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Answer :
a) The mean weight of the 80 randomly selected males is 163 pounds.
b) The standard deviation of the weight of the 80 randomly selected males is 3.92 pounds.
a) The mean of a sample is equal to the mean of the population. Therefore, the mean weight of the 80 randomly selected males is the same as the mean weight of males in the US, which is 163 pounds.
b) The standard deviation of a sample is calculated by dividing the standard deviation of the population by the square root of the sample size. In this case, the standard deviation of the population is 35 pounds, and the sample size is 80. Therefore, the standard deviation of the 80 randomly selected males is 35 divided by the square root of 80, which is approximately 3.92 pounds.
a) The mean weight of the 80 randomly selected males is equal to the mean weight of males in the US because we assume that the sample is representative of the population. This means that the average weight of the 80 males should be similar to the average weight of all males in the US, which is given as 163 pounds.
b) The standard deviation of the sample is calculated by dividing the standard deviation of the population by the square root of the sample size. In this case, the population standard deviation is 35 pounds, and the sample size is 80. Dividing 35 by the square root of 80 yields a standard deviation of approximately 3.92 pounds for the sample.
c) The distribution of the sample means follows a normal distribution. This is known as the Central Limit Theorem, which states that as the sample size increases, the distribution of sample means approaches a normal distribution regardless of the shape of the population distribution. In this case, the sample size is large (80), so we can assume that the distribution of the sample means follows a normal distribution.
d) To calculate the probability that the mean weight for the 50 males is less than 155 pounds, we need to use the standard normal distribution. We can standardize the value using the z-score formula: z = (x - μ) / σ, where x is the value we want to find the probability for, μ is the mean of the distribution, and σ is the standard deviation of the distribution. Plugging in the values, we have z = (155 - 163) / 3.92 ≈ -2.04. By looking up the z-score in the standard normal distribution table or using a statistical software, we can find the probability associated with this z-score.
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