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The t-test is a statistical test used to determine if there is a significant difference between the means of two groups. It is particularly useful when dealing with small sample sizes. In our exercise, we are examining whether the mean operation time of a worker differs from a standard time of 6.4 minutes.

A t-test calculates the t-score, which measures the size of the difference relative to the variation in your sample data. The formula for the t-score is:

• \( t = \frac{\bar{x} – μ_0}{(s/\sqrt{n})} \)

where \(\bar{x}\) is the sample mean, \(μ_0\) is the hypothesized mean, \(s\) is the sample standard deviation, and \(n\) is the sample size.

The result of this test statistic is compared against critical t-values to decide if the observed data satisfies the null hypothesis.

The level of significance, denoted as \( \alpha \), is a probability threshold used in hypothesis testing to determine when to reject the null hypothesis. It represents the risk of concluding that a difference exists when there actually is none.

In our example, the level of significance is set at 1%, or \( \alpha = 0.01 \). This implies that there is a 1% risk of making a Type I error, where we wrongly reject the null hypothesis assuming there is an effect when there is not. Thus, a lower \( \alpha \) offers more stringent evidence needed to conclude a difference.

In hypothesis testing, the \( \alpha \) level helps define the rejection region for the test statistic.

In hypothesis testing, the null hypothesis, denoted as \( H_0 \), is a statement that there is no effect or no difference. It serves as a default assumption with no deviation unless proven otherwise. In the problem at hand, the null hypothesis posits that the worker’s average operation time is equal to the standard time of 6.4 minutes.

In symbols, this is expressed as:

• \( H_0: \mu = 6.4 \)

The burden lies on the data to show enough evidence against this hypothesis. Our initial analysis centers on checking if deviations from the null hypothesis are significant enough beyond chance alone.

Failing to reject the null hypothesis, as evidenced in our calculations, indicates we have insufficient evidence to say there is a significant difference from the standard time.

The alternative hypothesis, denoted as \( H_1 \), is a statement that contradicts the null hypothesis, indicating that there is an effect or difference. It challenges the status quo defined by the null hypothesis.

In our scenario, the alternative hypothesis contends there is a difference in the worker's mean operation time from the standard time of 6.4 minutes.

• \( H_1: \mu eq 6.4 \)

An alternative hypothesis can be either one-tailed or two-tailed, depending on the direction of the inquiry. For our problem, a two-tailed test is used since we are interested in differences in both directions (either more or less than the standard).

To accept the alternative hypothesis, the test's calculated t-value must fall within the rejection region defined by the critical values, showing strong evidence against the null hypothesis.