Answer:
a) D.)H0: pF = pM versus Ha: pF ≠ pM
b) [[tex]z=\frac{0.537-0.384}{\sqrt{0.449(1-0.449)(\frac{1}{73}+\frac{1}{54})}}=1.7138[/tex]
c) [tex]p_v =2*P(Z>1.7138)=0.0866[/tex]
d) B.)The proportion of females that favor the war and the proportion of males that favor the war are not significantly different because the P-value is greater than 0.01.
e) [tex](0.537-0.384) - 2.58 \sqrt{\frac{0.537(1-0.537)}{54} +\frac{0.384(1-0.384)}{73}}=-0.0755[/tex]
f) [tex](0.537-0.384) + 2.58 \sqrt{\frac{0.537(1-0.537)}{54} +\frac{0.384(1-0.384)}{73}}=0.3815[/tex]
Step-by-step explanation:
1) Data given and notation
[tex]X_{M}=28[/tex] represent the number of men that favored war with Iraq
[tex]X_{W}=29[/tex] represent the number of women that favored war with Iraq
[tex]n_{M}=73[/tex] sample of male selected
[tex]n_{W}=54[/tex] sample of female selected
[tex]p_{M}=\frac{28}{73}=0.384[/tex] represent the proportion of men that favored war with Iraq
[tex]p_{W}=\frac{29}{54}=0.537[/tex] represent the proportion of women that favored war with Iraq
[tex]\alpha=0.01[/tex] represent the significance level
z would represent the statistic (variable of interest)
[tex]p_v[/tex] represent the value for the test (variable of interest)
Part a
We need to conduct a hypothesis in order to checkif the proportion of females that favored war with Iraq was significantly different from the proportion of males that favored war with Iraq , the system of hypothesis would be:
Null hypothesis:[tex]p_{M} = p_{W}[/tex]
Alternative hypothesis:[tex]p_{M} \new p_{W}[/tex]
The best option is:
D.)H0: pF = pM versus Ha: pF ≠ pM
Part b
We need to apply a z test to compare proportions, and the statistic is given by:
[tex]z=\frac{p_{W}-p_{M}}{\sqrt{\hat p (1-\hat p)(\frac{1}{n_{M}}+\frac{1}{n_{W}})}}[/tex] (1)
Where [tex]\hat p=\frac{X_{M}+X_{W}}{n_{M}+n_{W}}=\frac{28+29}{73+54}=0.449[/tex]
Calculate the statistic
Replacing in formula (1) the values obtained we got this:
[tex]z=\frac{0.537-0.384}{\sqrt{0.449(1-0.449)(\frac{1}{73}+\frac{1}{54})}}=1.7138[/tex]
Part c
We have a significance level provided [tex]\alpha=0.01[/tex], and now we can calculate the p value for this test.
Since is a one two sided test the p value would be:
[tex]p_v =2*P(Z>1.7138)=0.0866[/tex]
Part d
If we compare the p value and the significance level given [tex]\alpha=0.01[/tex] we see that [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and the best conclusion would be:
B.)The proportion of females that favor the war and the proportion of males that favor the war are not significantly different because the P-value is greater than 0.01.
Part e
The confidence interval for the difference of two proportions would be given by this formula
[tex](\hat p_W -\hat p_M) \pm z_{\alpha/2} \sqrt{\frac{\hat W_A(1-\hat p_W)}{n_W} +\frac{\hat p_M (1-\hat p_M)}{n_M}}[/tex]
For the 99% confidence interval the value of [tex]\alpha=1-0.99=0.01[/tex] and [tex]\alpha/2=0.005[/tex], with that value we can find the quantile required for the interval in the normal standard distribution.
[tex]z_{\alpha/2}=2.58[/tex]
And replacing into the confidence interval formula we got:
[tex](0.537-0.384) - 2.58 \sqrt{\frac{0.537(1-0.537)}{54} +\frac{0.384(1-0.384)}{73}}=-0.0755[/tex]
Part f
[tex](0.537-0.384) + 2.58 \sqrt{\frac{0.537(1-0.537)}{54} +\frac{0.384(1-0.384)}{73}}=0.3815[/tex]
The proper hypotheses are H0: pF = pM and Ha: pF ≠ pM. The test statistic is -0.087. The p-value is 0.1651.
Explanation:a) The proper hypotheses for this situation are:
H0: pF = pM
Ha: pF ≠ pM
Hence, the correct option is D.
b) The test statistic is computed by subtracting the male statistic from the female statistic:
Test statistic = pF - pM = 28/73 - 29/54 = -0.087
c) The p-value for the test is computed using the test statistic and the appropriate test statistic distribution. Based on the given values, the p-value is found to be 0.1651.
d) Using a 0.01 level of significance, the conclusion should be: The proportion of females that favor the war and the proportion of males that favor the war are not significantly different because the p-value is greater than 0.01. Hence, the correct option is D.
e) The lower endpoint of a 99% confidence interval for the difference between the proportion of females that favor the war and the proportion of males that favor the war is -0.2302.
f) The upper endpoint of a 99% confidence interval for the difference between the proportion of females that favor the war and the proportion of males that favor the war is 0.0566.
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Convert the integral ∫4√0∫x−xdydx to polar coordinates and evaluate it (use t for θ):
After converting the provided integral to polar coordinates, the value of integral is evaluated π/2.
What is polar form?When the Cartesian coordinates (x,y) are expressed in the polar coordinates (r, θ), then this form is called the polar form.
The given integral function in the problem is,
[tex]\int_0^{\sqrt{4}} \int\limits^x_{-x} dydx[/tex]
Let suppose, [tex]x=r\cos\theta[/tex] and [tex]y=r\sin\theta[/tex]. Thus,
[tex]\sin\theta=\dfrac{y}{r}\\\cos\theta=\dfrac{x}{r}[/tex]
Limits are y=x. From the trigonometry, the value of theta in the given triangle can be given as,
[tex]\dfrac{\sin\theta}{\cos\theta}=1\\\tan\theta=1\\\theta=\tan^{-1}1\\\theta=45^o\\\theta=\dfrac{\pi}{4}[/tex]
Similarly, for y=-x the value of angle,
[tex]\theta=-\dfrac{\pi}{4}[/tex]
Thus, the limits of theta are from -π/4 to π/4. From the Pythagoras theorem,
[tex]r^2=x^2+y^2\\r^2=(r\cos\theta)^2+(r\sin\theta)^2\\r^2=r^2(1)[/tex]
Thus, the limits of r is from 0 to 1. Convert the given integral in polar form as,
[tex]\int\limits^{\pi/4}_{-\pi/4} \int_0^{1} dt ds\\\int\limits^{\pi/4}_{-\pi/4} [1-0] dt \\\int\limits^{\pi/4}_{-\pi/4} dt \\\dfrac{\pi}{4}-\left(-\dfrac{\pi}{4} \right) \\\dfrac{\pi}{2} \\[/tex]
Hence, after converting the provided integral to polar coordinates, the value of integral is evaluated π/2.
Learn more about the polar form here;
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To convert the given Cartesian integral to polar coordinates, identify the bounds in polar terms, then rewrite the integral accordingly. After setting up the new limits for r and θ, use the relationship between Cartesian and polar coordinates to express the area element, and integrate step-by-step.
Explanation:To convert the integral ∫4√0∫x-xdydx to polar coordinates and evaluate it, we first need to describe the limits of integration and the region of integration in terms of polar coordinates (r, θ). The given integral ranges over a region bounded by the parabola y = √x and the x-axis from x=0 to x=4. Converted to polar coordinates, this region is bounded by the rays θ = 0 and θ = π/2 and the circles r = 0 and r = 4cos(θ).
So the double integral can be rewritten as ∫π/20∫4cos(θ)0 rdrdθ. To evaluate this integral, we integrate r from 0 to 4cos(θ), then integrate θ from 0 to π/2:
∫π/20 (∫4cos(θ)0 r dr) dθ = ∫π/20 [1/2 r^2]|^{4cos(θ)}_0 dθ = ∫π/20 8cos^2(θ) dθ
Using the double angle formula, cos^2(θ) = (1+cos(2θ))/2, the integral becomes:
8 ∫π/20 (1+cos(2θ))/2 dθ = 4 ∫π/20 (1+cos(2θ)) dθ
This can now be integrated directly to get the final result.
Find a formula for the described function.
A rectangle has perimeter 10 m. Express the area A of the rectangle as a function of the length, L, of one of its sides.
Answer:
The area of the rectangle is [tex]A=5L-L^2[/tex].
Step-by-step explanation:
The perimeter of a rectangle is
[tex]P=2(L+W)[/tex]
where, L is length and W is width.
It is given that perimeter of a rectangle is 10m.
[tex]10=2(L+W)[/tex]
Divide both sides by 2.
[tex]5=L+W[/tex]
Subtract L from both sides.
[tex]5-L=W[/tex]
Area of a rectangle is
[tex]A=L\times W[/tex]
Substitute W=(5-L) in the above formula.
[tex]A=L\times (5-L)[/tex]
[tex]A=5L-L^2[/tex]
Therefore, the area of the rectangle is [tex]A=5L-L^2[/tex].
Final answer:
The area A of a rectangle with a fixed perimeter of 10 meters is expressed as a function of the length L by the formula A(L) = L(5 - L), assuming 0 ≤ L ≤ 5 meters.
Explanation:
The question is asking us to express the area A of a rectangle as a function of the length L, one of its sides, given a fixed perimeter of 10 meters. The perimeter of a rectangle is given by the formula P = 2L + 2W, where L is the length and W is the width. For a rectangle with a perimeter of 10 meters, we have:
2L + 2W = 10
W = (10 - 2L) / 2 = 5 - L
The area A of the rectangle is A = L × W = L(5 - L)
This formula A = L(5 - L) gives the area as a function of the length.A: The rectangle's area is a function of its length, expressed as A(L) = L(5 - L), valid for 0 ≤ L ≤ 5, since the minimum possible width is 0 when the length equals 5, and the maximum possible length is 5 when the width equals 0.
Rewrite the expression ab dc ef such that each variable is replaced by a different non- zero digit and the value of expression is 1. (The answer may not be unique.)
Answer:
ab dc ef is replaced by 12 43 56
Step-by-step explanation:
ab dc ef are replaced by the position they take when writing in an alphabetical order. a is 1, b is 2, c is 3, and so on.
So, ab dc ed is written as
12 43 56
and
56 - 43 - 12 = 1