The GPA in an econometrics exam was normally distributed with a mean value of 80. In a sample of 20% of it was found that GPA was less than 75. Can you tell what the s.d. of the GPA was?
To determine the standard deviation (s.d.) of the GPA in the econometrics exam, we need some additional information. Specifically, we need either the standard deviation or the z-score corresponding to the given percentage value.
However, we can still provide you with some insights based on the information given.
First, let's assume that the GPA follows a normal distribution with a mean of 80. Since we are given that 20% of the sample had a GPA less than 75, we can consider this as the lower tail of the distribution.
To find the z-score corresponding to a cumulative probability of 20% in the lower tail, we can use a standard normal distribution table or a statistical calculator.
Using a standard normal distribution table, we find that the z-score corresponding to a cumulative probability of 20% is approximately -0.84.
Since the z-score formula is (X - μ) / σ, where X is the value you want to find the z-score for, μ is the mean, and σ is the standard deviation, we can rearrange the formula to solve for σ:
σ = (X - μ) / z
Now, substituting the known values into the formula:
σ = (75 - 80) / -0.84
Calculating this, we get:
σ = -5 / -0.84 ≈ 5.95
Therefore, based on the given information, the estimated standard deviation of the GPA in the econometrics exam is approximately 5.95. However, please note that this is an estimated value and may not precisely reflect the population standard deviation without further data or assumptions.