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When testing new cola recipes, it's important to measure any potential sweetness loss during storage. Sweetness loss is calculated as the difference between the sweetness rating before storage and after storage. This measurement helps companies understand if their formula maintains the desired taste over time.
In the exercise example, tasters rated batches of a new cola recipe both before and after storage. The individual differences, or sweetness losses, were: 2.0, 0.4, 0.7, 2.0, -0.4, 2.2, -1.3, 1.2, 1.1, and 2.3. These values are used to see if the cola experiences an overall loss in sweetness. Accurately measuring this helps companies decide if recipe adjustments are needed to ensure consistent taste.

The t-statistic is a key part of hypothesis testing. It helps determine if the sample provides enough evidence to conclude that the cola loses sweetness. Calculating the t-statistic involves the difference between the sample mean and the hypothesized population mean, all divided by the standard error.

In this case, the formula is:

• \[ t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}} \]
• \(\bar{x} = 1.02\): the sample mean sweetness loss
• \(\mu_0 = 0\): the population mean under the null hypothesis (no loss)
• \(s\): the sample standard deviation, and \(n = 10\): the sample size

Using this formula helps us decide if the observed sweetness loss can happen by chance or if it's statistically significant.

The sample mean is an average of the sweetness losses from all batches tested. It's a crucial measure because it represents the central tendency of the data. The sample mean is calculated by summing all individual sweetness losses and dividing by the number of samples.

In our example:

• \[\text{mean} = \frac{2.0 + 0.4 + 0.7 + 2.0 - 0.4 + 2.2 - 1.3 + 1.2 + 1.1 + 2.3}{10} = 1.02\]

This value, 1.02, indicates the average extent to which the cola's sweetness decreases during storage. Understanding the sample mean helps us assess whether the sweetness loss is significant enough to warrant changes in the formula.

Statistical significance helps us determine if the sweetness loss observed in our cola batches is meaningful or if it happened by random chance. We achieve this by comparing our calculated t-statistic to a critical value and examining the p-value.

Usually, a significance level \(\alpha\) of 0.05 is chosen. This means we're willing to accept a 5% chance of incorrectly rejecting the null hypothesis. For our exercise, with 9 degrees of freedom, the critical t-values in a two-tailed test are approximately \(\pm 2.262\).
We compare our t-statistic to these critical values:

• If \(|t|\) is larger than the critical value, the loss is statistically significant.
• If the p-value is less than \(0.05\), it strengthens our case to reject the null hypothesis.

This process decides if the evidence suggests that the cola has a significant loss of sweetness after storage.