To determine whether it is appropriate to use the normal approximation for the binomial distribution in this scenario, we need to check the conditions involving \(np\) and \(n(1-p)\).
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Calculate \(p\): The proportion of freshmen planning to major in a STEM discipline is given as \(p = 0.27\).
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Calculate \(n\): The sample size is \(n = 80\).
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Calculate \(np\): \[ np = 80 \times 0.27 = 21.6 \]
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Calculate \(n(1-p)\): \[ n(1-p) = 80 \times (1 - 0.27) = 80 \times 0.73 = 58.4 \]
Now we check the conditions:
- \(np = 21.6 > 10\)
- \(n(1-p) = 58.4 > 10\)
Since both \(np\) and \(n(1-p)\) are greater than 10, it is appropriate to use the normal approximation.
Therefore, it is appropriate to use the normal curve since \(np > 10\) and \(n(1-p) > 10\).