suppose a population of nursing students heights is normally distributed with a population mean of 166 cm and a population stanadr deviation of 7.5cm if a female is randomly selected from this population, what is the probability that she will have a height between 155 and 175 inches? if a female is randomly selected from the population, what is the probability that she will be atleast 175 inches tall?, find the 90th percentile of the heights of nursing students in the population.

To answer these questions, we can use the concept of the standard normal distribution. The standard normal distribution has a mean of 0 and a standard deviation of 1. By converting the values in the provided problem to z-scores, we can find the probabilities and percentiles.

1. Probability of height between 155 and 175 cm:
To find the probability that a randomly selected female student's height is between 155 and 175 cm, we need to calculate the area under the normal curve within this range. We convert the height values to z-scores using the formula: z = (x - μ) / σ, where x is the specific value, μ is the population mean, and σ is the population standard deviation.

For 155 cm:
z1 = (155 - 166) / 7.5

For 175 cm:
z2 = (175 - 166) / 7.5

Once we calculate these z-scores, we can use a standard normal distribution table or a calculator with the ability to calculate cumulative probabilities to find the probability associated with the z-scores.

P(155 < x < 175) = P(z1 < z < z2)

2. Probability that a female is at least 175 cm tall:
To find the probability that a randomly selected female student's height is at least 175 cm, we need to calculate the area under the normal curve to the right of this value.

P(x ≥ 175) = P(z ≥ z2)

3. 90th percentile of the heights of nursing students:
To find the 90th percentile of the nursing student heights, we need to find the z-score that corresponds to this percentile. We can then convert the z-score back to the original height scale.

We need to find the z-score such that P(z < z*) = 0.9. We can use a standard normal distribution table or a calculator to find the z-score corresponding to a cumulative probability of 0.9.

Finally, once we have the z-score for the 90th percentile, we can convert it back to the original height scale using the formula: x = μ + (z* * σ), where x is the height value, μ is the population mean, σ is the population standard deviation, and z* is the z-score for the desired percentile.

By following these steps, we can find the answers to the questions related to the given problem.