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The national average score on a mathematics test is known to be \(75\). A school principal introduces a new teaching method and wants to know if it has changed student performance.
A sample of 40 students taught with the new method achieves a mean score of \(78\) with a standard deviation of \(12\).
  1. Construct a \(95\pourcent\) confidence interval for the mean score under the new method.
  2. Can the principal conclude that the new method has significantly changed the average score compared to the national standard?

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