A certain test is designed to measure the satisfaction of an individual with his/her relationship. Suppose that the scores on this test are approximately normally distributed with a mean of 55 and a standard deviation of 8. An individual with a score of 45 or less is considered dissatisfied with his/her relationship. According to this criterion, what proportion of people in relationships are dissatisfied? Round your answer to at least four decimal places.
Question
A certain test is designed to measure the satisfaction of an individual with his/her relationship. Suppose that the scores on this test are approximately normally distributed with a mean of 55 and a standard deviation of 8. An individual with a score of 45 or less is considered dissatisfied with his/her relationship. According to this criterion, what proportion of people in relationships are dissatisfied? Round your answer to at least four decimal places.
Solution
To solve this problem, we need to use the concept of z-scores in statistics. A z-score measures how many standard deviations an element is from the mean.
Step 1: Calculate the z-score for a score of 45. The formula for the z-score is:
Z = (X - μ) / σ
where: X = individual score = 45 μ = mean = 55 σ = standard deviation = 8
So, Z = (45 - 55) / 8 = -1.25
Step 2: Look up the z-score in the standard normal distribution table or use a calculator with a normal distribution function. The value you get is the proportion of people who scored below 45.
The value for Z = -1.25 is approximately 0.1056.
So, about 10.56% of people in relationships are dissatisfied according to this criterion.
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