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Answer :
The probabilities are as follows:
(a) Probability for 1 woman's weight between 108 lb and 175 lb: P(108 lb ≤ X ≤ 175 lb) = P(Z1 ≤ Z ≤ Z2)
(b) Probability for 4 women's mean weight between 108 lb and 175 lb: P(108 lb ≤ X_bar ≤ 175 lb) = P(Z1' ≤ Z ≤ Z2')
(c) Probability for 89 women's mean weight between 108 lb and 175 lb: P(108 lb ≤ X_bar ≤ 175 lb) = P(Z1'' ≤ Z ≤ Z2'')
Let's analyze each section separately:
(a) Probability for 1 woman's weight between 108 lb and 175 lb:
To find the probability that a randomly selected woman's weight falls within the range of 108 lb to 175 lb, we need to standardize the values using the Z-score formula. The Z-score (Z) is calculated as (X - μ) / σ, where X is the weight value, μ is the mean, and σ is the standard deviation.
For the lower bound of 108 lb:
Z1 = (108 - 143) / 29 = -35 / 29 ≈ -1.2069
For the upper bound of 175 lb:
Z2 = (175 - 143) / 29 = 32 / 29 ≈ 1.1034
Using a Z-table or a calculator, we can find the corresponding probabilities associated with Z1 and Z2.
The probability of a woman's weight being between 108 lb and 175 lb is given by:
P(108 lb ≤ X ≤ 175 lb) = P(Z1 ≤ Z ≤ Z2)
Using the Z-table or a calculator, we can find these probabilities and calculate the difference between them.
(b) Probability for 4 women's mean weight between 108 lb and 175 lb:
To find the probability that the mean weight of 4 randomly selected women falls within the range of 108 lb to 175 lb, we need to consider the distribution of sample means. The mean of the sample means (μ') will still be the same as the population mean (μ), but the standard deviation of the sample means (σ') is calculated as σ / √n, where n is the sample size.
For n = 4, σ' = 29 / √4 = 29 / 2 = 14.5 lb.
We can then calculate the Z-scores for the lower and upper bounds using the formula mentioned earlier. Let's denote the Z-scores as Z1' and Z2'.
For the lower bound of 108 lb:
Z1' = (108 - 143) / 14.5 ≈ -2.4138
For the upper bound of 175 lb:
Z2' = (175 - 143) / 14.5 ≈ 2.2069
Using a Z-table or a calculator, we can find the probabilities associated with Z1' and Z2', which represent the probability of the mean weight falling between 108 lb and 175 lb.
(c) Probability for 89 women's mean weight between 108 lb and 175 lb:
Following the same approach as in (b), we can calculate the standard deviation of the sample means for a sample size of 89:
For n = 89, σ' = 29 / √89 ≈ 3.0755 lb.
We can then calculate the Z-scores for the lower and upper bounds using the formula mentioned earlier. Let's denote the Z-scores as Z1'' and Z2''.
For the lower bound of 108 lb:
Z1'' = (108 - 143) / 3.0755 ≈ -11.3405
For the upper bound of 175 lb:
Z2'' = (175 - 143) / 3.0755 ≈ 10.3904
Using a Z-table or a calculator, we can find the probabilities associated with Z1'' and Z2'', which represent the probability of the mean weight falling between 108 lb and 175 lb for a sample of 89 women.
To know more about normal distribution, refer here:
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