Final answer:
To determine the probabilities for the heights of female students at GIMPA modeled by a normal distribution, one must calculate Z-scores and use standard normal distribution tables. For individual heights, the specific Z-scores for the desired height ranges are needed. For the mean height of a sample, the sample's standard error is used to calculate the sample mean's Z-score.
Step-by-step explanation:
To solve the given problem regarding the heights of female students at GIMPA modeled by a normal distribution, we will first standardize the values of height and then use the standard normal distribution to calculate probabilities.
i. To find the probability that the height of a randomly selected female student is less than 172.5 cm, we calculate the Z-score:
Z = (X - μ) / σ
Where:
- X = 172.5 cm (the height in question)
- μ = 168.0 cm (mean height)
- σ = 4.5 cm (standard deviation)
Z = (172.5 - 168.0) / 4.5 ≈ 1
Using the standard normal distribution table, we find the probability corresponding to Z=1.
ii. To find the probability that the height is between 159 cm and 163.5 cm, we calculate the Z-scores for both values:
Z1 = (159 - 168) / 4.5
Z2 = (163.5 - 168) / 4.5
We then subtract the probability corresponding to Z1 from the probability corresponding to Z2 using the standard normal distribution table.
Next, to find the probability that the mean height of a random sample of 11 female students exceeds 172 cm, we use the standard error for the mean (SEM):
SEM = σ / √ n
Where:
- σ = 4.5 cm (standard deviation)
- n = 11 (sample size)
Calculate the Z-score for the sample mean using the SEM:
Z = (X - μ) / SEM
Substitute X with 172 cm, μ with 168.0 cm, and calculate the SEM to determine Z.
Use the standard normal distribution table to find the probability corresponding to the calculated Z-score.
Without providing the probabilities and Z-scores derived from standard normal distribution tables, the full accuracy of parts a and b cannot be determined. To accurately assign option A, B, C, or D, the actual calculations and probabiliti0es are required. The question statement does not contain specific calculations or results, so based only on the processes described, we cannot confirm which option is correct.