A new national survey found that Americans drink an average of 6 glasses of water daily, instead of the recommended 8. Assuming that the number of glasses of water is approximately normally distributed with a standard deviation of 1.7 glasses, what proportion of Americans drink more than the recommended amount?
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 amount of water, we need to calculate the area under the curve for the normal distribution to the right of 8 glasses.
First, let's calculate the z-score, which measures how many standard deviations away from the mean (average) a given value is.
The formula to calculate the z-score is:
z = (x - μ) / σ
Where:
x = value (in this case, 8 glasses)
μ = mean (average number of glasses, given as 6)
σ = standard deviation (given as 1.7 glasses)
Plugging in the values:
z = (8 - 6) / 1.7
z = 1.18
Now, we need to find the proportion of the normal distribution to the right of the z-score of 1.18. We can use a standard normal distribution table or a statistical software to find this proportion. For simplicity, let's assume we are using a standard normal distribution table.
Looking up the z-score of 1.18 in the table, we can find the corresponding area to the left of this z-score. Subtracting this area from 1 will give us the area to the right of the z-score.
Without an actual table, we can estimate the proportion using a calculator like the one provided by many statistics websites or statistical software. Entering the z-score of 1.18 and selecting the option to calculate the area to the right, it should give us the proportion.
Using a standard normal distribution table or a calculator, the proportion of Americans who drink more than the recommended amount of water can be approximately found as a decimal or percentage.