4) The amount of Jen’s monthly phone bill is normally distributed with a mean of $50 and a standard deviation of $10. Find the 25th percentile.
To find the 25th percentile, follow these steps:
Step 1: Convert the given mean and standard deviation to z-scores.
The formula to calculate the z-score is:
z = (x - mean) / standard deviation
Given:
mean = $50
standard deviation = $10
Step 2: Calculate the z-score corresponding to the 25th percentile.
Since the 25th percentile is one standard deviation below the mean, the z-score will be -1.
Step 3: Calculate the raw score (x) using the z-score formula:
z = (x - mean) / standard deviation
-1 = (x - $50) / $10
Step 4: Solve for x:
-1 * $10 = x - $50
-$10 = x - $50
x = -$10 + $50
x = $40
So, the 25th percentile of Jen's monthly phone bill is $40.
To find the 25th percentile of Jen's monthly phone bill, we need to use the standard normal distribution and z-scores.
Step 1: Convert the given values into z-scores:
The formula for a z-score is:
z = (x - μ) / σ
where
x = data value (in this case, the 25th percentile)
μ = mean of the distribution ($50)
σ = standard deviation of the distribution ($10)
z = (x - 50) / 10
Step 2: Look up the z-score in the standard normal distribution table. The table will give you the area to the left of that z-score.
From the standard normal distribution table, find the z-score corresponding to the 25th percentile, which is -0.674. This means that 25% of the data falls below this z-score (-0.674).
Step 3: Calculate the x-value (25th percentile) using the z-score:
Using the z-score formula:
z = (x - 50) / 10, rearrange it to solve for x:
x = (z * 10) + 50
Substituting the value of z = -0.674:
x = (-0.674 * 10) + 50
Calculating:
x = -6.74 + 50
x = 43.26
So, the 25th percentile of Jen's monthly phone bill is approximately $43.26.