91Ó°ÊÓ

A t-test is a statistical test that helps us compare the means of different groups to see if they are significantly different from each other. In the context of our original exercise, the t-test is used to determine if the sample mean thickness of the lenses significantly differs from the desired mean thickness, which is the parameter hypothesized under the null hypothesis.

The formula used for the t-test statistic is given by: \[t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}\]

• \( \bar{x} \) refers to the sample mean.
• \( \mu \) is the population mean as stated by the null hypothesis.
• \( s \) is the sample standard deviation.
• \( n \) represents the sample size.

Using this formula, we calculated the test statistic to be approximately \(-3.125\), which helps in determining whether the difference between the sample mean and the desired mean is statistically significant.

The sample mean represents the average value of a dataset obtained from a sample. In many cases, it is used to make inferences about the population mean. In our example, the sample mean thickness of the lenses is \(3.05\) mm. This value was derived from the 50 lens samples.

Calculating a sample mean is straightforward:\[\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i\]Here, \( \bar{x} \) is the sample mean, and \( n \) is the sample size, while each \( x_i \) represents the individual data points in the sample. The sample mean serves as an estimate of the population mean, allowing us to perform further analysis, such as hypothesis testing.

• Helps represent the central tendency of the data.
• Forms the basis for statistical tests like the t-test.

The null hypothesis is a fundamental concept in hypothesis testing. It is the assumption that there is no significant difference between specified populations and any observed difference is due to sampling or experimental error. In the context of the lens thickness exercise, the null hypothesis (\( H_0 \)) states that the true average thickness of the lenses is equal to the desired thickness, \( \mu = 3.20 \) mm.

When performing statistical tests, we desire to either reject or fail to reject the null hypothesis based on the evidence obtained from the sample data:

• If the null hypothesis is rejected, it indicates there is a significant effect or difference.
• Failing to reject implies insufficient evidence against the null.

In the given example, based on our calculations, the null hypothesis was rejected, suggesting a significant difference from the desired thickness.

The alternative hypothesis, often represented as \( H_a \) or \( H_1 \), posits that there is a statistically significant difference between the groups or conditions being studied. It is essentially the statement we want to prove. In our lens thickness test, the alternative hypothesis states that the true average thickness is different from the desired \( 3.20 \) mm.

• The alternative hypothesis can be one-sided or two-sided, reflecting the direction of the expected difference.
• In our exercise, a two-tailed test is used: \( H_a: \mu eq 3.20 \). This considers the possibility of being both greater or less than the desired thickness.

By rejecting the null hypothesis, we find support for the alternative hypothesis. In this case, the evidence suggested that the true mean was indeed different from the desired thickness.