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.
A. About $46.30
B. About $43.30
C. About $43.40
D. about $43.20
Answer : B---43.30
To find the 25th percentile of a normally distributed variable with a known mean and standard deviation, you can use the Z-score formula and the Z-table.
The Z-score formula is: Z = (X - μ) / σ
Where:
Z is the Z-score
X is the observed value
μ is the mean
σ is the standard deviation
In this case, the mean (μ) is $50, the standard deviation (σ) is $10, and we need to find the X value for the 25th percentile.
To find this X value, we need to find the corresponding Z-score using the Z-table. The Z-score for the 25th percentile is -0.6745. You can find this Z-score by looking for the area to the left of -0.675 in the standard normal distribution table.
Using the Z-score formula, we can rearrange it to solve for X:
X = Z * σ + μ
Plugging in the values, we have:
X = -0.6745 * $10 + $50
X = -$6.745 + $50
X = $43.255
To find the nearest cent, we round this value to the nearest cent, which gives us approximately $43.30.
Therefore, the 25th percentile of Jen's monthly phone bill is approximately $43.30.
So the correct answer is B. About $43.30.
To find the 25th percentile of Jen's monthly phone bill, we need to calculate the value that separates the lowest 25% of all bills from the higher 75%.
Since the phone bill amounts are normally distributed, we can use the standard normal distribution table or z-scores to find the desired percentile.
Step 1: Calculate the z-score for the 25th percentile.
The z-score formula is:
z = (x - μ) / σ
Where:
x = the value we want to find the percentile for (unknown)
μ = mean of the distribution (given as $50)
σ = standard deviation of the distribution (given as $10)
For the 25th percentile, we need to find the z-score that corresponds to an area of 0.25 to the left of the z-score. In other words, we need to find the z-score such that P(Z < z) = 0.25.
Using a standard normal distribution table or a z-score calculator, we can find that the z-score corresponding to an area of 0.25 is approximately -0.674.
Step 2: Solve for x.
Now that we have the z-score, we can solve for x using the z-score formula:
-0.674 = (x - 50) / 10
Multiplying both sides by 10, we get:
-6.74 = x - 50
Adding 50 to both sides, we get:
x = 43.26
Step 3: Round the result to the nearest cent.
Since the phone bill amounts are likely given to cents, we round the result to the nearest cent:
x ≈ $43.30
Therefore, the 25th percentile for Jen's monthly phone bill is approximately $43.30. So, the correct answer is option B.