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In statistics, a sample mean is a summary statistic that provides the average value of a sample data set. It is crucial in hypothesis testing as it helps to compare the sample with the population. Here, we calculate the sample mean by summing up all observed values and then dividing by the number of observations.
In our example, the sample mean is calculated based on the 12 senior citizens' coffee consumption values:

• 3.1, 3.3, 3.5, 2.6, 2.6, 4.3, 4.4, 3.8, 3.1, 4.1, 3.1, and 3.2

Adding these values yields 40.6. Divide this sum by 12, the number of observed samples, to get the sample mean: \( \bar{x} = \frac{40.6}{12} \approx 3.383 \).
By calculating the sample mean, we obtain a value that represents the average coffee consumption among the sampled senior citizens, which can then be compared with the national average of 3.1 cups per day.

The t-test is a statistical test used to analyze whether there is a significant difference between the means of two groups. It is particularly useful when the population standard deviation is unknown and the sample size is relatively small.
To perform a t-test, we follow these steps:

• Compute the sample mean \( \bar{x} \) and the standard deviation \( s \)
• Use the formula for the t-statistic: \\( t = \frac{\bar{x} - \mu}{s/\sqrt{n}} \)\, where:
• \( \mu \) is the population mean (3.1 cups in our exercise)
• \( s \) is the sample standard deviation
• \( n \) is the sample size (12 senior citizens)

For our exercise, after plugging in the values, the t-statistic comes out to approximately 1.6007. The t-value helps us determine how far our sample mean deviates from the national average in terms of the standard error.

The significance level, denoted by \( \alpha \), is a threshold that determines the cutoff for deciding whether to reject the null hypothesis. It is typically set at a conventional level such as 0.05, 0.01, or 0.10.
In the exercise, we've chosen a significance level of 0.05. This means that there is a 5% risk of concluding that there is a difference between the sample mean and the population mean when there is none.
The significance level helps to define the critical region boundaries for rejecting the null hypothesis. For our t-test:

• If the calculated t-value falls outside this critical region, we reject the null hypothesis.
• If it falls inside, as in our case (where the t-value is 1.6007 and within -2.201 to 2.201), we fail to reject the null hypothesis.

Thus, the significance level acts as a gatekeeper for making decisions about statistical evidence.

The null hypothesis, denoted as \( H_0 \), is a claim that there is no effect or no difference that we are testing against. It serves as a starting point for statistical analysis.
For this exercise, the null hypothesis states that the mean coffee consumption of senior citizens (sample mean) is equal to the national average (population mean), mentioned in mathematical terms as \( \mu = 3.1 \).
The alternative hypothesis, \( H_a \), against which the null is tested, suggests there is a difference, \( \mu eq 3.1 \).
To make a decision on the null hypothesis, statistical tests, like the t-test, are performed to determine if the observed data provides enough evidence to conclude a significant difference.

• If the test statistic falls into the critical region, we reject the null hypothesis.
• If not, as in the case with a t-value of 1.6007, we conclude that our sample does not provide sufficient evidence to suggest a difference.

Thus, the null hypothesis is crucial in hypothesis testing as it frames the basic question our analysis seeks to address.