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When conducting a hypothesis test, a two-tailed test is applicable when you want to determine if there is a significant difference in either direction from a specified value. Here, the primary question is whether the true mean thickness of the lenses is different from the desired thickness of 3.20 mm, meaning it could be either greater or smaller.
Thus, we are interested in deviations on both sides, hence the "two-tailed" terminology.
This approach is comprehensive because it considers the possibility of an average that is higher or lower than expected, allowing for more flexibility in testing hypotheses.

The t-distribution is used when the sample size is small, and the population standard deviation is unknown, which is the case in many real-world situations. For our problem, the sample size is 50, and the population standard deviation isn't known, so the t-distribution is appropriate.
The shape of the t-distribution is similar to the normal distribution but has heavier tails, allowing for greater variability with smaller samples.
As the sample size increases, the t-distribution approaches the normal distribution because estimates become more precise with larger datasets. The degrees of freedom, calculated as the sample size minus one, also affect the shape of the t-distribution, and in this example, there are 49 degrees of freedom.

In hypothesis testing, the significance level, denoted by \( \alpha \), is the probability of rejecting the null hypothesis when it is true. It is a predetermined threshold for significance.
For this exercise, a significance level of 0.05 is chosen, which is common in many fields of research.
This level represents a 5% risk of concluding that a difference exists when there is none.
Thus, it sets a high standard for evidence before concluding that the true mean thickness of the lenses differs from the anticipated 3.20 mm.

The sample mean \( \bar{x} \) is a key measure of central tendency that provides an estimate of the population mean. In this scenario, the sample mean is 3.05 mm.
This value was calculated from the 50 lenses sampled and forms the basis of comparison against the desired mean of 3.20 mm.
The difference between the sample mean and the specified population mean helps us compute the test statistic and decide the outcome of our hypothesis test.
A sample mean that differs significantly from the population mean suggests that the null hypothesis may not hold.

Standard deviation is a measure of the amount of variation or dispersion in a set of values. Here, the sample standard deviation is 0.34 mm.
This value indicates how much the individual lens thickness measurements deviate from the sample mean of 3.05 mm.
A small standard deviation implies that the sample data points are close to the mean, indicating lesser variability.
In hypothesis testing, the standard deviation is vital for calculating the test statistic, thus influencing the conclusions about the population mean.