The amount of rainfall in January in a certain city is normally distributed with a mean of 4.3 inches and a standard deviation of 0.3 inches. Find the value of the quartile Q1.
To find the value of the first quartile (Q1), we need to determine the z-score associated with it and then convert it back to the original measurement scale.
Step 1: Calculate the z-score for Q1.
The first quartile represents the 25th percentile. Since the data is normally distributed, we can use the z-score formula:
z = (x - μ) / σ
where:
x = value of interest (Q1)
μ = mean
σ = standard deviation
In this case, μ = 4.3 inches and σ = 0.3 inches. Let's calculate the z-score for Q1:
z = (x - 4.3) / 0.3
Step 2: Look up the z-score in the standard normal distribution table.
The z-score will give us the proportion of data below Q1. We need to find the corresponding value in the standard normal distribution table. The closest z-score to the 25th percentile is -0.6745 (rounding to four decimal places).
Step 3: Convert the z-score back to the original measurement scale.
To find the value of Q1, we can rearrange the z-score formula:
x = (z * σ) + μ
Using the values from previous steps:
x = (-0.6745 * 0.3) + 4.3
Calculating this expression gives:
x ≈ 4.1 inches
Therefore, the value of the first quartile (Q1) is approximately 4.1 inches.
To find the value of the quartile Q1, we need to determine the corresponding z-score and then use it to calculate the actual rainfall value.
Here's how you can solve this problem step by step:
Step 1: Calculate the z-score for the first quartile, Q1.
The first quartile corresponds to the 25th percentile, which in a standard normal distribution is a z-score of approximately -0.674. The formula to calculate the z-score is:
z = (x - μ) / σ
where:
z is the z-score,
x is the rainfall value,
μ is the mean, and
σ is the standard deviation.
Step 2: Substitute the known values into the formula.
In this case, the mean (μ) is 4.3 inches and the standard deviation (σ) is 0.3 inches.
z = (x - 4.3) / 0.3
Step 3: Solve for x.
Rearranging the formula, we get:
x = (z * σ) + μ
Substituting the value of the z-score (-0.674), mean (4.3), and standard deviation (0.3) into the formula, we can calculate the value of Q1.
x = (-0.674 * 0.3) + 4.3
x = -0.2022 + 4.3
x ≈ 4.0978
So, the value of the quartile Q1 for the amount of rainfall in January in the given city is approximately 4.0978 inches.
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.25) and its Z score. Insert data into above equation and solve for score.