Question

A college entrance exam company determined that a score of 2525 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 200200 students who completed this core set of courses results in a mean math score of 25.525.5 on the college entrance exam with a standard deviation of 3.13.1. Do these results suggest that students who complete the core curriculum are ready for​ college-level mathematics? That​ is, are they scoring above 2525 on the math portion of the​ exam? Complete parts​ a) through​ d) below.

Answer 1

There is enough evidence to support the claim that the students are ready for college-level mathematics (they get a score significantly higher than 25).

They are scoring significantly higher than 25 on the math portion of the exam.

P-value = 0.012

Explanation:

This is a hypothesis test for the population mean.

The claim is that the students are ready for college-level mathematics (they get a score significantly higher than 25).

Then, the null and alternative hypothesis are:

`H_0: \mu=25\\\\H_a:\mu> 25`

The significance level is 0.05.

The sample has a size n=200.

The sample mean is M=25.5.

As the standard deviation of the population is not known, we estimate it with the sample standard deviation, that has a value of s=3.1.

The estimated standard error of the mean is computed using the formula:

`s_M=(s)/(√(n))=(3.1)/(√(200))=0.219`

Then, we can calculate the t-statistic as:

`t=(M-\mu)/(s/√(n))=(25.5-25)/(0.219)=(0.5)/(0.219)=2.281`

The degrees of freedom for this sample size are:

`df=n-1=200-1=199`

This test is a right-tailed test, with 199 degrees of freedom and t=2.281, so the P-value for this test is calculated as (using a t-table):

`P-value=P(t>2.281)=0.012`

As the P-value (0.012) is smaller than the significance level (0.05), the effect is significant.

The null hypothesis is rejected.

There is enough evidence to support the claim that the students are ready for college-level mathematics (they get a score significantly higher than 25).