A survey of college students was conducted during final exam week to assess the number of cups of coffee consumed each day. The mean number of cups was 5 with a standard deviation of 1.5 cups. The distribution was normal.

Question

a. What proportion of students drank 7 or more cups of coffee per day

b. What proportion of students drank 2 or more cups per day

c. What proportion of students drank between 2 and 7 cups per day

d. How many cups of coffee would an individual at the 60th percentile rank drink?

e. What is the percentile rank for an individual who drinks 4 cups of coffee per day

f. What is the percentile rank for an individual who drinks 7.5 cups of coffee a day?

Answer 1

a. To determine the proportion of students who drank 7 or more cups of coffee per day, we first calculate the z-score for 7 cups of coffee:

z = (X - μ) / σ
z = (7 - 5) / 1.5
z = 1.33

Using a z-table, we find that the proportion of students who drank 7 or more cups of coffee per day is 0.092 (1 - 0.908).

b. To determine the proportion of students who drank 2 or more cups of coffee per day, we calculate the z-score for 2 cups of coffee:

z = (X - μ) / σ
z = (2 - 5) / 1.5
z = -2

Using a z-table, we find that the proportion of students who drank 2 or more cups of coffee per day is 0.977 (1 - 0.023).

c. To find the proportion of students who drank between 2 and 7 cups of coffee per day, we can subtract the proportion of students who drank less than 2 cups of coffee from the proportion of students who drank less than 7 cups of coffee:

Proportion drinking between 2 and 7 cups = P(X < 7) - P(X < 2)
Proportion drinking between 2 and 7 cups = 0.908 - 0.023
Proportion drinking between 2 and 7 cups = 0.885

d. To find out how many cups of coffee an individual at the 60th percentile rank drinks, we first find the corresponding z-score for the 60th percentile, which is about 0.25 (from a z-table).

Next, we convert this z-score back into cups of coffee:

X = μ + (z * σ)
X = 5 + (0.25 * 1.5)
X = 5.375

An individual at the 60th percentile rank drinks about 5.375 cups of coffee per day.

e. To find the percentile rank for an individual who drinks 4 cups of coffee per day, we first calculate the z-score for 4 cups of coffee:

z = (X - μ) / σ
z = (4 - 5) / 1.5
z = -0.67

Using a z-table, we find that the percentile rank for an individual who drinks 4 cups of coffee per day is approximately 25.1.

f. To find the percentile rank for an individual who drinks 7.5 cups of coffee per day, we first calculate the z-score for 7.5 cups of coffee:

z = (X - μ) / σ
z = (7.5 - 5) / 1.5
z = 1.67

Using a z-table, we find that the percentile rank for an individual who drinks 7.5 cups of coffee per day is approximately 95.2.