Question

a college entrance exam company determined that a score of 24 on the mathematics portion of the exam suggests that a student is ready for college level mathematics. to achieve this goal, the company recommends that students take a core curriculum of math courses in high school. suppose a random sample of 250 students who completed this core set of courses results in a mean math score of 24.6 on the college entrance exam with a standard deviation of 3.7. do these results suggest that students who complete the core curriculum are ready for college level mathematics? that is, are they scoring above 24 on the mathematics portion of the exam? complete parts a) through d) below
c) use the p - value approach at the \\( \alpha = 0.10 \\) level of significance to test the hypotheses in part (a)
identify the test statistic
\\( t _ { 0 } = 2.56 \\) (round to two decimal places as needed )
identify the p - value.
p - value = (round to three decimal places as needed.)

Explanation:

Step1: Determine the degrees of freedom

The degrees of freedom \( df=n - 1\), where \(n = 250\). So \(df=250-1 = 249\)

Step2: Use the t - distribution table or calculator

We know the test statistic \(t_0 = 2.56\) and \(df = 249\). Since this is a one - tailed test (we want to know if they are scoring above 24), we use the right - tailed area of the t - distribution.
Using a t - calculator or software (for a t - distribution with \(df = 249\) and \(t=2.56\)), the P - value is the probability \(P(T>t_0)\) where \(T\) follows a t - distribution with \(df = 249\)
\(P - value=P(T>2.56)\)
Using a calculator (e.g., in R: 1 - pt(2.56, 249) or in Excel: =T.DIST.RT(2.56,249)), we find that the P - value is approximately \(0.006\)

Answer:

\(0.006\)