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Answer:
The probability that the sample mean will be within 0.5 of the population mean is 0.3328.
Step-by-step explanation:
It is provided that a random variable <em>X</em> has mean, <em>μ</em> = 50 and<em> </em>standard deviation, <em>σ</em> = 7.
A random sample of size, <em>n</em> = 36 is selected.
According to the Central Limit Theorem if we have a population with mean μ and standard deviation σ and we take appropriately huge random samples (n ≥ 30) from the population with replacement, then the distribution of the sample mean will be approximately normally distributed.
Then, the mean of the distribution of sample mean is given by,
And the standard deviation of the distribution of sample mean is given by,
So, the distribution of the sample mean of <em>X</em> is N (50, 1.167²).
Compute the probability that the sample mean will be within 0.5 of the population mean as follows:
Thus, the probability that the sample mean will be within 0.5 of the population mean is 0.3328.