The amount of rainfall in January in a certain city is normally distributed with a mean of 4.2 inches and a standard deviation of 0.5 inches. Find the value of the quartile Q1
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 in the smaller portion) related to a Z score. Insert in above equation and solve for score.
To find the value of the quartile Q1, we need to calculate the z-score corresponding to the first quartile and then use it to find the corresponding value.
The first quartile (Q1) corresponds to the 25th percentile. In a normal distribution, we can find the z-score using the standard normal distribution table or formula.
The formula to calculate the z-score is:
z = (x - μ) / σ
Where:
z = z-score
x = the value we want to find the z-score for
μ = mean of the distribution
σ = standard deviation of the distribution
For Q1, the z-score will be negative because it is to the left of the mean.
Let's calculate the z-score using the given values:
z = (Q1 - μ) / σ
μ = 4.2 (mean)
σ = 0.5 (standard deviation)
Q1 = ?
Now, we rearrange the formula to solve for Q1:
Q1 = (z * σ) + μ
To find the z-score for the 25th percentile (Q1), we can refer to the standard normal distribution table. The cumulative probability for the 25th percentile is 0.25.
Looking up the value for 0.25 in the standard normal distribution table, we find that the closest z-score is approximately -0.674.
Now we can calculate the value of Q1:
Q1 = (-0.674 * 0.5) + 4.2
Q1 ≈ 4.2 - 0.337
Q1 ≈ 3.863
Therefore, the value of the first quartile (Q1) is approximately 3.863 inches.
To find the value of the quartile Q1, we can use the properties of the normal distribution. The first quartile, Q1, corresponds to the 25th percentile.
To solve this problem, we need to standardize the rainfall value by using the z-score formula:
z = (x - μ) / σ
where:
- x is the rainfall value
- μ is the mean of the distribution (4.2 inches)
- σ is the standard deviation (0.5 inches)
Let's calculate the z-score for Q1, which corresponds to the 25th percentile:
z = (Q1 - μ) / σ
Next, we need to find the z-value corresponding to the 25th percentile. We can use a standard normal distribution table or calculator to look up this value.
The z-value for the 25th percentile is approximately -0.674 (rounded to three decimal places).
Substituting this value into our z-score formula, we have:
-0.674 = (Q1 - 4.2) / 0.5
Now, let's solve for Q1:
-0.674 * 0.5 = Q1 - 4.2
-0.337 = Q1 - 4.2
Q1 = -0.337 + 4.2
Q1 ≈ 3.863
Therefore, the value of the quartile Q1 for the amount of rainfall in January in this city is approximately 3.863 inches.