An IQ test is designed so that the mean is 100 and the standard deviation is 25 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 5 IQ points of the true mean. Assume that $\sigma=25$ and determine the required sample size using technology. Then determine if this is a reasonable sample size for a real world calculation.
The required sample size is $\square$. (Round up to the nearest integer.)
Would it be reasonable to sample this number of students?
Yes. This number of IQ test scores is a fairly large number.
No. This number of IQ test scores is a fairly large number.
No. This number of $I Q$ test scores is a fairly small number.
Yes. This number of IQ test scores is a fairly small number.
Answer
Final Answer: The required sample size is \(\boxed{97}\). This number of IQ test scores is a fairly large number.
Step 1 ：The problem is asking for the sample size necessary to estimate the mean IQ score with a 95% confidence level and a margin of error of 5 IQ points. The standard deviation is given as 25.
Step 2 ：We can use the formula for the sample size in a confidence interval estimation, which is: \(n = (Z*σ/E)^2\) where: n is the sample size, Z is the Z-score (which corresponds to the desired confidence level, in this case 95%), σ is the standard deviation (25 in this case), E is the margin of error (5 in this case).
Step 3 ：The Z-score for a 95% confidence level is approximately 1.96. We can plug these values into the formula and solve for n.
Step 4 ：After calculating the sample size, we need to round up to the nearest integer because we can't have a fraction of a person.
Step 5 ：The calculated required sample size is 97. This means that we would need to sample 97 students to estimate the mean IQ score with a 95% confidence level and a margin of error of 5 IQ points.
Step 6 ：Now we need to determine if this sample size is reasonable. Given that this is a fairly large number, it may not be feasible to collect that many samples, especially if the population of statistics students is small. However, if the population is large, this sample size may be reasonable.
Step 7 ：Final Answer: The required sample size is \(\boxed{97}\). This number of IQ test scores is a fairly large number.
