When there is a linear relationship is observed between the variables, we use linear regression predict the relationship between them.
Also, we predict the values for dependent variable by modelling a linear model that best fits the data by drawing a line Y=a+bX, where X is the explanatory variable and Y is the dependent variable.
In other words: The line of best fit is a line through a scatter plot of data points that best describes the relationship between them.
That's why the regression line referred to as the line of best fit.
A mountain biker gets halfway up a hill and realizes he forgot his water, so he returns to his starting point.
Answer:
True
Explanation:
1/2 (up the hill) - 1/2 (down the hill) = 0
Write the algebraic expression without parentheses. - (- 7 + 7 y) 7 + 7y 7 - 7y -7 + 7y 49y
the answer is 7-7y i think if not the it should be 7+7y
sing the Divergence Theorem, find the outward flux of F across the boundary of the region D. F = x2i + y2j + zk; D: the solid cube cut by the coordinate planes and the planes x = 2, y = 2, and z = 2
Derivative Property [Multiplied Constant]:
[tex]\displaystyle (cu)' = cu'[/tex]
Derivative Property [Addition/Subtraction]:
[tex]\displaystyle (u + v)' = u' + v'[/tex]
Derivative Rule [Basic Power Rule]:
f(x) = cxⁿf’(x) = c·nxⁿ⁻¹Integration Rule [Reverse Power Rule]:
[tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]
Integration Rule [Fundamental Theorem of Calculus 1]:
[tex]\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)[/tex]
Integration Property [Multiplied Constant]:
[tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]
Integration Property [Addition/Subtraction]:
[tex]\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx[/tex]
Del (Operator):
[tex]\displaystyle \nabla = \hat{\i} \frac{\partial}{\partial x} + \hat{\j} \frac{\partial}{\partial y} + \hat{\text{k}} \frac{\partial}{\partial z}[/tex]
Div and Curl:
[tex]\displaystyle \text{div \bf{F}} = \nabla \cdot \textbf{F}[/tex][tex]\displaystyle \text{curl \bf{F}} = \nabla \times \textbf{F}[/tex]Divergence Theorem:
[tex]\displaystyle \iint_S {\big( \nabla \times \textbf{F} \big) \cdot \textbf{n}} \, d\sigma = \iiint_D {\nabla \cdot \textbf{F}} \, dV[/tex]
First, let's define what we are given:
[tex]\displaystyle \text{F} = x^2 \hat{\i} + y^2 \hat{\j} + z \hat{\text{k}}[/tex]
Region D is the solid cube cut by coordinate planes and planes [tex]x = 2[/tex]. [tex]y = 2[/tex], and [tex]z = 2[/tex]
Step 2: WorkIn order to use the Divergence Theorem, we first must find div F. We use partial differentiation and differentiation properties found under "Calculus" to attain div F:
[tex]\begin{aligned}\nabla \cdot \text{F} & = \frac{\partial}{\partial x}(x^2) + \frac{\partial}{\partial y}(y^2) + \frac{\partial}{\partial z}(z) \\& = 2x + 2y + 1 \\\textbf{div} \ \text{F} & = \boxed{2x + 2y + 1}\end{aligned}[/tex]
∴ [tex]\displaystyle \boxed{ \textbf{div} \ \text{F} = 2x + 2y + 1 }[/tex]
In order to find the outward flux of F across region D, we now must use the Divergence Theorem. Substitute our knowns into the Divergence Theorem Formula listed under "Multivariable Calculus":
[tex]\displaystyle \iiint_D \nabla \cdot \textbf{F} \, dV = \int\limits^2_0 \int\limits^2_0 \int\limits^2_0 {2x + 2y + 1} \, dx \, dy \, dz[/tex]
We can now evaluate the Divergence Theorem integral using basic + advanced integration techniques listed under "Calculus" and learned from "Multivariable Calculus":
[tex]\displaystyle\begin{aligned}\int\limits^2_0 \int\limits^2_0 \int\limits^2_0 {2x + 2y + 1} \, dx \, dy \, dz & = \int\limits^2_0 \int\limits^2_0 \int\limits^2_0 {2x + 2y + 1} \, dz \, dy \, dx \\& = \int\limits^2_0 \int\limits^2_0 {(2x + 2y + 1)z \bigg| \limits^2_0} \, dy \, dx \\& = 2 \int\limits^2_0 \int\limits^2_0 {2x + 2y + 1} \, dy \, dx \\& = 2 \int\limits^2_0 {\bigg( 2xy + y^2 + y \bigg) \bigg| \limits^2_0} \, dx \\\end{aligned}[/tex]
[tex]\displaystyle\begin{aligned}\int\limits^2_0 \int\limits^2_0 \int\limits^2_0 {2x + 2y + 1} \, dx \, dy \, dz & = 2 \int\limits^2_0 {4x + 6} \, dx \\& = 2 \bigg[ 2x^2 + 6x \bigg] \bigg| \limits^2_0 \\& = 2(20) \\& = \boxed{40} \\\end{aligned}[/tex]
∴ the integrals evaluates to 40.
Answer:[tex]\displaystyle \int\limits^2_0 \int\limits^2_0 \int\limits^2_0 {2x + 2y + 1} \, dx \, dy \, dz = \boxed{40}[/tex]
___
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___
Topic: Multivariable Calculus
Unit: Stokes' Theorem and Divergence Theorem
PLEASE HELP ME, I DON'T UNDERSTAND THIS! :(
Answer:
1/204
Step-by-step explanation:
6/18 * 5/17 * 4/16 * 3/15
= 1/3 * 5/17 * 1/4 * 1/5
= 1/204
Identify the property illustrated by the statement.
(3 + 6) + 6 = (6 + 3) + 6
Commutative
Associative
Distributive
Closure
Answer:
Associative
Step-by-step explanation:
2. Find the slope and y-intercept.
a. y = 3x - 4 b. 4x - 5y = 15
A company that prints Blue Books for exams makes a profit according to the number of books sold. Suppose that the Profit is
Answer:
The answer is not complete. I will explain the concept of profit to you.
Step-by-step explanation:
We can determine profit deducting direct costs (cost price) of commodities from sales (selling prices) of the commodities.
Profit = Selling Price - Cost Price
Example:
A trader buys some dresses for #2,500 in May and agrees to pay for it in three months’ time. He sells off all the dresses in August for #4,500. The profit for the month is #2,000.
The formula for percentage profit is [tex]\frac{profit * 100}{cost price}[/tex]
The formula for gross profit is Revenue – Cost of Sold Items
Profit Margin = [tex]\frac{Total Income}{Net Sales}[/tex] * 100
While Gross Profit Margin can be calculated as [tex]\frac{Gross Profit}{Net Sales}[/tex] * 100
Any of these formulas can be used to calculate profit-related questions.
i need help with this can someone help me
Answer:
#9 x=21
#10 a. QP b. 8 c. 18
Step-by-step explanation:
#9 use the theorem where the ratio between a midline and the base of the triangle is 1:2
as we know the midline and the base is 1:2
this means
2(4x-65)=2x-4
4x-65=x-2
3x=63
x=21
---------------------------------------------------------------
#10
a. RS|| QP (Remember to write the letters accordingly)
b. 8 (As mentioned before of the ratio 1:2)
c. 18
solve for p 7(p-9)=34.3
Answer: p = 13.9
Step-by-step explanation: 7(p-9) = 34.3 multiply 7 into p and -9
7p - 63 = 34.3 add 63 into both sides
7p = 97.3 divide 7 into both sides.
p = 13.9
This question can be solved in two ways
the equation given is 7(p-9)=34.3
The first step to take is to divide both sides of the equation by 7
p - 9 = 34.3 / 7
p - 9 = 4.9
The second step is to combine similar terms
p = 4.9 + 9 = 13.9
A second method
the equation given is 7(p-9)=34.3
1. expand the bracket
7p - 63 = 34.3
2. Combine similar terms
7p = 34.3 + 63
7p = 97.3
3. Divide both sides of the equation by 7
p = 13.9
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Find the least whole number that can replace _
to make the statement true.
_ ÷ 17 > 99
The _ can be replaced with?
Answer:
1684
Step-by-step explanation:
x ÷ 17 > 99
x > 17 * 99
x > 1683
The least whole number that is greater than 1683 is 1684.
Answer: 1684
Answer:
1683
Step-by-step explanation:
To undo the problem you do 99 times 17 which is 1683. Then you know that 1683 divided by 17 is 99.
given directed line segment AB find the coordinates of p such that the ratio of ap to pb is 2:1 plot point p
Answer:
[tex]P(\frac{10}{3}, \frac{-2}{3})[/tex]
Step-by-step explanation:
The question is incomplete; However
[tex]A = (-2, -4)[/tex]
[tex]B = (6, 1)[/tex]
Required
Determine coordinates of P
When line segment is divided in ratios, the following formula calculates the coordinates;
[tex]P(x,y) = \{\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\}[/tex]
In this case;
[tex]m:n = 2:1[/tex]
[tex](x_1,y_1) = (-2, -4)[/tex]
[tex](x_2,y_2) = (6, 1)[/tex]
So, the coordinates of P is calculated as thus
[tex]P(x,y) = \{\frac{2 * 6 + 1 * (-2)}{2+1}, \frac{2 * 1 + 1 * (-4)}{2+1}\}[/tex]
[tex]P(x,y) = \{\frac{12 -2}{3}, \frac{2 -4}{3}\}[/tex]
[tex]P(x,y) = \{\frac{10}{3}, \frac{-2}{3}\}[/tex]
Hence, the coordinates of P is
[tex]P(\frac{10}{3}, \frac{-2}{3})[/tex]
The question is incomplete, the points A and B are:
A = (-2, -4)
B = (6, 1)
We want to find a point P in the segment AB such that the ratio of AP to PB is 2:1
We will find that:
[tex]P = (\frac{2}{3} , \frac{7}{3} )[/tex]
The general formula for two points (x₁, y₁) and (x₂, y₂), the coordinates of a point that separates the segment in a ratio j:k is
[tex]P = (\frac{j*x_1 + k*x_2}{j + k} , \frac{j*y_1 + k*y_2}{j + k})[/tex]
So we only need to use that general formula:
[tex]P = P = (\frac{2*(-2) + 1*6}{3} , \frac{2*(-4) + 1*1}{3})\\\\P = (\frac{2}{3} , \frac{7}{3} )[/tex]
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If you have 200 cupcakes and you give 90 how many do you have left
Answer:
110 cupcakes are left
Step-by-step explanation:
200-90 is 110
Answer:
110 cupcakes
Step-by-step explanation:
If you have 200 cupcakes and you give away 90, that means you only have [tex]200-90[/tex] cupcakes left.
[tex]200 - 90 = 110[/tex]
So you have 110 cupcakes left.
Hope this helped!
How many atoms in human body?
16. In the adjoining figure, if ll|m, then find the value of angle x
Step-by-step explanation:
Hey, there!!
Let's solve it simply,
angle ONP = 60°+55° { as exterior angle is equal to the sum of opposite interior angle}.
Therefore, angle ONP = 115°.
Now, let's solve for x,
x+ 115°=180° { because the sum of co-interior angle is 180°}.
x= 180°-115°
Therefore, the measure of angle x is 65°.
Hope it helps...
Answer:
[tex]\Huge \boxed{{x=65\° }}[/tex]
Step-by-step explanation:
l and m are parralel lines.
The third angle in the triangle is also x, because of corresponding angles.
Angles in a triangle add up to 180 degrees.
Create an equation and solve for x.
x + 60 + 55 = 180
Add the numbers.
x + 115 = 180
Subtract 115 from both sides.
x = 65
Write an equation of the line below.
Answer:
y = x - 1
Step-by-step explanation:
Need help!!! A rocket scientist is designing a rocket to visit the planets in the solar system. The velocity that is needed to escape a planet’s gravitational pull is called the escape velocity. The escape velocity depends on the planet’s radius and its mass, according to the equation V escape=square root(2gR where R is the radius and g is the gravitational constant for the particular planet. The rocket’s maximum velocity is exactly double Earth’s escape velocity. The earth’s gravitational pull is 9.8 m/s^2 The earth’s radius is 6.37 x10 ^6 For which planets will the rocket have enough velocity to escape the planet’s gravity?
Answer:
Mercury, Venus, Earth, Mars, and Uranus
Step-by-step explanation:
Calculate the escape velocity for each planet, using the equation v = √(2gR).
[tex]\left[\begin{array}{cccc}Planet&R(m)&g(m/s^{2})&v(m/s)\\Mercury&2.43\times10^{6}&3.61&4190\\Venus&6.07\times10^{6}&8.83&10400\\Earth&6.37\times10^{6}&9.80&11200\\Mars&3.38\times10^{6}&3.75&5030\\Jupiter&6.98\times10^{7}&26.0&60200\\Saturn&5.82\times10^{7}&11.2&36100\\Uranus&2.35\times10^{7}&10.5&22200\\Neptune&2.27\times10^{7}&13.3&24600\end{array}\right][/tex]
The rocket's maximum velocity is double the Earth's escape velocity, or 22,400 m/s. So the planets the rocket can escape from are Mercury, Venus, Earth, Mars, and Uranus.
A spherical hot-air balloon has a diameter of 55 feet. When the balloon is inflated, the radius increases at a rate of 1.5 feet per minute. Approximately how long does it take to inflate the balloon to Two-thirds of its maximum volume? Use π = 3.14 and V = four-thirds pi r cubed.
Answer: A
16 minutes
Step-by-step explanation:
The required time to inflate the balloon to Two-thirds of its maximum Volume is 16 minutes.
The rate of change is defined as the change in value with the rest of the time is called rate of change.
Here,
radius = 55 / 2 = 27.5
Since volume is directly proportional to the cube of the radius,
So,
For maximum volume,
V ∝ R³
and
2/3V = v
2/3R³ = r³
r = 27.5 ∛2/3
r = 24
Now,
the radius increases at a rate of 1.5 feet per minute.
Time to reach a radius of 24 feet,
= 24 / 1.5
= 16 minutes
Thus, the required time to inflate the balloon to Two-thirds of its maximum Volume is 16 minutes.
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Micah was given $200 for his birthday. Each week he spends $15 on comic books. In how many weeks will his birthday money be gone?
Answer:
13 weeks
Step-by-step explanation:
V = -15 x + 200 x 213.33 weeks 6 = -15x + 200 after 13 weeks.
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Tysm!
The number of weeks Micah needs to spend so that her birthday money of $200 will be gone is 13 weeks.
Given,
Micah was given $200 for his birthday.
Each week he spends $15 on comic books.
We need to find in how many weeks will his birthday money be gone.
How to compare two units in proportion?We need to make sure we get the required value on the left side.
If 1 week = 7 days
Find how many days in 2 weeks
2 weeks = 2 x 7 days = 14 days
We have 2 weeks on the left side.
If 2 items = $5
5/2 x 2 items = 5/2 x $5
5 items = $25/2 = $12.5
Find the total amount Micah has.
= $200
Find the amount spend each week.
= $15
Find how many weeks she must spend to spend $200.
We have,
$15 = 1 week
Multiply 200/15 on both sides.
200/15 x $15 = 200/15 x 1 week
$200 = 13.33 weeks
This means 13 weeks.
Thus the number of weeks Micah needs to spend so that her birthday money of $200 will be gone is 13 weeks.
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Which of the following expressions does not represent "four more than one-third x"?
Options:
1/3x + 4
x/3 + 4
1/4x + 3
Please explain well so I can understand it. Thanks! :)
Answer:
1/4x + 3
Step-by-step explanation:
The question said one-third(1/3) not one-fourth(1/4).
1/3x + 4 can also be written as x/3 + 4.
Answer:
1/4x + 3
Step-by-step explanation:
expressions does not represent "four more than one-third x"
1/3x + 4 --- No.
x/3 + 4 --- No.
1/4x + 3 --- Yes. it does not represent 4 + 1/3x. you can tell it by +3
After a shipwreck, 120 rats manage to swim from the wreckage to a deserted island. The population on the island grows exponentially according to the model n(t)-ne", where n(t)is the number of rats at time t in months. After 15 months there are 280 rats on the island a. Find a function that models the population t months after the arrival of the rats. b. What will the population be 3 years after the shipwreck? Round to a whole number. c. When will the population reach 2000 rats?
Complete Question
The complete question is shown on the first uploaded image
Answer:
a
[tex]n(t) = 120 e^{(0.056t)}[/tex]
b
[tex]n(36) = 901 \ rats[/tex]
c
[tex]t = 50 \ months[/tex]
Step-by-step explanation:
From the question we are told that
The original number of rats that swim from the wreckage to a deserted island is
The population on the island grows exponentially according to the model
[tex]n(t) = n_o e^{rt}[/tex]
The number of rats after t = 15 months is 280
So
[tex]n(15) = 120 e^{15 * r } = 280[/tex]
=> [tex]120 e^{15 * r } = 280[/tex]
=> [tex]e^{15r} = 2.33[/tex]
=> [tex]15r = 0.8459[/tex]
=> [tex]r = 0.056[/tex]
Therefore the population t months after the arrival of the rats is mathematically represented as
[tex]n(t) = 120 e^{(0.056t)}[/tex]
Here t is in months
Considering question b
For t = 3 years = 12 * 3 = 36 months
Then the number of rats that will be present is mathematically represented as
[tex]n(36) = 120 e^{0.056 * 36 }[/tex]
[tex]n(36) = 901 \ rats[/tex]
Considering question c
Now the number of rats considered is n(t) = 2000
So
[tex]n(t) = 2000 = 120e^{0.056t}[/tex]
=> [tex]2000 = 120e^{0.056t}[/tex]
=> [tex]16.67 = e^{0.056t}[/tex]
Taking natural log of both sides
=> [tex]0.056 t = 2.81[/tex]
=> [tex]t = 50 \ months[/tex]
Your math professor gives you a set of 10 problems and tells you that the final exam will consist of a random selection of 5 of them. If you figured out how to do 7 of the problems, what is the probability that you answer correctly (a) all 5 problems
Answer:
0.083
Step-by-step explanation:
To solve the above question, we use the combination method.
C(n, r) = nCr = n!/r!(n - r)!
Probability of answering (all 5 correctly)
= Probability of answering ( 5 out of 7 correctly)/ Probability of answering (5 out of 10 problems)
Probability of answering ( 5 out of 7 correctly) = 7C5
nCr = n!/r!(n - r)!
= 7!/5! (7 - 5)!
= 7 × 6 × 5 × 4 × 3 × 2 × 1/( 5 × 4 × 3 × 2 × 1) × (2 × 1)
= 21 ways
Probability of answering (5 out of 10 problems) = 10C5
nCr = n!/r!(n - r)!
= 10!/5! (10 - 5)!
= 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1/( 5 × 4 × 3 × 2 × 1) × (5 × 4 × 3 × 2 × 1)
= 252 ways
Probability of answering (all 5 correctly)
= 21/252
= 0.0833333333
Approximately = 0.083
The average age of Iowa residents is 37 years. Amy believes that the average age in her hometown in Iowa is not equal to this average and decided to sample 30 citizens in her neighborhood. Using the alternative hypothesis that µ ≠ 531, Amy found a t-test statistic of 1.311. What is the p-value of the test statistic? Answer choices are rounded to the hundredths place.
Answer:
0.10
Step-by-step explanation:
Average age of Iowa residents = 37 years
Amy believes that the average age in her hometown in Iowa is not equal to this average
Alternative hypothesis : µ ≠ 37
Sample size = 30
t - test statistic = 1.311
The p-value of the test statistic can be obtained using the online p-value calculator :
The p- value obtained using a t-test statistic value of 1.311 and a degree of freedom (df =N-1; 30 - 1 = 29) at a 0.05 significance level = 0.100072 = 0.10 ( nearest hundredth)
Answer:
0.20
Step-by-step explanation:
In order to find the p-value we first need the degrees of freedom because we are running a t-test. The df = n-1 = 30-1 = 29. If Amy is using the alternative hypothesis, µ ≠ 531, this is a two-tailed test. In order to find the p-value, we simply need to find the probability of being as extreme or more extreme as our current test statistic in one-tail and then multiply it by 2. Another way to describe this is it is the tail probability/area. If we go to the df row of 29 and then scan to the right we see 1.311 is in the 0.10 column. So if we multiply this by 2, we find 0.20 or 20% in both of the tails. So the p-value is 0.20.
If "a" is a negative number, and "ab" is a positive
answer, then what is the sign of "b”?
Answer:
b would be a negative number
Answer: The sign of b is negative or -
Step-by-step explanation:
If you multiply two negative numbers you will have a positive number. And if you multiply a positive and negative numbers you will have a negative answer.
Solve and graph the following inequality
Answer:
x is greater than 4
Step-by-step explanation:
4x + 5 is greater than 13
you want to find out what x is
4 - 4 x + 5 is greater than 13 - 4 ( whatever you do to one side you have to do to the other)
x + 5 is greater than 9
x + 5 - 5 is greater than 9 - 5
x is greater than 4
The price of a technology stock was $9.69 yesterday. Today, the price fell to $9.58 . Find the percentage decrease. Round your answer to the nearest tenth of a percent.
Answer:
1.1% decrease
Step-by-step explanation:
Use the formula, % change = (difference/original) x 100
Plug in the values:
% change = ((9.69 - 9.58) / 9.69) x 100
= 1.1% decrease
Kara needs to fence her yard. How many feet of fencing is needed?
A. 370 feet
B. 500 feet
C. 13, 700 feet
D. 15, 000 feet
Answer:
newzz new new me he new new
Answer:
oiuhj90pkomnjhb vcfgtyu789i0op,lmkn bvghyu7890opklmnjbhgvfty67890-p[lkjhngvty67u890-op[l;',
Step-by-step explanation:
Find the value of 83 - [59 - (22 - 18)].
Answer:
28
Step-by-step explanation:
83 - [59 - (22-18)]
= 83 - [59 - 4]
= 83 - 55
= 28
would you use the Law of Sines or the law of cosines to solve a triangle if. given all three sides of the triangle?
a. the Law of Sines
b. the Law of Cosines
Answer:
b. [tex]\displaystyle Law\:of\:Cosines[/tex]
Step-by-step explanation:
You would use this law under two conditions:
One angle and two edges defined, while trying to solve for the third edgeALL three edges definedHence, you have your answer.
* Just make sure to use the inverse function towards the end, or elce you will throw your answer off!
_______________________________________________
Now, you would use the Law of Sines under three conditions:
Two angles and one edge defined, while trying to solve for the second edgeOne angle and two edges defined, while trying to solve for the second angleALL three angles defined [of which does not occur very often, but it all refers back to the first bullet]* I HIGHLY suggest you keep note of all of this significant information. You will need it going into the future.
I am delighted to assist you at any time.
Answer:
Conditions for SINE RULE:
If two angles and any side is given.If two sides and an angle opposite one of the side is given.Conditions for COSINE RULE:
If two sides and an inclined angle is given.If all the three sides are give.Step-by-step explanation:
Side is the number in m/cm.
Angle is the number with degree/in degree. Like this: ° ° °
With all these, it means you should COSINE rule.
If a data set produces SSR = 400 and SSE = 100, then the coefficient of determination is a. .10. b. .80. c. .25. d. .40.
Answer: 0.8
Step-by-step explanation:
Given the following :
Sum of Square Error Error (SSE) = 100 which is the difference between the actual and predicted value.
Sum of Square due to regression (SSR) = 400
The Coefficient of determination (R^2) :
(Sum of Square Regression(SSR)) / Sum of Squared total (SST))
Sum of Squared total (SST) = SSE + SSR
Sum of Squared Total = (100 + 400) = 500
R^2 = 400 / 500
R^2 = 0.8
Test the claim that the proportion of men who own cats is smaller than the proportion of women who own cats at the .10 significance level. Let p_M and p_F be the proportion of Men and Women who own cats respectively.
Based on a sample of 80 men, 40% owned cats
Based on a sample of 80 women, 51% owned cats
What is the test statistic and the critical value? Reject or Fail to Reject Null hypothesis?
Answer:
The test statistics is [tex]t = -1.40[/tex]
The critical value is [tex]Z_{\alpha } = 2.33[/tex]
The null hypothesis is rejected
Step-by-step explanation:
From the question we are told that
The sample size for men is [tex]n_1 = 80[/tex]
The sample proportion of men that own a cat is [tex]\r p _M = 0.40[/tex]
The sample size for women is [tex]n_2 = 80[/tex]
The sample proportion of women that own a cat is [tex]\r p_F = 0.51[/tex]
The level of significance is [tex]\alpha = 0.10[/tex]
The null hypothesis is [tex]H_o : \r p _M = \ r P_F[/tex]
The alternative hypothesis is [tex]H_a : \r p _M < \r p_F[/tex]
Generally the test statistic is mathematically represented as
[tex]t = \frac{(\r p_M - \r p_F)}{\sqrt{\frac{(p_M*(1-p_M)}{n_1 } } + \frac{p_F*(1-pF)}{n_2 } }[/tex]
=> [tex]t = \frac{(0.40 - 0.51)}{\sqrt{\frac{(0.40 *(1-0.41)}{80} } + \frac{0.51*(1-0.51)}{80 } }[/tex]
=> [tex]t = -1.40[/tex]
The critical value of [tex]\alpha[/tex] from the normal distribution table is
[tex]Z_{\alpha } = 2.33[/tex]
The p-value is obtained from the z-table ,the value is
[tex]p-value = P( Z < -1.40) = 0.080757[/tex]
=> [tex]p-value = 0.080757[/tex]
Given that the [tex]p-value < \alpha[/tex] then we reject the null hypothesis