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

a) state the appropriate null and alternative hypotheses. fill in the correct answers below
the appropriate null and alternative hypotheses are ( h_0: mu = 24 ) versus ( h_1: mu>24 )
b) verify that the requirements to perform the test using the t - distribution are satisfied. check all that apply
□ a. the sample data come from a population that is approximately normal
□ b. the students were randomly sampled
□ c. the sample size is larger than 30
□ d. a boxplot of the sample data shows no outliers
□ e. the students test scores were independent of one another
□ f. none of the requirements are satisfied

Explanation:

• For part b:
• Random sampling (B): A random sample of 250 students is mentioned in the problem statement. Random sampling is a fundamental requirement for hypothesis - testing as it helps to ensure that the sample is representative of the population.
• Sample size (C): The sample size \(n = 250>30\). When the sample size is large (\(n\geq30\)), the Central Limit Theorem (CLT) comes into play. The CLT states that for a large sample size, the sampling distribution of the sample mean \(\bar{x}\) is approximately normal, regardless of the shape of the population distribution. This is important for performing a \(t\) - test (in fact, for large \(n\), the \(t\) - distribution is very close to the standard normal distribution).
• Independence (E): We assume that the students' test scores are independent of one another. In a random sample, if the sample is drawn without replacement (which is the case here, as we are sampling students), if \(n\leq0.1N\) (where \(N\) is the population size, and we assume the population of all students who could take the exam is much larger than \(n = 250\)), we can assume independence.

• For part a:
• The null hypothesis \(H_0\) is a statement of no effect or no difference. Here, it is that the population mean score \(\mu\) is equal to the benchmark score of 24. The alternative hypothesis \(H_1\) is what we are trying to find evidence for. Since we want to know if students are scoring above 24, the alternative hypothesis is that \(\mu>24\).

Answer:

• a) \(H_0:\mu = 24\) versus \(H_1:\mu>24\)
• b) B. The students were randomly sampled; C. The sample size is larger than 30; E. The students' test scores were independent of one another.