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The weight of a small Starbucks coffee is a normally distributed random variable with a mean of 350 grams and a standard deviation of 11 grams. Find the weight that corresponds to each event. (Use Excel or Appendix C to calculate the z-value. Round your final answers to 2 decimal places.)
a. Highest 10 percent
b. Middle 50 percent to
c. Highest 80 percent
d. Lowest 10 percent
a. 364.10 grams
b.
c. 340.74 grams
d.335.90 grams
Step-by-step explanation:
a. Highest 10 percent
Highest 10% = 90th percentile
The z score for the 90th percentile = 1.282
Using the z score formula
The formula for calculating a z-score is is z = (x-μ)/σ
where x is the raw score
μ is the population mean
σ is the population standard deviation
z = (x-μ)/σ
Mean = 350 grams
Standard deviation = 11 grams
1.282 = x -350/11
1.282 × 11 = x - 350
x = 14.102 + 350
x = 364.102 grams
The weight that corresponds in 2 decimal places to the highest 10% ≈ 364.10grams
b. Middle 50 percent
Middle 50 percent is found between the 25th and 75th percentile
Z score for 25th percentile = -0.674
Using the z score formula
The formula for calculating a z-score is is z = (x-μ)/σ
where x is the raw score
μ is the population mean
σ is the population standard deviation
z = (x-μ)/σ
Mean = 350 grams
Standard deviation = 11 grams
-0.674 = x -350/11
-0.674 × 11 = x - 350
x = -7.414 + 350
x = 342.586 grams
Approximately in 2 decimal ≈ 342.59 grams
Z score for 75th percentile = 0.674.
Using the z score formula
The formula for calculating a z-score is is z = (x-μ)/σ
where x is the raw score
μ is the population mean
σ is the population standard deviation
z = (x-μ)/σ
Mean = 350 grams
Standard deviation = 11 grams
0.674 = x - 350/11
0.674 × 11 = x - 350
7.414 = x - 350
x = 350 + 7.414
= 357.414 grams
Approximately in 2 decimal places is 357.41 grams
Therefore, the weight that corresponds to the middle 50 percent is between
342.59 grams to 357.41 grams
c. Highest 80 percent
Highest 80 percent = 20th percentile
Z score for 20th percentile = -0.842
Using the z score formula
The formula for calculating a z-score is is z = (x-μ)/σ
where x is the raw score
μ is the population mean
σ is the population standard deviation
z = (x-μ)/σ
Mean = 350 grams
Standard deviation = 11 grams
-0.842 = x -350/11
-0.842 × 11 = x - 350
x = -9.262 + 350
x = 340.738 grams
The weight that corresponds in 2 decimal places to the highest 80% ≈ 340.74 grams
d. Lowest 10 percent = 10th percentile
Z score for 10th percentile = -1.282
Using the z score formula
The formula for calculating a z-score is is z = (x-μ)/σ
where x is the raw score
μ is the population mean
σ is the population standard deviation
z = (x-μ)/σ
Mean = 350 grams
Standard deviation = 11 grams
-1.282 = x -350/11
-1.282 × 11 = x - 350
x = -14.102 + 350
x = 335.898 grams
The weight that corresponds in 2 decimal places to the lowest 10% ≈ 335.90 grams
