# 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, we need to find the value below which 25% of the data falls. In other words, we need to find the value x for which P(X ≤ x) = 0.25.

Since we know the distribution is normal with a mean of $50 and a standard deviation of $10, we can use the standard normal distribution (Z-distribution) to find the corresponding z-score.

The z-score is a measure of how many standard deviations an individual data point is from the mean. In this case, we want to find the z-score for which P(Z ≤ z) = 0.25.

Using a standard normal distribution table or a calculator, we find that the z-score corresponding to a cumulative probability of 0.25 is approximately -0.674.

Now, we can use the formula for converting a z-score back to the raw data value:

x = μ + (z * σ)

where x is the raw data value, μ is the mean, z is the z-score, and σ is the standard deviation.

Plugging in the known values, we have:

x = $50 + (-0.674 * $10)

x = $50 - $6.74

x = $43.26

Therefore, the 25th percentile of Jen's monthly phone bill is approximately $43.26.

## To find the 25th percentile of a normally distributed variable, we can use the z-score formula and the standard normal distribution table.

The z-score formula is:

z = (x - μ) / σ

Where:

- x is the value (in this case, the 25th percentile) we want to find the z-score for,

- μ is the mean of the distribution, and

- σ is the standard deviation of the distribution.

In this case, the mean (μ) is $50 and the standard deviation (σ) is $10.

Let's calculate the z-score for the 25th percentile:

z = (x - μ) / σ

To find x, we need to find the corresponding z-score in the standard normal distribution table. For the 25th percentile, we need to find z such that the area to the left of z under the standard normal curve is 0.25.

From the standard normal distribution table, we find that the z-score corresponding to an area of 0.25 is approximately -0.674.

Substituting the known values into the z-score formula:

-0.674 = (x - 50) / 10

Now, solve for x:

-0.674 * 10 = x - 50

-6.74 + 50 = x

x = 43.26

Therefore, the 25th percentile of Jen's monthly phone bill is approximately $43.26.