Final answer:
Based on the calculated test statistic and the critical value, we can make a conclusion about whether there is evidence that the average score of the class is significantly different from the national average.
Step-by-step explanation:
To determine if there is evidence that the average score of a class is significantly different from the national average on a vocabulary test, we can follow a four-step hypothesis testing process:
Step 1): State: The null hypothesis (νo) states that there is no significant difference between the class mean and the national mean, so νo: μ = 68.
The alternative hypothesis (α) posits that there is a significant difference, so α: μ ≠ 68.
Step 2): Formulate: We will use a two-tailed t-test to test the hypothesis since the population standard deviation is unknown and the sample size is less than 30.
The t-score is calculated using the formula (μx - μ) / (s/ √n), where μx is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the sample size.
Step 3): Solve: For this class: μx = 64, s = 13, n = 20, and μ = 68. Substituting these values into the formula gives us a t-score.
We then consult a t-distribution table to find the p-value associated with this t-score at a 95% confidence level.
Step 4): Conclude: If the p-value is less than the significance level (typically 0.05), we reject the null hypothesis, concluding that there is a significant difference in scores.
Otherwise, we fail to reject the null hypothesis, indicating that there is no significant evidence of a difference in scores.