of lift (MWOL, in kilograms) for a frequency of 4 lifts per minute were reported in the article "The Effects of Speed, Frequency, and Load on Measured Hand Forces for a Floor-to-Knuckle Lifting Task" (Ergonomics \([1992]: 833-843)\) : \(\begin{array}{lllll}25.8 & 36.6 & 26.3 & 21.8 & 27.2\end{array}\) Suppose that it is reasonable to regard the sample as a random sample from the population of healthy males, age \(18-30\). Do the data suggest that the population mean MWOL exceeds 25 ? Carry out a test of the relevant hypotheses using a \(.05\) significance level.

Step by step solution

01

Formulate the hypotheses

The null hypothesis \(H_0\) is that the population mean MWOL is equal to 25kg. The alternative hypothesis \(H_a\) is that the population mean MWOL is greater than 25kg. This means we are performing a right-tailed test.

02

Calculate the sample mean and sample standard deviation

Using the provided data, calculate the mean and standard deviation. The mean \(\bar{X}\) is the sum of the observations divided by the number of observations. The standard deviation \(s\) is the square root of variance, which is the average of the squared differences from the mean.

03

Determine the test statistic

Use the sample mean, sample standard deviation, and the size of the sample to calculate the test statistic. The test statistic can be calculated using a t-statistic formula: \(t = (\bar{X} - \mu_0)/(s / \sqrt{n})\), where \(\mu_0\) is the population mean under the null hypothesis, in this case, 25, s is the sample standard deviation, and n is the size of the sample.

04

Determine the p-value

Once the test statistic is calculated, determine the p-value associated with it. Since this is a right-tailed test, the p-value is the probability that a t-distributed random variable is greater than the calculated test statistic.

05

Compare the p-value to the significance level and make a decision

If the p-value is less than or equal to the significance level, reject the null hypothesis. If the p-value is greater than the significance level, fail to reject the null hypothesis. In this case, the significance level is 0.05.

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Key Concepts

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

Population Mean

The population mean is a key parameter in hypothesis testing. It represents the average value of a specific characteristic across a whole population. In our example, the characteristic of interest is the Maximum Weight of Lift (MWOL). The population mean is denoted by the Greek letter mu (\(\mu\)), and it tells us what the average MWOL might be if we could measure every healthy male aged 18-30.
Understanding the difference between the sample mean and the population mean is crucial. The sample mean is calculated from a limited group of individuals within the population, while the population mean is the average across the entire group.

• This distinction guides us in hypothesis testing as we derive conclusions about the population mean based on sample data.
• In hypothesis tests, we are generally interested in whether a sample provides enough evidence to support a claim about the population mean.

Sample Standard Deviation

Sample standard deviation is a measure of how spread out the numbers in a data sample are from the sample mean. It is an essential statistic when performing hypothesis tests involving a sample. In our context, s
is the sample standard deviation of the MWOL data points.

• It offers insights into the variability within a particular sample.
• Calculated as the square root of the variance, which is the average of the squared differences from the mean.

This statistic helps us understand how much the sampled values deviate from the average. In hypothesis testing, a larger sample standard deviation means greater variability within the sample observations, affecting the reliability and outcome of the test.

Significance Level

The significance level is a threshold used in hypothesis testing to determine when to reject the null hypothesis. It is denoted by alpha (\(\alpha\)) and often set at 0.05 for many scientific experiments.
In this example, \(\alpha = 0.05\), meaning there is a 5% risk of concluding that the population mean MWOL exceeds 25 kg when it doesn't.

• It represents the probability of making a Type I error, which involves incorrectly rejecting a true null hypothesis.
• A lower significance level indicates stricter criteria for rejecting the null hypothesis while reducing the risk of Type I errors.

Ultimately, the significance level helps balance the risk of making incorrect conclusions from sample data, guiding decision-making within the hypothesis testing framework.

t-Statistic

The t-statistic is a crucial value in hypothesis testing when dealing with small sample sizes. It's derived from the sample data to determine whether the sample provides enough evidence to reject the null hypothesis.
The t-statistic formula is \[ t = \frac{\bar{X} - \mu_0}{s / \sqrt{n}} \]where \(\bar{X}\) is the sample mean, \(\mu_0\) is the population mean under the null hypothesis, \(s\) is the sample standard deviation, and \(n\) is the sample size.

• The t-statistic helps assess how much the sample mean differs from the hypothesized population mean.
• In our exercise, it seeks to determine if the observed mean MWOL significantly differs from 25 kg.

The calculated t-statistic is then compared to a critical value from the t-distribution table, guiding conclusions about the hypothesis. A t-statistic beyond the critical value leads to rejecting the null hypothesis, suggesting the sample provides strong evidence for the alternative hypothesis.