Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the Z scores.
For c, work backwards from the Z score for that proportion.
a. What proportion drinks 2 cups per day or more?
b. What proportion drink no more than 4 cups per day?
c. If the top 5% of coffee drinkers are considered heavy coffee drinkers, what is the minimum number of cups consumed by a heavy coffee drinker? find the 95th percentile.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the Z scores.
For c, work backwards from the Z score for that proportion.
a. To find the proportion of coffee drinkers who drink 2 cups per day or more, we will calculate the z-score for 2 cups and find the proportion to the right of that value.
Z-score formula: Z = (x - μ) / σ
Where:
x = value
μ = mean
σ = standard deviation
Z = (2 - 3.2) / 0.8
Z = -1.5
Now, let's find the proportion.
Using the z-table, the proportion to the right of -1.5 is approximately 0.9332.
Therefore, approximately 93.32% of coffee drinkers drink 2 cups per day or more.
b. To find the proportion of coffee drinkers who drink no more than 4 cups per day, we will calculate the z-score for 4 cups and find the proportion to the left of that value.
Z = (4 - 3.2) / 0.8
Z = 1
Using the z-table, the proportion to the left of 1 is approximately 0.8413.
Therefore, approximately 84.13% of coffee drinkers drink no more than 4 cups per day.
c. To find the minimum number of cups consumed by a heavy coffee drinker (the 95th percentile), we need to find the z-score that corresponds to the proportion of 0.95.
Using the z-table, we find that the closest z-score to 0.95 is approximately 1.645.
Now, let's find the minimum number of cups:
Z = (x - 3.2) / 0.8
1.645 = (x - 3.2) / 0.8
1.316 = x - 3.2
x = 4.516
Therefore, the minimum number of cups consumed by a heavy coffee drinker (the 95th percentile) is approximately 4.516 cups per day.
To do this, we will use the z-score formula:
z = (x - μ) / σ
where x is the value we want to convert, μ is the mean, and σ is the standard deviation.
a. To find the proportion of coffee drinkers who drink 2 cups per day or more, we need to find the area under the normal curve to the right of 2 cups.
First, we calculate the z-score for 2 cups:
z = (2 - 3.2) / 0.8 = -1.5
Next, we look up the area to the right of -1.5 in the standard normal distribution table (or use a calculator). The value we find is 0.9332.
Since we want the area to the right of -1.5, we subtract this value from 1:
Proportion = 1 - 0.9332 = 0.0668 or 6.68% of coffee drinkers drink 2 cups per day or more.
b. To find the proportion of coffee drinkers who drink no more than 4 cups per day, we need to find the area under the normal curve to the left of or up to 4 cups.
First, we calculate the z-score for 4 cups:
z = (4 - 3.2) / 0.8 = 1
Next, we look up the area to the left of 1 in the standard normal distribution table (or use a calculator). The value we find is 0.8413.
Proportion = 0.8413 or 84.13% of coffee drinkers drink no more than 4 cups per day.
c. To find the minimum number of cups consumed by a heavy coffee drinker, we need to find the value that corresponds to the 95th percentile of the standard normal distribution.
The 95th percentile corresponds to a z-score that leaves 5% of the area to the right.
We look up the z-score corresponding to a cumulative probability of 0.95 in the standard normal distribution table (or use a calculator). The z-score we find is approximately 1.645.
Next, we use the z-score formula to calculate the minimum number of cups:
x = μ + (z * σ)
x = 3.2 + (1.645 * 0.8)
x ≈ 4.916
Therefore, the minimum number of cups consumed by a heavy coffee drinker (95th percentile) is approximately 4.916 cups.