Answer :
The probability of P(S <= 10) for a sample of size 8 is practically zero.
To calculate P(S <= 10) for a sample of size 8, we need to first standardize the variable. The formula for standardizing a variable is z = (x - μ) / (σ / sqrt(n)), where z is the standardized value, x is the given value, μ is the mean, σ is the standard deviation, and n is the sample size.
In this case, the given value is 10, the mean is 176 cm, the standard deviation is 12 cm, and the sample size is 8.
Using the formula, we can calculate the standardized value:
z = (10 - 176) / (12 / sqrt(8))
z = -166 / (12 / sqrt(8))
z = -166 / (12 / 2.8284)
z = -166 / 4
z = -41.5
The probability associated with this standardized value. Since the height of the population group follows a normal distribution, we can use a standard normal distribution table or a calculator to find the probability.
However, the standardized value of -41.5 is extremely unlikely and falls far beyond the normal distribution curve. Thus, the probability of P(S <= 10) for a sample of size 8 is practically zero.
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