Faculty in the psychology department at State University consume an average of 5 cups of coffee per day with a standard deviation of 1.5. The distribution is normal. What proportion of faculty consumes an amount between 4 and 6 cups?Choose one0.54680.49720.50none of these
Question
Faculty in the psychology department at State University consume an average of 5 cups of coffee per day with a standard deviation of 1.5. The distribution is normal. What proportion of faculty consumes an amount between 4 and 6 cups?Choose one0.54680.49720.50none of these
Solution
To solve this problem, we need to convert the coffee consumption amounts to z-scores. The z-score is a measure of how many standard deviations an element is from the mean.
Step 1: Calculate the z-scores for 4 and 6 cups of coffee.
The formula for calculating a z-score is:
Z = (X - μ) / σ
where: X is the value we are comparing to the mean, μ is the mean, and σ is the standard deviation.
For 4 cups of coffee:
Z = (4 - 5) / 1.5 = -0.67 (approximately)
For 6 cups of coffee:
Z = (6 - 5) / 1.5 = 0.67 (approximately)
Step 2: Look up these z-scores in a standard normal distribution table or use a calculator that can calculate the area under the standard normal curve.
The table or calculator will give the proportion of data below the given z-score.
For Z = -0.67, the proportion is approximately 0.2514. For Z = 0.67, the proportion is approximately 0.7486.
Step 3: Subtract the proportion for 4 cups from the proportion for 6 cups.
0.7486 - 0.2514 = 0.4972
So, the proportion of faculty that consumes between 4 and 6 cups of coffee per day is approximately 0.4972, or 49.72%.
So, the answer is 0.4972.
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