Questions: An IQ test is designed so that the mean is 100 and the standard deviation is 18 for the population of normal adults. Find the sample size necessary to estimate the mean IQ score of statistics students such that it can be said with 95% confidence that the sample mean is within 7 IQ points of the true mean. Assume that σ=18 and determine the required sample size using technology. The required sample size is 26. (Round up to the nearest integer.) Would it be reasonable to sample this number of students? No. This number of IQ test scores is a fairly large number. Yes. This number of IQ test scores is a fairly small number. No. This number of IQ test scores is a fairly small number. Yes. This number of IQ test scores is a fairly large number.

An IQ test is designed so that the mean is 100 and the standard deviation is 18 for the population of normal adults. Find the sample size necessary to estimate the mean IQ score of statistics students such that it can be said with 95% confidence that the sample mean is within 7 IQ points of the true mean. Assume that σ=18 and determine the required sample size using technology. The required sample size is 26. (Round up to the nearest integer.) Would it be reasonable to sample this number of students? No. This number of IQ test scores is a fairly large number. Yes. This number of IQ test scores is a fairly small number. No. This number of IQ test scores is a fairly small number. Yes. This number of IQ test scores is a fairly large number.
Transcript text: An IQ test is designed so that the mean is 100 and the standard deviation is 18 for the population of normal adults. Find the sample size necessary to estimate the mean IQ score of statistics students such that it can be said with $95 \%$ confidence that the sample mean is within 7 IQ points of the true mean. Assume that $\sigma=18$ and determine the required sample size using technology. The required sample size is 26 . (Round up to the nearest integer.) Would it be reasonable to sample this number of students? No. This number of IQ test scores is a fairly large number. Yes. This number of IQ test scores is a fairly small number. No. This number of IQ test scores is a fairly small number. Yes. This number of IQ test scores is a fairly large number.
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Solution

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Solution Steps

Step 1: Calculate the Z-score

To find the Z-score for a \(95\%\) confidence level, we use the formula:

\[ Z = \Phi^{-1}\left(1 - \frac{\alpha}{2}\right) \]

where \(\alpha = 0.05\). Thus, we have:

\[ Z \approx 1.959964 \]

Step 2: Calculate the Required Sample Size

The formula for the required sample size \(n\) to estimate the mean with a specified margin of error \(E\) is given by:

\[ n = \left(\frac{Z \cdot \sigma}{E}\right)^2 \]

Substituting the values:

  • \(Z \approx 1.959964\)
  • \(\sigma = 18\)
  • \(E = 7\)

We calculate:

\[ n = \left(\frac{1.959964 \cdot 18}{7}\right)^2 \approx 25.999 \]

Rounding up, we find:

\[ n = 26 \]

Step 3: Assess the Reasonableness of the Sample Size

To determine if a sample size of \(26\) is reasonable, we consider the context of IQ testing. A sample size of \(26\) is generally considered small in statistical studies, especially for a population as large as statistics students.

Final Answer

The required sample size is \(n = 26\) and it is reasonable to sample this number of students.

Thus, the answer is:

\[ \boxed{n = 26} \]

And the assessment of the sample size is:

\[ \text{Yes. This number of IQ test scores is a fairly small number.} \]

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