The weight of certain population of young females is approximately normally distributed with the mean of 66 kg and a standard deviation of 7.5 kg. If an individual is selected at random from this population, find the probability that she will weigh between 52.5 and 72.5 kg
To find the probability that an individual will weigh between 52.5 and 72.5 kg, we need to calculate the z-scores for these weights and use the standard normal distribution table.
The z-score formula is given by:
z = (x - μ) / σ
where:
x = the individual weight
μ = the mean weight (66 kg)
σ = the standard deviation (7.5 kg)
For 52.5 kg:
z1 = (52.5 - 66) / 7.5
z1 = -13.5 / 7.5
z1 = -1.8
For 72.5 kg:
z2 = (72.5 - 66) / 7.5
z2 = 6.5 / 7.5
z2 = 0.87
Now, using the standard normal distribution table, we can find the probability for z-scores.
From the table, the probability of a z-score between -1.8 and 0.87 is the same as the probability of a z-score between -0.87 and 1.8.
So, P(-0.87 ≤ z ≤ 1.8) = P(z ≤ 1.8) - P(z ≤ -0.87)
Using the table or a calculator, we find:
P(z ≤ 1.8) ≈ 0.9641
P(z ≤ -0.87) ≈ 0.1915
Now, we can calculate the probability:
P(52.5 ≤ x ≤ 72.5) ≈ P(-0.87 ≤ z ≤ 1.8) ≈ P(z ≤ 1.8) - P(z ≤ -0.87)
≈ 0.9641 - 0.1915
≈ 0.7726
Therefore, the probability that an individual randomly selected from this population will weigh between 52.5 and 72.5 kg is approximately 0.7726 or 77.26%.
To find the probability that a randomly selected individual from this population weighs between 52.5 and 72.5 kg, we need to use the properties of the normal distribution.
First, we need to standardize the values using the formula:
z = (x - μ) / σ
where:
z = the z-score
x = the value we want to find the probability for
μ = the mean of the population
σ = the standard deviation of the population
For the lower value of 52.5 kg:
z1 = (52.5 - 66) / 7.5
For the upper value of 72.5 kg:
z2 = (72.5 - 66) / 7.5
Then, we need to find the probability associated with these z-scores using a standard normal distribution table or calculator.
P(52.5 < x < 72.5) = P(z1 < z < z2)
Let's calculate the z-scores first:
z1 = (52.5 - 66) / 7.5 = -1.8
z2 = (72.5 - 66) / 7.5 = 0.87
Using a standard normal distribution table or calculator, we can find the probability associated with these z-scores:
P(-1.8 < z < 0.87) = P(z < 0.87) - P(z < -1.8)
Using the standard normal distribution table or calculator, we find:
P(z < 0.87) = 0.8078
P(z < -1.8) = 0.0359
Therefore, the probability that a randomly selected individual weighs between 52.5 and 72.5 kg is:
P(52.5 < x < 72.5) = P(-1.8 < z < 0.87) = P(z < 0.87) - P(z < -1.8) = 0.8078 - 0.0359 = 0.7719
So, the probability is approximately 0.7719 or 77.19%.
To find the probability that a randomly selected individual from this population will weigh between 52.5 and 72.5 kg, we need to use the properties of the normal distribution.
The first step is to standardize the values using the standard normal distribution, which has a mean of 0 and a standard deviation of 1.
To standardize a value, we use the formula:
z = (x - μ) / σ
where z is the standardized value, x is the given value, μ is the mean, and σ is the standard deviation.
In this case, to find the probability that the weight is between 52.5 and 72.5 kg, we need to find the z-scores for these values.
For 52.5 kg:
z1 = (52.5 - 66) / 7.5
For 72.5 kg:
z2 = (72.5 - 66) / 7.5
Next, we use a standard normal distribution table or a statistical calculator to find the probability associated with each z-score.
Let's assume that you have access to a standard normal distribution table. Consulting this table, you will find the cumulative probabilities associated with the z-scores.
For example, for z1, you would look up the probability associated with the z-score of -1.8 (rounded to one decimal place). This will give you the cumulative probability of the weight being below 52.5 kg, denoted as P(Z < -1.8).
Similarly, for z2, look up the probability associated with the z-score of +0.67 (rounded to two decimal places). This will give you the cumulative probability of the weight being below 72.5 kg, denoted as P(Z < +0.67).
Finally, subtract the probability associated with the lower z-score from the probability associated with the higher z-score to find the probability between the two values.
P(52.5 kg < X < 72.5 kg) = P(-1.8 < Z < 0.67)
= P(Z < 0.67) - P(Z < -1.8)
Using the standard normal distribution table, you can find these probabilities.
Alternatively, if you have access to a statistical calculator such as Excel or Python, you can directly calculate the probabilities using the cumulative distribution function (CDF) for the normal distribution.