Questions: The graph shown is based on more than 170,000 interviews done by Gallup that took place from January through December 2012. The sample consists of employed Americans 18 years of age or older. The Emotional Health Index Scores are the sample space. We randomly sample one Emotional Health Index Score. What is the probability that an Emotional Health Index Score is less than 80.2 given that it is already less than 81 ? (Round your answer to four decimal places.)

Solution Steps

The problem asks for the probability that an Emotional Health Index Score is less than 80.2 given that it is already less than 81.

From the graph, the scores less than 81 are:

• Service: 79.6
• Transportation: 80.0
• Manufacturing or production: 80.0
• Clerical or office: 80.1
• Installation and repair: 80.7

Count the number of scores less than 81:

• Total scores less than 81: 5

Count the number of scores less than 80.2:

• Service: 79.6
• Transportation: 80.0
• Manufacturing or production: 80.0
• Clerical or office: 80.1

Total scores less than 80.2: 4

The probability is the number of scores less than 80.2 divided by the number of scores less than 81: \[ P(X < 80.2 \mid X < 81) = \frac{\text{Number of scores less than 80.2}}{\text{Number of scores less than 81}} = \frac{4}{5} = 0.8 \]

Final Answer