123k views
1 vote
The weight of a small Starbucks coffee is a normally distributed random variable with a mean of 415 grams and a standard deviation of 23 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. Lowest 15 percent
b. Highest 20 percent
c. Middle 60 percent
d. Highest 80 percent ​

1 Answer

3 votes

Answer:

To find the weight that corresponds to each event, we will use the concept of z-scores, which represent the number of standard deviations a particular value is away from the mean.

a. Lowest 15 percent:

To find the weight that corresponds to the lowest 15 percent, we need to find the z-score associated with that percentile. This z-score corresponds to the point where the cumulative probability is 0.15. Using a standard normal distribution table or functions in Excel, we can find the z-score to be approximately -1.04.

Next, we can use the formula for z-scores to find the corresponding weight:

z = (X - mean) / standard deviation

-1.04 = (X - 415) / 23

Solving for X, we get:

X = -1.04 * 23 + 415 ≈ 391.88 grams

Therefore, the weight corresponding to the lowest 15 percent is approximately 391.88 grams.

b. Highest 20 percent:

Similar to the previous case, we need to find the z-score associated with the highest 20 percent. This z-score corresponds to the point where the cumulative probability is 0.80 (1 - 0.20). Using a standard normal distribution table or functions in Excel, we find the z-score to be approximately 0.84.

Again, using the formula for z-scores:

0.84 = (X - 415) / 23

Solving for X, we get:

X = 0.84 * 23 + 415 ≈ 434.92 grams

Therefore, the weight corresponding to the highest 20 percent is approximately 434.92 grams.

c. Middle 60 percent:

The middle 60 percent corresponds to the range between the 20th and 80th percentiles. To find the z-scores associated with these percentiles, we can subtract and add half the range from the mean.

Lower z-score:

20th percentile = 0.20

Lower z-score = (1 - 0.60) / 2 = -0.20

Using the formula for z-scores:

-0.20 = (X - 415) / 23

Solving for X, we get:

X = -0.20 * 23 + 415 ≈ 410.60 grams

Upper z-score:

80th percentile = 0.80

Upper z-score = (1 + 0.60) / 2 = 0.80

Using the formula for z-scores:

0.80 = (X - 415) / 23

Solving for X, we get:

X = 0.80 * 23 + 415 ≈ 419.40 grams

Therefore, the weight corresponding to the middle 60 percent ranges approximately from 410.60 grams to 419.40 grams.

d. Highest 80 percent:

To find the weight that corresponds to the highest 80 percent, we can subtract half the remaining range from the mean.

Remaining range:

100% - 80% = 20%

Half of the remaining range:

20% / 2 = 10%

Lower z-score:

80th percentile = 0.80

Lower z-score = (1 - 0.10) = 0.90

Using the formula for z-scores:

0.90 = (X - 415) / 23

Solving for X, we get:

X = 0.90 * 23 + 415 ≈ 432.70 grams

Therefore, the weight corresponding to the highest 80 percent is approximately 432.70 grams.

In summary:

a. Lowest 15 percent: Approximately 391.88 grams.

b. Highest 20 percent: Approximately 434.92 grams.

c. Middle 60 percent: Approximately ranging from 410.60 grams to 419.40 grams.

d. Highest 80 percent: Approximately 432.70 grams.

User SteveR
by
7.6k points