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.
The following site takes the place of tables in a textbook
http://davidmlane.com/hyperstat/z_table.html
click on:
Value from an area
enter:
area 0.7
mean 2.9
sd .6
click on 'below' to get3.214
To find the 70th percentile of a normal distribution with a given mean and standard deviation, you can follow these steps:
Step 1: Start with the given mean and standard deviation
Mean (μ) = 2.9
Standard Deviation (σ) = 0.6
Step 2: Find the z-score corresponding to the desired percentile using a z-table or a calculator.
For the 70th percentile, you need to find the z-score that cuts off the lower 70% of the distribution.
P(Z ≤ z) = 0.70
Step 3: Use the z-score formula to calculate the z-score.
z = (x - μ) / σ
Step 4: Substitute the given values into the formula and solve for x.
(x - 2.9) / 0.6 = z
Step 5: Rearrange the formula to solve for x.
x = (z * σ) + μ
Step 6: Substitute the z-score value from step 2 and the given mean and standard deviation into the formula and calculate x.
x = (z * σ) + μ
Once you find the value of x, that will be the GPA cutoff for the 70th percentile.
To find the 70th percentile of a normal distribution, you can use a Z-table or a calculator. Here's how you can calculate it using a Z-table:
Step 1: Convert the percentile value to a Z-score.
The percentile (70th percentile) corresponds to the area under the curve to the left of the Z-score. To find the Z-score, you can use a standard normal distribution table (also known as a Z-table) or a calculator. Look up the Z-score that corresponds to a cumulative area of 0.70.
Step 2: Use the Z-score to find the raw score.
Once you have the Z-score, you can use it to find the corresponding raw score using the formula:
raw score = (Z-score × standard deviation) + mean
Now let's calculate the 70th percentile step by step:
Step 1: Finding the Z-score
Using a Z-table or a calculator, we find that the Z-score corresponding to a cumulative area of 0.70 is approximately 0.5244.
Step 2: Finding the raw score
Using the formula:
raw score = (Z-score × standard deviation) + mean
raw score = (0.5244 × 0.6) + 2.9
raw score = 0.3146 + 2.9
raw score ≈ 3.2146
Therefore, the 70th percentile of GPA at this college is approximately 3.2146.