The calculated first quartile is approximately 68.255, which differs from Flavia's expected value of 65. This suggests that the sample's first quartile score is higher than the expected value.
How to solve
Given:
Mean (μ) = 75
Standard Deviation (σ) = 10
First Quartile = 65
In a normal distribution:
The first quartile (Q1) corresponds to the z-score -0.6745.
The formula for obtaining a value from a z-score in a normal distribution is:
Value = Mean + (Z-score * Standard Deviation)
Calculating the first quartile:
Z-score for the first quartile = -0.6745
First Quartile = Mean + (Z-score * Standard Deviation)
First Quartile = 75 + (-0.6745 * 10)
First Quartile = 75 - 6.745
First Quartile = 68.255
Therefore, the calculated first quartile is approximately 68.255, which differs from Flavia's expected value of 65. This suggests that the sample's first quartile score is higher than the expected value.
The Complete Question
Flavia was curious if a sample of test scores with a mean of 75 and a standard deviation of 10 had a first quartile score of 65. Calculate the first quartile for this sample and determine if it aligns with Flavia's expectation. Show the step-by-step calculation process, and explain how the first quartile is obtained from the mean and standard deviation in a normally distributed set of scores