a. To find the probability that an individual distance is greater than 218.00 cm, we'll utilize the Z-score formula, which standardizes the individual distance to a standard normal distribution with a mean of 0 and a standard deviation of 1.
The Z-score formula is given by:
Z = (X - μ) / σ
Where:
- X = 218.00 cm (individual distance)
- μ = 205.5 cm (mean)
- σ = 8.6 cm (standard deviation)
Substituting the given values into the formula, we find:
Z ≈ 1.453
We then use the standard normal distribution table to find the probability corresponding to Z ≈ 1.453, which is approximately 0.9265.
Therefore, the probability that an individual distance is greater than 218.00 cm is approximately 0.9265.
b. To find the probability that the mean for 20 randomly selected distances is greater than 204.00 cm, we'll employ the Central Limit Theorem (CLT). Given that the sample size is 20, we proceed with the assumption that the CLT holds. Therefore, the sampling distribution of the sample mean will be approximately normal, even though the sample size is less than 30.
We calculate the standard error of the mean (SE) using the formula:
SE = σ / √n
Where:
- σ = 8.6 cm (standard deviation)
- n = 20 (sample size)
SE ≈ 1.924
Next, we calculate the Z-score using the formula:
Z ≈ -0.779
We then use the standard normal distribution table to find the probability corresponding to Z ≈ -0.779, which is approximately 0.2181.
Therefore, the probability that the mean for 20 randomly selected distances is greater than 204.00 cm is approximately 0.2181.
c. The normal distribution can be used in part (b) despite the sample size being less than 30 due to the Central Limit Theorem (CLT). The CLT ensures that as the sample size increases, the sampling distribution of the sample mean approaches a normal distribution, regardless of the shape of the population distribution. In this case, even though the sample size is 20, which is less than 30, it is still large enough for the CLT to hold, assuming that the underlying population distribution is approximately normal. Therefore, the normal distribution can be reasonably used to analyze the sampling distribution of the sample mean.
