Question

you are performing a two - tailed t - test with test statistic ( t=-2.65 ) and a sample of size 30, find the p - value to 4 decimal places.

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: 2pt(-2.65,29) or in Python with scipy.stats library: 2scipy.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:

\(0.0118\)