Questions: A college entrance exam company determined that a score of 20 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 20.7 on the college entrance exam with a standard deviation of 3.5. Do these results suggest that students who complete the core curriculum are ready for college-level mathematics? That is, are they scoring above 20 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 H0 : μ = 20 versus H1 : μ > 20. 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 students' test scores were independent of one another. D. The sample size is larger than 30. E. A boxplot of the sample data shows no outliers. F. None of the requirements are satisfied.

A college entrance exam company determined that a score of 20 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 20.7 on the college entrance exam with a standard deviation of 3.5. Do these results suggest that students who complete the core curriculum are ready for college-level mathematics? That is, are they scoring above 20 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 H0 : μ = 20 versus H1 : μ > 20. 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 students' test scores were independent of one another. D. The sample size is larger than 30. E. A boxplot of the sample data shows no outliers. F. None of the requirements are satisfied.
Transcript text: A college entrance exam company determined that a score of 20 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 20.7 on the college entrance exam with a standard deviation of 3.5. Do these results suggest that students who complete the core curriculum are ready for college-level mathematics? That is, are they scoring above 20 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 $\mathrm{H}_{0}$ : $\square$ 20 versus $\mathrm{H}_{1}$ : $\mu$ > 20. 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 students' test scores were independent of one another. D. The sample size is larger than 30. E. A boxplot of the sample data shows no outliers. F. None of the requirements are satisfied.
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Solution

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Solution Steps

Step 1: State the appropriate null and alternative hypotheses

The null hypothesis (\(H_0\)) represents the claim that the mean math score of students who completed the core curriculum is equal to 20. The alternative hypothesis (\(H_1\)) represents the claim that the mean math score is greater than 20.

Thus, the hypotheses are: \[ H_0: \mu = 20 \quad \text{versus} \quad H_1: \mu > 20. \]

Step 2: Verify the requirements to perform the test using the \(t\)-distribution

To use the \(t\)-distribution, the following requirements must be satisfied:

  1. The sample data come from a population that is approximately normal.
  2. The students were randomly sampled.
  3. The students' test scores were independent of one another.
  4. The sample size is larger than 30.
  5. A boxplot of the sample data shows no outliers.

From the problem statement:

  • The sample size is 250, which is larger than 30, so the Central Limit Theorem applies, and the distribution of the sample mean is approximately normal.
  • The students were randomly sampled.
  • The students' test scores are independent of one another.
  • No information is provided about outliers, but with a large sample size, the impact of outliers is minimized.

Thus, the requirements are satisfied. The correct options are:

  • B. The students were randomly sampled.
  • C. The students' test scores were independent of one another.
  • D. The sample size is larger than 30.

Final Answer

a) The appropriate null and alternative hypotheses are: \[ H_0: \mu = 20 \quad \text{versus} \quad H_1: \mu > 20. \] \[ \boxed{H_0: \mu = 20 \quad \text{versus} \quad H_1: \mu > 20} \]

b) The requirements to perform the test using the \(t\)-distribution are satisfied by: \[ \boxed{\text{B, C, and D}} \]

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