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Answer:
The probability that the sample mean weight will be more than 262 lb is 0.0047.
Step-by-step explanation:
The random variable <em>X</em> can be defined as the weight of National Football League (NFL) players now.
The mean weight is, <em>μ</em> = 252.8 lb.
The standard deviation of the weights is, <em>σ</em> = 25 lb.
A random sample of <em>n</em> = 50 NFL players are selected.
According to the Central Limit Theorem if we have an unknown population with mean <em>μ</em> and standard deviation <em>σ</em> and appropriately huge random samples (<em>n</em> > 30) are selected from the population with replacement, then the distribution of the sample means will be approximately normally distributed.
Then, the mean of the sample means is given by,
And the standard deviation of the sample means is given by,
The sample of players selected is quite large, i.e. <em>n</em> = 50 > 30, so the central limit theorem can be used to approximate the distribution of sample means.
Compute the probability that the sample mean weight will be more than 262 lb as follows:
*Use a <em>z</em>-table for the probability.
Thus, the probability that the sample mean weight will be more than 262 lb is 0.0047.