Answer :
The probability that 12 randomly selected people will have a mean weight greater than 172 pounds is approximately 0.2454 (rounded to four decimal places).
To find the probability that 12 randomly selected people will have a mean weight greater than 172 pounds, we can use the Central Limit Theorem. The Central Limit Theorem states that the distribution of sample means approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution.
First, let's find the mean and standard deviation of the sample mean weight. The mean of the sample mean weight can be found by taking the mean of the population weight, which is 176 pounds. The standard deviation of the sample mean weight can be found by dividing the standard deviation of the population weight by the square root of the sample size. In this case, the standard deviation of the population weight is 29 pounds, and the sample size is 12.
Mean of the sample mean weight = 176 pounds
Standard deviation of the sample mean weight = 29 pounds / √12 = 8.3545 pounds (rounded to four decimal places)
Now, we can calculate the z-score for a sample mean weight of 172 pounds using the formula:
z = (x - μ) / (σ / √n)
Where:
x = sample mean weight (172 pounds)
μ = population mean weight (176 pounds)
σ = standard deviation of the population weight (29 pounds)
n = sample size (12)
Plugging in the values, we get:
z = (172 - 176) / (29 / √12) ≈ -0.6923 (rounded to four decimal places)
Next, we can use a z-table or a calculator to find the probability associated with this z-score. Looking up the z-score of -0.6923 in the z-table, we find that the probability is approximately 0.2454 (rounded to four decimal places).
Therefore, the probability that 12 randomly selected people will have a mean weight greater than 172 pounds is approximately 0.2454 (rounded to four decimal places).
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