Question

A psychological introvert-extrovert test produced scores that had a normal distribution with a mean and standard deviation of 75 and \(12,\) respectively. If we wish to designate the highest \(15 \%\) as extroverts, what would be the proper score to choose as the cutoff point?

Short answer

Answer: 87 or 88

Step-by-step solution

Understand the information provided

We are given the mean (\(\mu\)) and standard deviation (\(\sigma\)) of the normal distribution. These values are \(\mu = 75\) and \(\sigma = 12\). We also know that the highest 15% of extroverts are to be identified, that is, the cutoff point for the 85th percentile (\(100 - 15 = 85\)).

Calculate the z-score for the 85th percentile

In order to find the desired z-score, we need to look up the z-score table or use an online calculator. Using the z-score table, we search for the z-score corresponding to the given percentile (85%). For this percentile, we find a z-score of approximately 1.04.

Calculate the score corresponding to the z-score

The z-score is used to measure the distance between a specific value, x, and the mean, in terms of standard deviations. To calculate the corresponding score (x) for the z-score (1.04), we can use the z-score formula: $$ x = \mu + z \cdot \sigma $$ Plug in the given values: $$ x = 75 + 1.04 \cdot 12 $$

Compute the cutoff score

Calculate the value of x based on the values entered in the equation: $$ x = 75 + 1.04 \cdot 12 \approx 87.48 $$ Since a psychological score usually is given as a whole number, we can round this to either 87 or 88. This means that a score of 87 or 88 would be the cutoff score for designating the top 15% as extroverts.