Students in the psychology department consume an average of 5 cups of coffee per day with a standard deviation of 1.75 cups. The number of cups of coffee consumed is normally distributed.
• What proportion of students consume an amount equal to or less than 6 cups a day?
• How many cups would an individual at the 80th percentile drink?
Use z-scores.
Formula:
z = (x - mean)/sd
In the first part, find z.
z = (6 - 5)/1.75
Calculate, then use a z-table to find the proportion.
In the second part, find z using a z-table for the 80th percentile. Plug into the formula, along with mean and standard deviation. Solve for x.
I hope this will help get you started.
z = (6 - 5)/1.75 = 0.57
.2157+.50= 0.7157x100=71.57%
i.e 71.57%
In the second part, find z using a z-table for the 80th percentile. Plug into the formula, along with mean and standard deviation. Solve for x.
z = (x - mean)/sd
z (sd)= X-mean
z (sd)+mean= X
0.20= (X-5)/0.57
(0.20 x 0.57)+ 5= 5.11
b)= 5.11?
The closest in the table for 80th percentile is z=0.20 or 0.21 if I am looking at the right place? Is the above correct by any chance?
Thanks
Well, that depends on how much coffee you've had today. Ready for some caffeinated humor?
• To find the proportion of students who consume an amount equal to or less than 6 cups a day, we can use a z-score and the standard normal distribution. By calculating the z-score for 6 cups using the formula (𝑥 – 𝜇) / 𝜎, where 𝑥 is the value, 𝜇 is the mean, and 𝜎 is the standard deviation, we can then look up the proportion from the z-score table.
• Now, for the 80th percentile, we need to find the z-score that corresponds to that percentile. Then, we can use the same formula to calculate the value of cups of coffee from the z-score.
But hey, you know what they say, "A cup of coffee shared is a cup of happiness doubled." So, let's calculate these and spread some coffee-induced joy!
To answer these questions, we can use the z-score formula and standard normal distribution.
1. Proportion of students consuming 6 cups or less per day:
- We need to find the z-score of 6 cups using the formula: z = (x - μ) / σ
- where x = 6 cups, μ = mean = 5 cups, σ = standard deviation = 1.75 cups
- z = (6 - 5) / 1.75 = 0.5714
- Now we can find the proportion by using the z-score in the standard normal distribution table or calculator.
- The proportion is the area to the left of the z-score of 0.5714 in the standard normal distribution. Let's assume this proportion is represented by P(z ≤ 0.5714).
2. Cups consumed by an individual at the 80th percentile:
- We need to find the z-score that corresponds to the 80th percentile.
- The 80th percentile is equivalent to a z-score of 0.84 (based on standard normal distribution table or calculator).
- We can use the z-score formula to calculate the value at the 80th percentile: x = μ + (z * σ)
- where μ = mean = 5 cups, σ = standard deviation = 1.75 cups, z = 0.84
- x = 5 + (0.84 * 1.75)
Now, let's calculate the answers.
1. Proportion of students consuming 6 cups or less per day:
- Using a standard normal distribution table or calculator, find the area to the left of z = 0.5714.
- The proportion P(z ≤ 0.5714) is the answer.
2. Cups consumed by an individual at the 80th percentile:
- Calculate x = 5 + (0.84 * 1.75) to determine the number of cups an individual at the 80th percentile would drink.
To answer these questions, we need to use the concept of the standard normal distribution, also known as the Z-distribution. This distribution allows us to find proportions or percentiles based on the mean and standard deviation of a normally distributed variable.
First, we need to calculate the Z-score for each question. The Z-score measures the number of standard deviations a particular value is away from the mean.
1. Proportion of students consuming 6 cups or less per day:
To find the proportion of students consuming 6 cups or less, we need to calculate the Z-score for 6 cups and then look up the corresponding proportion in the Z-table.
The formula to calculate Z-score is: Z = (X - μ) / σ
Where:
- X is the value we are calculating the Z-score for (6 cups)
- μ is the mean (average) of consumption in cups per day (5 cups)
- σ is the standard deviation of consumption in cups per day (1.75 cups)
Plugging in the values, we get:
Z = (6 - 5) / 1.75 = 0.571
Looking up the Z-score of 0.571 in the Z-table, we find the proportion corresponding to it. In this case, it is approximately 0.714.
Therefore, approximately 71.4% of students consume an amount equal to or less than 6 cups a day.
2. Cups consumed by an individual at the 80th percentile:
To find the cups an individual at the 80th percentile would drink, we need to find the Z-score that corresponds to the 80th percentile and then convert it back to the original value using the formula:
X = Z * σ + μ
In this case, we want to find the cups corresponding to the Z-score that corresponds to the 80th percentile. We refer to the Z-table to find this Z-score.
The Z-score for the 80th percentile is approximately 0.8416. Plugging this value into the formula:
X = 0.8416 * 1.75 + 5 = 6.47
Therefore, an individual at the 80th percentile would drink approximately 6.47 cups of coffee per day.
By using the Z-score and the Z-table, we can find the proportions and percentiles in a standard normal distribution.