In detail: Based on the central limit theorem, the distribution appears normal due to the large sample size. The confidence interval is presented in the format: (Sample mean - margin of error, sample mean + margin of error). The sample mean, denoted as x, serves as the point estimate for the population mean. The confidence interval is computed as: mean ± z × σ/√n, where σ represents the population standard deviation. The formula transforms into confidence interval = x ± z × σ/√n, with specific values: x = $75, σ = $24. To find the z score, we subtract the confidence level from 100% which gives α as 1 - 0.96 = 0.04; halving this results in α/2 = 0.02, signifying the tail areas. To ensure we account for the center area, we have 1 - 0.02 = 0.98, corresponding to a z score of 2.05 for the 96% confidence level. The confidence interval becomes 75 ± 2.05 × 24/√64 = 75 ± 2.05 × 3 = 75 ± 6.15. The lower limit is 75 - 6.15 = 68.85, while the upper limit stands at 75 + 6.15 = 81.15. For n = 400, with x = $75 and σ = $24, the z score remains 2.05, resulting in the confidence interval calculated as 75 ± 2.05 × 24/√400 = 75 ± 2.05 × 1.2 = 75 ± 2.46. Subsequently, the lower bound becomes 75 - 2.46 = 72.54, and the upper limit adds up to 75 + 2.46 = 77.46. Lastly, when n = 400, x = $200, and σ = $80, the z score tied to a 94% confidence level is 1.88. Thus, the confidence interval is expressed as 200 ± 1.88 × 80/√400 = 200 ± 1.88 × 4 = 200 ± 7.52, giving us a margin of error of 7.52.
Each 16-ounce serving contains 330 mg of caffeine. For 8-ounce servings: 330 mg divided by 2 equals 165 mg. The limit for caffeine intake is capped at 600 mg. Calculating for three servings: 165 mg * 3 = 495 mg, which is under the limit. Calculating for four servings: 165 mg * 4 = 660 mg, exceeding the maximum amount. Thus, a person can consume 3 full 8-ounce servings daily while remaining within the safety limit.
Given that the water in the tank doubles every minute, it implies that when the tank was full, it was half full just one minute prior. If it reaches fullness after 60 minutes, it was indeed double what it was a minute earlier. Hence, the opposite of doubling is halving. A minute before the tank filled completely, it was at half capacity. Therefore, it confirms that it was half full after 59 minutes.
