Suppose that weights of 5th grade elementary boys are normally distributed with mean µ =90 lbs and variance ó2 = 4 lbs. Find the weight that corresponds to Q3 and interpret this measure of position.
Q3 = 75th percentile
Z = (score-mean)/SD
Standard deviation = square root of variance
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the Z score = mean + .25. Insert values into above equation and solve.
I'll leave the interpretation to you.
To find the weight that corresponds to Q3, we need to calculate the third quartile of the weight distribution.
Step 1: Calculate the standard deviation (σ) by taking the square root of the variance (σ^2):
σ = √4 = 2 lbs
Step 2: Determine the z-score corresponding to the third quartile (Q3) using the standard normal distribution table or a z-score calculator. The z-score associated with Q3 is approximately 0.6745.
Step 3: Use the z-score formula to find the weight that corresponds to Q3:
z = (X - µ) / σ
Rearranging the formula, we have:
X = z * σ + µ
Substituting the known values, we have:
X = 0.6745 * 2 + 90
X ≈ 1.349 + 90
X ≈ 91.349 lbs
Therefore, the weight that corresponds to Q3 is approximately 91.349 lbs.
Interpretation:
Q3, also known as the third quartile, represents the weight value below which approximately 75% of the weights of the 5th grade elementary boys fall. In this case, it means that 75% of the boys weigh less than or equal to 91.349 lbs.
To find the weight that corresponds to Q3, we first need to understand what Q3 represents. Q3, also known as the third quartile, is the value that divides the upper 75% of the data from the lower 25%.
To calculate Q3, we can use the cumulative distribution function (CDF) of the normal distribution. The CDF gives the probability that a value is less than or equal to a given point.
Since the weights of 5th grade elementary boys are normally distributed with a mean of µ = 90 lbs and a variance of σ^2 = 4 lbs, we can calculate the standard deviation (σ) by taking the square root of the variance: σ = √(4) = 2 lbs.
Now we can use the CDF to find Q3. Let's assume X represents the weight of a 5th grade elementary boy.
P(X ≤ Q3) = 0.75
Using the standardized normal distribution (with a mean of 0 and a standard deviation of 1), we can find the corresponding z-score for the 75th percentile. We can then unstandardize this z-score to find the weight (X) that corresponds to Q3.
P(Z ≤ z) = 0.75
We can use a standard normal distribution table, or a statistical software, to find the z-score that corresponds to the cumulative probability of 0.75. The z-score is approximately 0.674.
To unstandardize the z-score and find the weight (X), we can use the formula:
X = µ + (σ × z)
Plugging in the values:
X = 90 + (2 × 0.674) = 90 + 1.348 = 91.348 lbs
Therefore, the weight that corresponds to Q3 is approximately 91.348 lbs.
Interpretation:
The weight that corresponds to Q3 represents the weight at which 75% of the 5th grade elementary boys weigh less than or equal to. In this case, it means that 75% of the boys weigh less than or equal to approximately 91.348 lbs.