Kane is preparing for a marathon and initially runs 3 miles each training session, planning to extend his distance by a quarter mile each week. Denote w as the number of weeks; construct an expression that reflects the distance Kane covers during a training session after w weeks. The solution is given by: (11 + w) / 4. Explanation: He starts with 3 miles per session and adds 1/4 mile weekly. Thus, after w weeks, the distance becomes: = 3 + (w - 1) × 1/4 = 3 + (w/4 - 1/4) = (12 + w - 1) / 4 = (11 + w) / 4.
Customer arrivals at a fast-food outlet conform to a Poisson distribution with an average rate of 16 customers per hour. In statistical probability analysis, the Poisson distribution is a commonly utilized discrete probability distribution. Employing the formula, it has been calculated that 0.0661 represents the probability of there being precisely 12 arrivals in the next hour.
To determine the perimeter, we sum the lengths of all sides.
Looking at the diagram, we find the perimeter is calculated as follows:
P = 11 + (x - 2) + (11 - 3) + [(x - 2) - (x - 11)] + (x - 11)
= 11 + x - 2 + 8 + 9 + x - 11
= 2x + 15.
Final expression: 2x + 15
The risk of down syndrome, in terms of the percentage of births per year, is changing at a rate given by the equation r(x) = 0.004641x² - 0.3012x + 4.9 for the range 20 ≤ x ≤ 45, where x signifies the maternal age at delivery. To derive the risk function as a percentage of births relative to maternal age x, we integrate r(x), leading to the function f(x) = 0.001547x³ - 0.1506x² + 4.9x + c. When x is 30, f evaluates to 0.14%. This means that 0.001547(30³) - 0.1506(30²) + 4.9(30) + c equals 0.14. Solving gives 41.769 - 135.54 + 147 + c = 0.14, which simplifies to c = -53.089. As a result, we establish that f(x) = 0.001547x³ - 0.1506x² + 4.9x - 53.089 for 20 ≤ x ≤ 45. The graph corresponding to this function is illustrated below.
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.
