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The weight of a small Starbucks coffee is a normally distributed random variable with a mean of 340 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 30 percent
b. Middle 70 percent to
c. Highest 90 percent
d. Lowest 20 percent

User Guglhupf
by
8.7k points

1 Answer

3 votes

Explanation:

To find the weight that corresponds to each event, we need to use the z-score formula and standard normal distribution tables. The z-score formula is given by:

z = (x - μ) / σ

Where:

z = z-score

x = observed value

μ = mean

σ = standard deviation

a. Highest 30 percent:

To find the weight corresponding to the highest 30 percent, we need to find the z-score that corresponds to a cumulative probability of 0.30 from the standard normal distribution table.

Using the z-table, we find that the z-score corresponding to a cumulative probability of 0.30 is approximately -0.52.

Now we can calculate the weight:

z = (x - μ) / σ

-0.52 = (x - 340) / 11

Solving for x:

x - 340 = -0.52 * 11

x - 340 = -5.72

x = 334.28

Therefore, the weight corresponding to the highest 30 percent is approximately 334.28 grams.

b. Middle 70 percent:

To find the weight corresponding to the middle 70 percent, we need to find the z-scores that correspond to the lower and upper percentiles. The lower percentile would be (100 - 70) / 2 = 15 percent, and the upper percentile would be 100 - 15 = 85 percent.

Using the z-table, we find that the z-score corresponding to a cumulative probability of 0.15 is approximately -1.04, and the z-score corresponding to a cumulative probability of 0.85 is approximately 1.04.

Now we can calculate the weights:

Lower weight:

z = (x - μ) / σ

-1.04 = (x - 340) / 11

x - 340 = -1.04 * 11

x - 340 = -11.44

x = 328.56

Upper weight:

z = (x - μ) / σ

1.04 = (x - 340) / 11

x - 340 = 1.04 * 11

x - 340 = 11.44

x = 351.44

Therefore, the weight corresponding to the middle 70 percent is approximately between 328.56 grams and 351.44 grams.

c. Highest 90 percent:

To find the weight corresponding to the highest 90 percent, we need to find the z-score that corresponds to a cumulative probability of 0.90 from the standard normal distribution table.

Using the z-table, we find that the z-score corresponding to a cumulative probability of 0.90 is approximately 1.28.

Now we can calculate the weight:

z = (x - μ) / σ

1.28 = (x - 340) / 11

Solving for x:

x - 340 = 1.28 * 11

x - 340 = 14.08

x = 354.08

Therefore, the weight corresponding to the highest 90 percent is approximately 354.08 grams.

d. Lowest 20 percent:

To find the weight corresponding to the lowest 20 percent, we need to find the z-score that corresponds to a cumulative probability of 0.20 from the standard normal distribution table.

Using the z-table, we find that the z-score corresponding to a cumulative probability of 0.20 is approximately -0.84.

Now we can calculate the weight:

z = (x - μ) / σ

-0.84 = (x - 340) / 11

Solving for x:

x - 340 = -0.84 *

11

x - 340 = -9.24

x = 330.76

Therefore, the weight corresponding to the lowest 20 percent is approximately 330.76 grams.

User Ajay Chaudhary
by
8.1k points