Response:
PQ's slope is 0
MN's slope equals infinity
The lines PQ and MN are perpendicular to one another
Detailed explanation:
For two points in the coordinate plane, denoted as (x1, y1) and (x2, y2), the slope is determined as follows:
If a line has a slope of zero, it runs parallel to the X-axis and stands perpendicular to the Y-axis
If a line's slope is infinite, it is parallel to the Y-axis and perpendicular to the X-axis
Moreover, it is established that X and Y are perpendicular to each other.
As PQ's slope is zero, it runs parallel to the X-axis and perpendicular to the Y-axis
With MN having an infinite slope, it runs parallel to the Y-axis and perpendicular to the X-axis.
Therefore, lines PQ and MN are indeed perpendicular.
Let P(3) denote the probability of landing on 3 in a spin, and P(5) denote the probability of landing on 5. In probability terms, "AND" signifies multiplication while "OR" indicates addition. We aim to find the probability that the first number is "3" AND the second number is "5." Thus, we identify the individual probabilities and MULTIPLY them. The spinner has numbers ranging from 1 to 8, each appearing once. Therefore, since there is one instance of "3," we have P(3) = 1/8 and similarly P(5) = 1/8. Consequently, the overall probability of P(3 and 5) is 1/8 multiplied by 1/8, which equals 1/64.
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.
