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.
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?
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.
To solve these questions, we can use the Z-score formula and the standard normal distribution table. The formula for calculating the Z-score is:
Z = (X - μ) / σ
Where:
- X is the value we are interested in (number of cups of coffee),
- μ is the mean (5 cups),
- σ is the standard deviation (1.5 cups).
Let's calculate the answers step-by-step:
a. Proportion of students who drank 7 or more cups of coffee per day:
To find this proportion, we need to find the area under the normal curve to the right of 7 cups. We calculate the Z-score for 7 cups:
Z = (7 - 5) / 1.5 = 1.33
From the standard normal distribution table, we find that the area to the right of 1.33 is approximately 0.0918.
Therefore, the proportion of students who drank 7 or more cups of coffee per day is approximately 0.0918.
b. Proportion of students who drank 2 or more cups per day:
To find this proportion, we need to find the area under the normal curve to the right of 2 cups. We calculate the Z-score for 2 cups:
Z = (2 - 5) / 1.5 = -2
From the standard normal distribution table, we find that the area to the right of -2 is approximately 0.9772.
Therefore, the proportion of students who drank 2 or more cups of coffee per day is approximately 0.9772.
c. Proportion of students who drank between 2 and 7 cups per day:
To find this proportion, we need to find the area under the normal curve between 2 and 7 cups. We calculate the Z-scores for 2 cups and 7 cups:
Z1 = (2 - 5) / 1.5 = -2
Z2 = (7 - 5) / 1.5 = 1.33
From the standard normal distribution table, we find the area to the right of -2 (0.9772) and subtract the area to the right of 1.33 (0.0918):
Proportion = 0.9772 - 0.0918 = 0.8854
Therefore, the proportion of students who drank between 2 and 7 cups of coffee per day is approximately 0.8854.
d. Cups of coffee for an individual at the 60th percentile rank:
To find this value, we need to find the Z-score that corresponds to the 60th percentile. From the standard normal distribution table, we find the Z-score that corresponds to a cumulative area of 0.60, which is approximately 0.25.
We use this Z-score to find the corresponding X-value:
Z = (X - μ) / σ
0.25 = (X - 5) / 1.5
X - 5 = 0.25 * 1.5
X - 5 = 0.375
X = 5 + 0.375
X ≈ 5.375
Therefore, an individual at the 60th percentile rank would drink approximately 5.375 cups of coffee.
e. Percentile rank for an individual who drinks 4 cups of coffee per day:
We calculate the Z-score for 4 cups:
Z = (4 - 5) / 1.5 = -0.67
From the standard normal distribution table, we find that the area to the left of -0.67 is approximately 0.2514.
Therefore, the percentile rank for an individual who drinks 4 cups of coffee per day is approximately 25.14%.
f. Percentile rank for an individual who drinks 7.5 cups of coffee per day:
We calculate the Z-score for 7.5 cups:
Z = (7.5 - 5) / 1.5 = 1.67
From the standard normal distribution table, we find that the area to the left of 1.67 is approximately 0.9525.
Therefore, the percentile rank for an individual who drinks 7.5 cups of coffee per day is approximately 95.25%.
To solve these questions, we will use the concept of the standard normal distribution.
Before we begin, it's important to note that the mean and standard deviation provided in the question are for the number of cups of coffee consumed per day.
The standard normal distribution has a mean of 0 and a standard deviation of 1. In order to use the standard normal distribution for our calculations, we need to standardize the data using the following formula:
Standardized value (z) = (X - mean) / standard deviation
where X is the value we are interested in, mean is the mean of the distribution, and standard deviation is the standard deviation of the distribution.
a. To find the proportion of students who drank 7 or more cups of coffee per day, we need to find the area under the standard normal distribution curve to the right of the standardized value for 7 cups.
Standardized value for 7 cups:
z = (7 - 5) / 1.5
z = 1.33
Using a standard normal distribution table or a calculator, we can find the area to the right of z = 1.33. This will give us the proportion of students who drank 7 or more cups of coffee per day.
b. To find the proportion of students who drank 2 or more cups per day, we need to find the area under the standard normal distribution curve to the right of the standardized value for 2 cups.
Standardized value for 2 cups:
z = (2 - 5) / 1.5
z = -2
Using a standard normal distribution table or a calculator, we can find the area to the right of z = -2. This will give us the proportion of students who drank 2 or more cups of coffee per day.
c. To find the proportion of students who drank between 2 and 7 cups per day, we need to find the area under the standard normal distribution curve between the standardized values for 2 cups and 7 cups.
Standardized value for 2 cups:
z = (2 - 5) / 1.5
z = -2
Standardized value for 7 cups:
z = (7 - 5) / 1.5
z = 1.33
Using a standard normal distribution table or a calculator, we can find the area between z = -2 and z = 1.33. This will give us the proportion of students who drank between 2 and 7 cups of coffee per day.
d. To find the number of cups of coffee an individual at the 60th percentile rank would drink, we need to find the standardized value (z) corresponding to the 60th percentile.
Using a standard normal distribution table or a calculator, we can find the z-value that corresponds to the 60th percentile. Then, we can use the formula to find the number of cups:
Number of cups = (z * standard deviation) + mean
e. To find the percentile rank for an individual who drinks 4 cups of coffee per day, we need to find the area under the standard normal distribution curve to the left of the standardized value for 4 cups.
Standardized value for 4 cups:
z = (4 - 5) / 1.5
z = -0.67
Using a standard normal distribution table or a calculator, we can find the area to the left of z = -0.67. This will give us the percentile rank for an individual who drinks 4 cups of coffee per day.
f. To find the percentile rank for an individual who drinks 7.5 cups of coffee per day, we need to find the area under the standard normal distribution curve to the left of the standardized value for 7.5 cups.
Standardized value for 7.5 cups:
z = (7.5 - 5) / 1.5
z = 1.67
Using a standard normal distribution table or a calculator, we can find the area to the left of z = 1.67. This will give us the percentile rank for an individual who drinks 7.5 cups of coffee per day.