An IQ test is designed so that the mean is 100 and the standard deviation is 19 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 2 IQ points of the true mean. Assume that Ơ = 19 and determine the required sample size using technology. Then determine if this is a reasonable sample size for a real world calculation.
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To find the required sample size, we can use the formula for calculating the sample size for estimating a population mean with a specified margin of error:
n = (Z * σ / E)²
Where:
n is the required sample size,
Z is the Z-score corresponding to the desired level of confidence,
σ is the population standard deviation, and
E is the desired margin of error.
In this case, the desired level of confidence is 95%, which corresponds to a Z-score of 1.96 (obtained from a standard normal distribution table). The population standard deviation (σ) is given as 19, and the desired margin of error (E) is 2.
Substituting these values into the formula:
n = (1.96 * 19 / 2)²
n = (37.24 / 2)²
n = 18.62²
n ≈ 346.53
Rounding up to the nearest whole number, the required sample size is approximately 347.
To determine if this is a reasonable sample size for a real-world calculation, we need to consider factors such as population size, available resources, and practical constraints. The required sample size of 347 seems reasonable for a study involving IQ scores, as it provides a good balance between statistical accuracy and practical feasibility. However, additional considerations may be needed depending on the specific context of the study.