Hypothesis Testing: Test Statistics Calculation

During a study on the effectiveness of a new drug, researchers collected data from a sample of 100 patients. They found that the sample mean reduction in symptoms was 12 points with a standard deviation of 4 points. The researchers want to determine if the new drug results in a mean reduction in symptoms that is statistically significantly different from the known population mean reduction of 10 points at a 5% significance level.

To conduct this hypothesis test, they formulated the following:

Null Hypothesis (H₀): The mean reduction is equal to the population mean, $ar{x} = 10$

Alternative Hypothesis (H₁): The mean reduction is not equal to the population mean, $ar{x} eq 10$

Using these parameters, the test statistic can be calculated using the formula:

$T = rac{ar{x} - ext{Population Mean}}{s / ext{sqrt}(n)}$ where:

- $ar{x} = 12$ (sample mean)

- Population Mean = 10

- $s = 4$ (sample standard deviation)

- $n = 100$ (sample size)

Calculate the test statistic to determine which of the following is true: