Answer :
By answering the presented question, we may conclude that As a result, equation the area above the x-axis and below the curve t2 + 4t + 2, y = e-t with 0 t 2 is roughly 18.49 square units.
What is equation?
A mathematical equation is a formula that connects two statements and denotes equivalence with the equals symbol (=). An equation is a mathematical statement that shows the equality of two mathematical expressions in algebra. In the equation 3x + 5 = 14, for example, the equal sign separates the variables 3x + 5 and 14. A mathematical formula describes the connection between the two sentences that occur on opposite sides of a letter. The symbol and the single variable are frequently the same. As in 2x - 4 Equals 2, for instance.
[tex]t = (-4 ± √(4^2 - 412))/2*1\st = (-4 + √12)/2\st = (-4 + 2√3)/2\\t = (-4 + 2√3)/2 = -2 + √3\\[/tex]
We can now integrate the difference between the curve and the x-axis as t increases from 0 to 2:
[tex]Area = ∫[0,2] (e^-t - (t^2 + 4t + 2)) dt[/tex]
For the first term, we may use integration by parts to get:
[tex]Area equals [-e-t - 1/2(t2 + 4t + 2)2 + 2(t2 + 4t + 2)]\\Area = (-e^-2 - 1/2(18 + 16√3)^2 + 2(18 + 16√3)) - (-1 - 2) (-1 - 2)\\Area = -e^-2 + 17 + 16√3 ≈ 18.49[/tex]
As a result, the area above the x-axis and below the curve t2 + 4t + 2, y = e-t with 0 t 2 is roughly 18.49 square units.
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All the conditions for inference are met, and we can proceed with constructing a confidence interval for the difference in proportions.
What is statistical inference?
The technique of assessing the outcome and drawing conclusions from data with random variation is known as statistical inference. Additionally known as inferential statistics. Applications of statistical inference include hypothesis testing and confidence intervals.
To check the conditions for inference, we need to verify the following:
1. Randomness condition: The problem states that both samples were selected randomly, so this condition is met.
2. Independence condition: Since each household is sampled independently from each other, this condition is met.
3. Large Counts Condition: We need to check if the sample sizes are large enough to use normal approximation. The expected counts for each category should be at least 10.
For households with school-aged children:
- Sample size: n1 = 40
- Proportion of households in the sample who responded "yes": p1 = 38/40 = 0.95
- Expected count of households who responded "yes": n1p1 = 40*0.95 = 38
For households without school-aged children:
- Sample size: n2 = 45
- Proportion of households in the sample who responded "yes": p2 = 25/45 = 0.56
- Expected count of households who responded "yes": n2p2 = 45*0.56 = 25.2
Since both expected counts are greater than 10, the Large Counts Condition is met.
Therefore, all the conditions for inference are met, and we can proceed with constructing a confidence interval for the difference in proportions.
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