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 _________

Answer:

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