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Answer :
Final answer:
The distribution of the sample mean IQ, for a sample size of 200 from a population with mean 118 and standard deviation 20, is approximately normal with a mean of 118 and standard deviation 1.414, in line with the Central Limit Theorem.
Explanation:
When considering the distribution of sample means, particularly in a scenario involving IQ scores among college-educated adults, it's essential to apply the Central Limit Theorem. This theorem posits that for a sufficiently large sample size, the distribution of the sample mean will be approximately normal, regardless of the population's distribution shape. Given a population with a mean IQ of 118 and a standard deviation of 20, when we select a sample of 200 adults, the distribution of the sample mean IQ will adhere to this theorem.
The formula for calculating the standard deviation (σ) of the sample mean (μ) is given by σ/√n, with σ being the standard deviation of the population and n the sample size. Thus, for our case, the calculation would be 20/√200, which simplifies to 20/14.14, approximately equal to 1.414. Therefore, the correct answer is that the distribution of the sample mean IQ is approximately Normal with mean 118 and standard deviation 1.414, making option 'd' the correct choice.
This principle is fundamental in inferential statistics, allowing us to make predictions and conclusions about population parameters based on sample statistics. The Central Limit Theorem guarantees that with a large enough sample size, we can assume a normal distribution for the sample mean, facilitating further statistical analysis and hypothesis testing.
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The distribution of the sample mean IQ is approximately Normal with mean 118 and standard deviation 1.414. This is due to the Central Limit Theorem which states that as the sample size increases, the distribution of the sample means becomes approximately normal regardless of the shape of the population distribution. The correct answer is option (d).
In this case, since the sample size is large (n=200), the distribution of the sample means will be approximately normal.
The standard deviation of the sample mean is calculated by dividing the population standard deviation by the square root of the sample size. In this case, the standard deviation of the sample mean is 20/sqrt(200) = 1.414. Therefore, the distribution of the sample mean IQ is approximately Normal with mean 118 and standard deviation 1.414.Hence the right answer is option (d).
It is important to note that while the sample mean is likely to be close to the population mean, it is not guaranteed to be exactly equal. Therefore, option c is incorrect as it assumes the sample mean is equal to the population mean. Option b is also incorrect as it assumes a very small standard deviation for the sample mean, which is not realistic. Option a is partially correct in stating that the distribution of the sample mean is normal, but it does not provide the correct standard deviation.
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