Final answer:
The mean (μ) and standard deviation (σ) of SAT scores for Penn State's 2020 incoming class are estimated to be 1325 and 126.11, respectively, calculated using the 25th and 75th percentile scores and the corresponding z-scores from a standard normal distribution.
Step-by-step explanation:
The mean (μ) and standard deviation (σ) of SAT scores for Penn State's incoming class can be found by using the given 25th and 75th percentile scores. These percentiles correspond to specific z-scores on the standard normal distribution.
Firstly, we need to find the z-scores that correspond to the given percentiles, which can be looked up in a z-table or calculated using statistical software.
However, since the z-scores for the percentiles were not provided in the question, we can approximate them: the 25th percentile is roughly -0.674 and the 75th percentile is roughly 0.674. Using the formulas μ + z * σ = X, where X is the SAT score, we can set up two equations:
Equation 1: μ - 0.674σ = 1240
Equation 2: μ + 0.674σ = 1410
By solving these equations simultaneously, we can find the actual values of μ and σ for the SAT scores.
Let's add both equations:
μ - 0.674σ + μ + 0.674σ = 1240 + 1410
2μ = 2650
μ = 1325
Now, substituting μ into one of the equations to find σ:
1325 - 0.674σ = 1240
0.674σ = 1325 - 1240
0.674σ = 85
σ = 85 / 0.674
σ ≈ 126.11
Therefore, for Penn State's incoming 2020 class, the estimated mean SAT score is 1325 and the estimated standard deviation is approximately 126.11.