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A t-test is a statistical method used to determine if there's a significant difference between the means of two groups. It's particularly helpful when working with small sample sizes and unknown population variances. There are different variations of the t-test, such as one-sample, independent, and paired t-tests. In this exercise, a one-sample t-test is used to compare a single sample mean against a known population mean. This helps to assess whether the new sample mean of daily revenue after the price increase significantly differs from the past revenue mean of $75.00.

In a t-test, you'll first set up two hypotheses:

• The null hypothesis ( $H_0 $), which suggests no effect or difference (e.g., mean revenue has not decreased).
• The alternative hypothesis ( $H_a $), which suggests a potential difference (e.g., mean revenue has decreased).

Next, you'll calculate the t-statistic using the sample data, which helps you decide whether to reject the null hypothesis. This process involves comparing the calculated t-value to a critical value derived from a t-distribution table.

The sample mean, often denoted as \(\bar{x}\), is the average of all values in a sample. It is calculated by adding up all the data points and dividing by the number of data points in the sample. In our context, the sample mean of the daily revenue collected over 20 days after the price change is $70.00.

The sample mean plays a key role in hypothesis testing because it serves as the best estimate of the true population mean when individual population data is not available. When comparing the sample mean to the known population mean, you can determine if there is a statistically significant difference. A sample mean that significantly deviates from the hypothesized population mean provides evidence to reject the null hypothesis.

Standard deviation measures the amount of variability or dispersion in a set of data. It's a crucial concept because it gives context to the mean, showing how spread out the sample values are around the mean. In hypothesis testing, the sample standard deviation is used to understand the variability within the data set.

In our example, the sample standard deviation is $4.20. This value tells us how much the daily revenue fluctuates around the mean of $70.00 over the sample period. Lower standard deviation indicates that data points are closer to the mean, while a larger standard deviation signifies more dispersion. Standard deviation is also used to calculate the standard error, which is crucial for determining the t-statistic, allowing you to measure how "standard" the sample mean is compared to the hypothesized mean.

The critical value is a threshold in hypothesis testing that helps determine whether to accept or reject the null hypothesis. It is derived from a probability distribution—in this case, the t-distribution—based on a desired level of significance, denoted as \(\alpha\).

For our test, with a chosen significance level \(\alpha = 0.05\), and 19 degrees of freedom (calculated as the sample size minus one), we find a critical t-value of -1.729. This value is the cutoff point for accepting or rejecting the null hypothesis.

When the calculated t-statistic is more extreme than the critical value, it indicates that the observed data is unlikely under the null hypothesis. In our example, since the calculated t-value of -4.76 is less than -1.729, it falls into the rejection region. Thus, we reject the null hypothesis and conclude that the mean daily revenue probably decreased after the price increase.