The international bottled water association says that Americans on the average drink 4.6 (8oz) servings of water a day. Assume that the number of 8oz. servings of water is approximately normally distributed with a standard deviation of 1.8 servings what proportion of Americans drink more than the recommended 8 servings?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score.
To find the proportion of Americans who drink more than the recommended 8 servings, we need to calculate the z-score and find the corresponding area under the normal distribution curve.
The z-score formula is: z = (x - μ) / σ,
where x is the value we want to find the proportion for, μ is the mean, and σ is the standard deviation.
Given:
Mean (μ) = 4.6 servings/day
Standard deviation (σ) = 1.8 servings/day
To find the z-score for 8 servings:
z = (8 - 4.6) / 1.8
z = 3.4 / 1.8
z ≈ 1.89
Using a z-table or a calculator with a normal distribution function, we can find the proportion of values greater than 1.89.
The area to the right of a z-score of 1.89 corresponds to the proportion of Americans who drink more than the recommended 8 servings.
Looking up the z-score of 1.89 in a standard normal distribution table, we find that the area to the left is approximately 0.9706.
Since we want the area to the right, we subtract this value from 1:
1 - 0.9706 ≈ 0.0294
Therefore, approximately 0.0294 or 2.94% of Americans drink more than the recommended 8 servings of water per day.
To find the proportion of Americans who drink more than the recommended 8 servings of water, we need to calculate the area under the normal distribution curve to the right of 8 servings. Here's how to do it step by step:
Step 1: Convert the average number of servings to the standard normal distribution. We subtract the mean (4.6 servings) and divide it by the standard deviation (1.8 servings) to get the standard score (z-score).
z = (8 - 4.6) / 1.8
z = 3.4 / 1.8
z ≈ 1.89
Step 2: Use a z-table (also known as a standard normal distribution table) or a statistical software to find the proportion of the distribution to the right of the z-score calculated in step one. A z-table provides the area under the curve to the left of the z-score, so we need to subtract that value from 1 to get the area to the right.
Using a z-table, the area to the left of 1.89 is 0.9706.
The area to the right is 1 - 0.9706 = 0.0294.
So, approximately 2.94% of Americans drink more than the recommended 8 servings of water per day.