To determine the sample size needed for the sample proportion \(\hat{p}\) to be approximately normally distributed, you can use the rule of thumb based on the Central Limit Theorem. The sample size \(n\) should be sufficiently large such that both \(np\) and \(n(1-p)\) are greater than or equal to 10, where \(p\) is the true population proportion.
Given that the proportion \(p\) of people under the age of 18 is 0.07, we can calculate:
- \(np \geq 10\)
- \(n(1 - p) \geq 10\)
Let's solve for \(n\):
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For \(np \geq 10\): \[ n \cdot 0.07 \geq 10 \ n \geq \frac{10}{0.07} \ n \geq 142.86 \]
So, \(n\) must be at least 143 for this condition.
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For \(n(1-p) \geq 10\): \[ n(1 - 0.07) \geq 10 \ n \cdot 0.93 \geq 10 \ n \geq \frac{10}{0.93} \ n \geq 10.75 \]
So, \(n\) must be at least 11 for this condition.
The stricter condition is the first one (\(n \geq 143\)), so you need to draw a sample size of at least 143 kidney transplant patients to ensure that the sample proportion \(\hat{p}\) is approximately normally distributed.
Using a TI-84 Calculator
If you'd like to verify this or perform calculations involving sample size on the TI-84:
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Step 1: Access the Calculator
- Turn on your TI-84 calculator.
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Step 2: Perform the Calculation for n
- Use the
nvalue calculated previously to find \(n\), which is at least 143 in this case.
- Use the
The calculator itself does not have a built-in function specifically for this calculation, but this calculation can be performed manually, as shown above, or you can use statistical software or programming to find sample sizes.
So, you can conclude the sufficient sample size needed for the sample proportion to be normally distributed is \(n \geq 143\).
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