IQ test is designed so that the mean is 100 and the standard deviation is 12 for the population of normal adults. Find the sample size necessary to estimate the mean IQ score of statistics students such that it be said with $$ 99 \% $$ confidence that the sample mean is within 7 IQ points of the true mean. Assume that $$ \sigma=12 $$ and determine the required sample size using technology. Then determine if this is a reasonable nple size for a real world calculation. e required sample size is $$ \square $$. (Round up to the nearest integer.)

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We are given that the population IQ scores follow a normal distribution with
$$
\mu = 100 \quad \text{and} \quad \sigma = 12.
$$
We wish to estimate the mean IQ score of statistics students so that the sample mean is within 7 IQ points of the true mean with $$99\%$$ confidence.

The margin of error $$ E $$ is given by
$$
E = z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}},
$$
where $$ z_{\alpha/2} $$ is the critical value for a $$99\%$$ confidence level. For a $$99\%$$ confidence level, $$ \alpha = 0.01 $$ and thus $$ \alpha/2 = 0.005 $$. The corresponding critical value is approximately
$$
z_{0.005} \approx 2.576.
$$

We set up the equation for the margin of error:
$$
7 = 2.576 \cdot \frac{12}{\sqrt{n}}.
$$

Solving for $$ n $$, we first isolate $$ \sqrt{n} $$:
$$
\sqrt{n} = 2.576 \cdot \frac{12}{7}.
$$

Now calculate the right-hand side:
$$
\sqrt{n} = 2.576 \cdot \frac{12}{7} \approx 2.576 \cdot 1.714 \approx 4.416.
$$

Squaring both sides gives:
$$
n \approx (4.416)^2 \approx 19.52.
$$

Since the sample size must be an integer and we round up to ensure the margin of error does not exceed 7, we have:
$$
n = 20.
$$

For a real-world calculation, a sample size of $$20$$ is fairly small. Although it may be sufficient for estimating the mean IQ of statistics students under the assumption that the students' IQ scores are normally distributed with the given standard deviation of $$12$$, in practice one should consider potential deviations from the assumed normality and possible sampling biases. Therefore, while the calculation gives a required sample size of $$20$$, one might opt for a larger sample to account for such factors in real-world situations.

The required sample size is $$20$$.
The required sample size is 20.

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