you are performing a two - tailed t - test with test statistic ( t=-2.65 ) and a sample of size 30, find the…

Explanation:

Step1: Calculate degrees of freedom

Degrees of freedom (df=n - 1), where (n = 30). So (df=30-1 = 29).

Step2: Use t - distribution property for two - tailed test

For a two - tailed t - test, (p - value=2\times P(T>\vert t\vert)) (where (T) follows a t - distribution with (df) degrees of freedom). Here (t=- 2.65), so (\vert t\vert = 2.65). Using a t - distribution table or a statistical software (e.g., in R: 2*pt(-2.65,29) or in Python with scipy.stats library: 2*scipy.stats.t.cdf(-2.65,29)), we find the probability. (P(T>2.65)) with (df = 29). The cumulative distribution function (P(T\leq x)) for (x = 2.65) and (df=29) is (P(T\leq2.65)\approx0.9941). Then (P(T > 2.65)=1 - 0.9941=0.0059).

Step3: Calculate p - value

Since (p - value=2\times P(T>2.65)), then (p - value=2\times0.0059 = 0.0118).

Answer:

(0.0118)