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Answer :
The probability that the sample mean weight is above 176 lbs can be calculated using the standard normal distribution. Similarly, when 70 women are randomly selected, the distribution of the sample mean weight follows a normal distribution with the same mean as the population and a smaller standard deviation due to the larger sample size. The probability that the sample mean weight is above 176 lbs can be calculated using the standard normal distribution.
(a) To find the probability that a randomly selected woman's weight is above 176 lbs, we can standardize the weight using the z-score formula. The z-score is calculated as (x - μ) / σ, where x is the weight, μ is the population mean, and σ is the population standard deviation. We can then use the standard normal distribution table or a statistical software to find the corresponding probability.
(b) When 3 women are randomly selected, the mean weight of the sample (x) follows a normal distribution with a mean equal to the population mean (μ) and a standard deviation equal to the population standard deviation (σ) divided by the square root of the sample size (n). In this case, the standard deviation of the sample mean weight would be σ / √n. We can then calculate the z-score for the sample mean weight and find the corresponding probability using the standard normal distribution.
(c) Similarly, when 70 women are randomly selected, the mean weight of the sample (x) follows a normal distribution with a mean equal to the population mean (μ) and a smaller standard deviation due to the larger sample size. We can calculate the z-score for the sample mean weight and find the corresponding probability using the standard normal distribution.
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