The Weight of a small Starbucks coffee is normally distributed random variable with a mean of 360 grams and a standard deviation of 9 grams. Find the weight that corresponds to each event.
a. highest 10%
b. highest 50%
c. highest 5%
D. highest 80%
e. lowest 10%
f. middle 50%
g. lowest 90%
h. middle 95%
I. highest 4%
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions/probabilities and their corresponding Z scores. Enter the relevant data in the above equation and calculate the score.
To find the weight that corresponds to each event, we need to use the Z-score formula and the standard normal distribution table.
The Z-score formula is given by:
Z = (X - μ) / σ
where:
Z is the standard score,
X is the observation (weight in this case),
μ is the mean of the distribution, and
σ is the standard deviation of the distribution.
Let's calculate the weight for each event:
a. highest 10%:
We need to find the weight that corresponds to the Z-score when the area under the curve to the right is 10%. Using the standard normal distribution table, we find that the Z-score for a right-tail area of 10% is approximately 1.28.
Z = 1.28
Now we can solve for X:
1.28 = (X - 360) / 9
1.28 * 9 = X - 360
11.52 = X - 360
X = 371.52 grams
Therefore, the weight that corresponds to the highest 10% is 371.52 grams.
b. highest 50%:
This means we need to find the weight that corresponds to the median of the distribution, which is the value that separates the highest 50% from the lowest 50%.
The median is equal to the mean of the distribution.
Therefore, the weight that corresponds to the highest 50% is 360 grams.
c. highest 5%:
We need to find the weight that corresponds to the Z-score when the area under the curve to the right is 5%. Using the standard normal distribution table, we find that the Z-score for a right-tail area of 5% is approximately 1.65.
Z = 1.65
Now we can solve for X:
1.65 = (X - 360) / 9
1.65 * 9 = X - 360
14.85 = X - 360
X = 374.85 grams
Therefore, the weight that corresponds to the highest 5% is 374.85 grams.
d. highest 80%:
This means we need to find the weight that corresponds to the Z-score when the area under the curve to the right is 20%. Using the standard normal distribution table, we find that the Z-score for a right-tail area of 20% is approximately 0.84.
Z = 0.84
Now we can solve for X:
0.84 = (X - 360) / 9
0.84 * 9 = X - 360
7.56 = X - 360
X = 367.56 grams
Therefore, the weight that corresponds to the highest 80% is 367.56 grams.
e. lowest 10%:
This means we need to find the weight that corresponds to the Z-score when the area under the curve to the left is 10%. Using the standard normal distribution table, we find that the Z-score for a left-tail area of 10% is approximately -1.28.
Z = -1.28
Now we can solve for X:
-1.28 = (X - 360) / 9
-1.28 * 9 = X - 360
-11.52 = X - 360
X = 348.48 grams
Therefore, the weight that corresponds to the lowest 10% is 348.48 grams.
f. middle 50%:
We already know this corresponds to the mean of the distribution, which is 360 grams.
g. lowest 90%:
This means we need to find the weight that corresponds to the Z-score when the area under the curve to the left is 90%. Using the standard normal distribution table, we find that the Z-score for a left-tail area of 90% is approximately -1.28.
Z = -1.28
Now we can solve for X:
-1.28 = (X - 360) / 9
-1.28 * 9 = X - 360
-11.52 = X - 360
X = 348.48 grams
Therefore, the weight that corresponds to the lowest 90% is 348.48 grams.
h. middle 95%:
For the middle 95%, we are looking for the range between the 2.5th percentile and the 97.5th percentile. These percentiles correspond to Z-scores of -1.96 and 1.96 respectively.
Z1 = -1.96
Z2 = 1.96
Now we can solve for X1 and X2:
-1.96 = (X1 - 360) / 9
-1.96 * 9 = X1 - 360
-17.64 = X1 - 360
X1 = 342.36 grams
1.96 = (X2 - 360) / 9
1.96 * 9 = X2 - 360
17.64 = X2 - 360
X2 = 377.64 grams
Therefore, the weight that corresponds to the middle 95% ranges between 342.36 grams and 377.64 grams.
i. highest 4%:
We need to find the weight that corresponds to the Z-score when the area under the curve to the right is 4%. Using the standard normal distribution table, we find that the Z-score for a right-tail area of 4% is approximately 1.75.
Z = 1.75
Now we can solve for X:
1.75 = (X - 360) / 9
1.75 * 9 = X - 360
15.75 = X - 360
X = 375.75 grams
Therefore, the weight that corresponds to the highest 4% is 375.75 grams.
To find the weights corresponding to each event, we need to use the standard normal distribution (also known as the Z-distribution) and convert the given probabilities to Z-scores.
The Z-score formula is given as: Z = (x - μ) / σ
Where,
Z is the Z-score,
x is the value we want to find the corresponding weight for,
μ is the mean of the distribution,
σ is the standard deviation of the distribution.
1. Highest 10% (a):
To find the weight that corresponds to the highest 10%, we need to find the Z-score that corresponds to a cumulative probability of 0.9 (1 - 0.1). Using a Z-table or calculator, we find the Z-score to be approximately 1.28.
Plug the values into the Z-score formula:
1.28 = (x - 360) / 9
Solving for x, we find:
x = 1.28 * 9 + 360 ≈ 371.52 grams
Therefore, the weight that corresponds to the highest 10% is approximately 371.52 grams.
2. Highest 50% (b):
Finding the weight that corresponds to the highest 50% means finding the Z-score that will give us a cumulative probability of 0.5. Using the Z-table or calculator, we find that a Z-score of 0 corresponds to a cumulative probability of 0.5.
Plug the values into the Z-score formula:
0 = (x - 360) / 9
Solving for x, we get:
x = 0 * 9 + 360 = 360 grams
Therefore, the weight that corresponds to the highest 50% is 360 grams.
3. Highest 5% (c):
To find the weight that corresponds to the highest 5%, we need to find the Z-score for a cumulative probability of 0.95 (1 - 0.05). Using the Z-table or calculator, we find the Z-score to be approximately 1.645.
Using the Z-score formula:
1.645 = (x - 360) / 9
Solving for x, we find:
x = 1.645 * 9 + 360 ≈ 375.81 grams
Therefore, the weight that corresponds to the highest 5% is approximately 375.81 grams.
4. Highest 80% (D):
Finding the weight that corresponds to the highest 80% means finding the Z-score that corresponds to a cumulative probability of 0.2 (1 - 0.8). Using the Z-table or calculator, we find the Z-score to be approximately -0.84 (the negative sign indicates that it's on the lower side of the mean).
Using the Z-score formula:
-0.84 = (x - 360) / 9
Solving for x, we get:
x = -0.84 * 9 + 360 ≈ 352.76 grams
Therefore, the weight that corresponds to the highest 80% is approximately 352.76 grams.
5. Lowest 10% (e):
To find the weight that corresponds to the lowest 10%, we can use the Z-score for the highest 10% and negate it. We determined earlier that the Z-score for the highest 10% is approximately 1.28.
Using the Z-score formula:
-1.28 = (x - 360) / 9
Solving for x, we find:
x = -1.28 * 9 + 360 ≈ 348.88 grams
Therefore, the weight that corresponds to the lowest 10% is approximately 348.88 grams.
6. Middle 50% (f):
The middle 50% corresponds to the range that falls between the 25th and 75th percentiles. Since the Z-score for the 25th percentile is -0.674 and the Z-score for the 75th percentile is 0.674 (using the Z-table or calculator), we can find the corresponding weights.
Using the Z-score formula for -0.674:
-0.674 = (x - 360) / 9
Solving for x, we find:
x = -0.674 * 9 + 360 ≈ 353.07 grams
Therefore, the weight that corresponds to the lower bound of the middle 50% is approximately 353.07 grams.
Using the Z-score formula for 0.674:
0.674 = (x - 360) / 9
Solving for x, we find:
x = 0.674 * 9 + 360 ≈ 366.07 grams
Therefore, the weight that corresponds to the upper bound of the middle 50% is approximately 366.07 grams.
7. Lowest 90% (g):
To find the weight that corresponds to the lowest 90%, we can use the Z-score for the highest 10% (1.28) and negate it.
Using the Z-score formula:
-1.28 = (x - 360) / 9
Solving for x, we find:
x = -1.28 * 9 + 360 ≈ 348.88 grams
Therefore, the weight that corresponds to the lowest 90% is approximately 348.88 grams.
8. Middle 95% (h):
To find the weight that corresponds to the middle 95%, we can use the Z-scores for the lowest 2.5% (-1.96) and the highest 2.5% (1.96) from the Z-table or calculator.
Using the Z-score formula for -1.96:
-1.96 = (x - 360) / 9
Solving for x, we find:
x = -1.96 * 9 + 360 ≈ 341.56 grams
Therefore, the weight that corresponds to the lower bound of the middle 95% is approximately 341.56 grams.
Using the Z-score formula for 1.96:
1.96 = (x - 360) / 9
Solving for x, we find:
x = 1.96 * 9 + 360 ≈ 378.24 grams
Therefore, the weight that corresponds to the upper bound of the middle 95% is approximately 378.24 grams.
9. Highest 4% (i):
To find the weight that corresponds to the highest 4%, we need to find the Z-score for a cumulative probability of 0.96 (1 - 0.04).
Using the Z-score formula:
Z = (x - μ) / σ
Z = (x - 360) / 9
0.96 = (x - 360) / 9
Solving for x, we find:
x = 0.96 * 9 + 360 ≈ 368.64 grams
Therefore, the weight that corresponds to the highest 4% is approximately 368.64 grams.
Note: When using Z-tables or calculators, the values may vary slightly depending on the level of precision used in the calculations.