among coffee drinkers, men drink a mean of 3.2 cups per day with a standard deviation of 0.8 cups. Assume the number of coffee drinks per day follows a normal distribution.

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.

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.

To answer these questions, we need to calculate the z-scores and then use the z-table to find the corresponding proportions.

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 answer these questions, we need to convert the given information into a standard normal distribution, where the mean is 0 and the standard deviation is 1.

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.