Final answer:
1. Approximately 95.4% of the K-Cups will fall within two standard deviations of the mean. 2. The probability that a K-Cup will weigh more than 0.33 oz is approximately 15.87%. 3. Approximately 2.28% of the K-Cups are considered weak (more than two standard deviations below the mean). 4. Approximately 81.86% of the K-Cups will weigh between 0.23 and 0.38 oz in a standard box of 44 K-Cups. 5. All of the K-Cups in a box of 128 will weigh less than 23 oz. 6. Approximately 95.4% of the K-Cups in a box of 128 from Sam's Club will be considered ideal.
Step-by-step explanation:
1. To determine the percentage of K-Cups that can be expected to fall within two standard deviations of the mean, we need to calculate the z-score. The formula for the z-score is:
z = (X - μ) / σ
where X is the value, μ is the mean, and σ is the standard deviation. In this case, the mean is 0.28 oz and the standard deviation is 0.05 oz.
Using the z-table or a calculator, we can find that the area under the normal distribution curve within two standard deviations of the mean is approximately 95.4%.
Therefore, we can expect that approximately 95.4% of the K-Cups will fall within two standard deviations of the mean.
2. To calculate the probability that a K-Cup will have more than 0.33 oz, we need to calculate the z-score for 0.33 oz. Using the formula mentioned above, the z-score is:
z = (0.33 - 0.28) / 0.05 = 1
Using the z-table or a calculator, we can find that the area under the normal distribution curve to the right of 1 is approximately 0.1587. Therefore, the probability that a K-Cup will have more than 0.33 oz is approximately 0.1587 or 15.87%.
3. To find the percentage of K-Cups that are considered weak (more than two standard deviations below the mean), we need to calculate the z-score for two standard deviations below the mean. The z-score is:
z = -2
Using the z-table or a calculator, we can find that the area under the normal distribution curve to the left of -2 is approximately 0.0228. Therefore, the percentage of K-Cups that are considered weak is approximately 0.0228 or 2.28%.
4. To find the number of K-Cups that will weigh between 0.23 and 0.38 oz in a standard box of 44 K-Cups, we need to calculate the z-scores for these values.
The z-score for 0.23 oz is:
z_1 = (0.23 - 0.28) / 0.05 = -1
The z-score for 0.38 oz is:
z_2 = (0.38 - 0.28) / 0.05 = 2
Using the z-table or a calculator, we can find that the area under the normal distribution curve between -1 and 2 is approximately 0.8186. Therefore, approximately 81.86% of the K-Cups will weigh between 0.23 and 0.38 oz in a standard box of 44 K-Cups.
5. To find the number of K-Cups in a box of 128 that will weigh less than 23 oz, we need to calculate the z-score for 23 oz. The z-score is:
z = (23 - 0.28) / 0.05 = 457.44
The area under the normal distribution curve to the left of 457.44 is extremely close to 1, which means that all of the K-Cups in a box of 128 will weigh less than 23 oz.
6. To find the number of K-Cups that are considered ideal (within one standard deviation of the mean) in a box of 128 from Sam's Club, we can use the percentage we calculated in question 1. Approximately 95.4% of the K-Cups will fall within one standard deviation of the mean. Therefore, approximately 95.4% of the K-Cups in a box of 128 from Sam's Club will be considered ideal.