# At one college, GPA’s are normally distributed with a mean of 2.9 and a standard deviation of 0.6. Find the 70th percentile.

## 3.2

## To find the 70th percentile of a normally distributed variable, we need to find the z-score corresponding to that percentile and then use the z-score to find the corresponding value.

The formula for calculating the z-score is:

z = (x - μ) / σ

Where:

z = z-score

x = value in question (unknown)

μ = mean

σ = standard deviation

In this case, the percentile is 70%, which means we need to find the z-score corresponding to that percentile.

Using the standard normal distribution, we can find the z-score corresponding to the 70th percentile by using a z-table or a calculator. For the 70th percentile, the z-score is approximately 0.5244.

Now, we can use the z-score formula to find the value (x):

0.5244 = (x - 2.9) / 0.6

Now, we can solve for x:

0.5244 * 0.6 = x - 2.9

0.31464 = x - 2.9

x = 0.31464 + 2.9

x ≈ 3.21464

Therefore, the 70th percentile of GPA at this college is approximately 3.21464.

## To find the 70th percentile of a normally distributed variable, we need to determine the value that separates the lowest 70% of the data from the highest 30%.

First, we need to convert the college GPA's mean and standard deviation into a standardized z-score, which tells us how many standard deviations an observation is from the mean.

The formula for calculating the z-score is:

z = (X - μ) / σ

where:

X = the value we want to find the percentile for,

μ = mean of the distribution, and

σ = standard deviation of the distribution.

In this case, since we are looking for the 70th percentile, we need to find the z-score that corresponds to a cumulative probability of 0.70.

Using a standard normal distribution table or a calculator, we can find that the z-score corresponding to a cumulative probability of 0.70 is approximately 0.524.

Now, we can rearrange the original formula to solve for X:

X = z * σ + μ

Plugging in the values:

X = 0.524 * 0.6 + 2.9

X = 0.3144 + 2.9

X ≈ 3.2144

Therefore, the 70th percentile of GPA at this college is approximately 3.214.