Solve the problem.


The library is to be given 3 books as a gift. The books will be selected from a list of 16 titles. If each book selected must have a different title, how many possible selections are there?



48



560



3360



4096

Answer:

see explanation

Step-by-step explanation:

To determine the magnitude of the scale factor, calculate the ratio of corresponding sides of image to original, that is

scale factor = [tex]\frac{A'B'}{AB}[/tex] = [tex]\frac{2}{5}[/tex]

ΔA''B''C'' is a reflection of ΔA'B'C' in the y- axis ( corresponding vertices are equidistant from the y- axis )

Answer:

[tex]0.060[/tex]

Step-by-step explanation:

In a two tailed test the probability of occurrence is the total area under the critical range of values on both the sides of the curve (negative side and positive side)

Thus, the probability values for a two tailed test as compared to a one tailed test is given by the under given relation -

[tex]p-value = P(Z< -\frac{\alpha }{2} )+P(Z >\frac{\alpha}{2})[/tex]\

Here [tex]P\frac{\alpha}{2} = 0.030[/tex]

Substituting the given value in above equation, we get -

probability values for a two tailed test

=[tex]0.030 + 0.030\\= 0.060[/tex]

Answer:

0.68

Step-by-step explanation:

Given

Mean = μ = 6 cm

SD = σ = 1.5 cm

We have to find the z-scores for 4.5 and 7.5

z-score for 4.5 = z_1 = (x-μ)/σ = (4.5-6)/1.5 = -1.5/1.5 = -1

z-score for 4.5 = z_2 = (x-μ)/σ = (7.5-6)/1.5 = 1.5/1.5 = 1

We have to find area to the left of z-scores

Using the rule of thumb for SD from mean, 68% of data lies between one standard deviation from mean. So the probability of choosing a band with length between 4.5 and 7.5 cm is 0.68 ..

Answer:

  D.  X-7

Step-by-step explanation:

The table tells you that when x=7, g(x) = 0. In order for g(7) to be zero, at least one factor must be zero when x=7. The only factor on the list that is zero when x=7 is (x-7).

To see if (x - a) is a factor of g(x), a polynomial function, you check if g(a) = 0 from the values in the table. Unfortunately, without the table of values, we cannot definitively determine which of the options must be a factor of the function g(x).

In order to determine which of the options given must be a factor of the function g(x), we need to understand a property about polynomial functions and their factors. If (x - a) is a factor of a polynomial, then the function g(a) = 0. This means that (a,0) is a point on the graph of the function.

Unfortunately, without the values of x and g(x) from the table above we cannot definitively conclude which of the options A, B, C, or D must be a factor of the polynomial g(x). However, if for example, g(1) = 0 in the table, then (x - 1) or option A would be a factor of g(x). The same logic applies to the other options.

Remember to always check any similar question using the table of values provided to determine if a given expression is a factor of a function!

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Answer:

11 years old

Step-by-step explanation:

Let Jimmy's age be represented as x. His big brother's age is x+5 and his sister's age is 2x. Adding these gives us 4x+5=29. Solving for x gives us 6. His brother's age is 6+5=11.

11 years old is Jimmy's brother.

Let Jimmy's age be represented as x. His big brother's age is x+5 and his sister's age is 2x. Adding these gives us 4x+5=29. Solving for x gives us 6. His brother's age is 6+5=11.

J=Jimmy's age; S=sister's age=2J; B=brother's age=J+5

.

J+S+B=29

J+(2J)+(J+5)=29

4J+5=20

4J=24

J=6

Jimmy is 6 years old.

B=J+5=6+5=11

ANSWER: Jimmy's brother is 11 years old.

.

CHECK:

S=2J=2(6)=12

Jimmy's sister is 12 years old.

.

J+S+B=29

6+12+11=29

29=29.

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Answer:

Step-by-step explanation:

speed x time = distance

s (1.5) = 105

1.5s=105

s = 70mph

d(t) = 70t

Answer:

The equation that relates y and x is [tex]y=x+612 mi[/tex], and the equation that relates x and d is [tex]4d=x[/tex].

Step-by-step explanation:

Step 1: First we know that the total distance is equal to y. The distance traveled in 4 hours equals x, and the distance from point x to y equals 612 miles. Adding x to the remaining 612 miles gives the total distance y.

[tex]y=x+612 mi[/tex]

Step 2: To know the relationship between x and d, we must first raise the speed during the journey to x.

[tex]v=\frac{x}{4h}[/tex]

Then, we set the speed d e equal v:

[tex]v=\frac{d}{h}[/tex]

[tex]\frac{x}{4h} = \frac{d}{h}[/tex]

Clearing x we get:

[tex]x=4h * \frac{d}{h}[/tex]

[tex]x=4d[/tex]

Have a nice day!

Answer:

Step-by-step explanation:

The vertex order ABC is clockwise in the original figure and also in the first image: A'B'C'. The altitude from AC to B is up in the original and left in the first image, indicating a rotation 90° CCW.

  The first transformation is a rotation 90° CCW.

The vertex order of A''B''C'' is CCW, indicating a reflection. The direction of the altitude from A''C'' to B'' is still to the left, so the reflection must be over a horizontal line. We find the x-axis bisects the segments A'A'', B'B'', and C'C'', confirming that it is the line of reflection.

  The second transformation is reflection across the x-axis.

_____

Algebraically, the transformations are ...

  1st: (x, y) ⇒ (-y, x)

  2nd: (x, y) ⇒ (x, -y)

Both together: (x, y) ⇒ (-y, -x).

Answer:

S = -9,372 ⇒ 1st answer

Step-by-step explanation:

* Lets revise the geometric series

- There is a constant ratio between each two consecutive numbers

- Ex:

# 5  ,  10  ,  20  ,  40  ,  80  ,  ………………………. (×2)

# 5000  ,  1000  ,  200  ,  40  ,  …………………………(÷5)

* General term (nth term) of a Geometric series:

 U1 = a  ,  U2  = ar  ,  U3  = ar2  ,  U4 = ar3  ,  U5 = ar4

 Un = ar^(n-1), where a is the first term, r is the constant ratio between

 each two consecutive terms

- The sum of first n terms of a geometric series is calculate from

  [tex]S_{n}=\frac{a(1-r^{n})}{1-r}[/tex]

* Lets solve the problem

∵ The series is geometric

∵ a1 = -12

∴ a = -12

∵ a5 = -7500

∵ a5 = ar^4

∴ -7500 = -12(r^4) ⇒ divide both sides by -12

∴ 625 = r^4 take root four to both sides

∴ r = ± 5

∵ r = 5 ⇒ given

∵ [tex]Sn=\frac{a(1-r^{n})}{1-r}[/tex]

∵ n = 5

∴ [tex]S_{5}=\frac{-12[1-(5)^{5}]}{1-5}=\frac{-12[1-3125]}{-4}=3[-3124]=-9372[/tex]

* S = -9,372

Answer:

  18 °F

Step-by-step explanation:

The means are calculated in the usual way: add up the numbers and divide by the number of them. When there are a bunch of numbers, it is convenient to let a calculator or spreadsheet compute the mean for you.

In the attached, we see the mean for week 1 is 52°, and in week 2, it is 70°. The mean is 70° -52° = 18° greater in week 2.

Answer:

The correct option is 1.

Step-by-step explanation:

Given information: AD = 25, CD = 5, DB = 1 and CD⊥AB.

According to the Pythagoras theorem,

[tex]hypotenuse^2=base^2+perpendicular^2[/tex]

In triangle BCD,

[tex]CB^2=DB^2+CD^2[/tex]

[tex]CB^2=1^2+5^2[/tex]

[tex]CB^2=26[/tex]

Taking square root both sides.

[tex]CB=\sqrt{26}[/tex]

In a right angled triangle,

[tex]\sin\theta=\frac{opposite}{hypotenuse}[/tex]

[tex]\sin B=\frac{CD}{CB}[/tex]

[tex]\sin B=\frac{5}{\sqrt{26}}[/tex]

[tex]\sin B=0.980580675691[/tex]

[tex]\sin B\approx 0.9806[/tex]

Therefore the correct option is 1.

Answer:

0.9806 is the correct answer.

Step-by-step explanation:

Answer:

  B.)  32

Step-by-step explanation:

The diagonals of a parallelogram bisect each other, so ...

  BE = DE

  y+10 = 4y - 8 . . . substitute the given expressions

  18 = 3y . . . . . . . . add 8-y

  6 = y . . . . . . . . . . divide by 3

Then BE = y+10 = 16 and ...

  BD = 2×BE = 2×16

  BD = 32

Answer:

B.) 32

Step-by-step explanation:

Given parallelogram ABCD, diagonals AC and BD intersect at point E, AE = 2x, BE=y+10, CE=x+2 and DE=4y - 8, the length of BD is 32.

BD = 2×BE = 2×16

Answer:

So the approximate relative minimum is (0.4,-58.5).

Step-by-step explanation:

Ok this is a calculus approach.  You have to let me know if you want this done another way.

Here are some rules I'm going to use:

[tex](f+g)'=f'+g'[/tex]       (Sum rule)

[tex](cf)'=c(f)'[/tex]          (Constant multiple rule)

[tex](x^n)'=nx^{n-1}[/tex] (Power rule)

[tex](c)'=0[/tex]               (Constant rule)

[tex](x)'=1[/tex]                (Slope of y=x is 1)

[tex]y=4x^3+13x^2-12x-56[/tex]

[tex]y'=(4x^3+13x^2-12x-56)'[/tex]

[tex]y'=(4x^3)'+(13x^2)'-(12x)'-(56)'[/tex]

[tex]y'=4(x^3)'+13(x^2)'-12(x)'-0[/tex]

[tex]y'=4(3x^2)+13(2x^1)-12(1)[/tex]

[tex]y'=12x^2+26x-12[/tex]

Now we set y' equal to 0 and solve for the critical numbers.

[tex]12x^2+26x-12=0[/tex]

Divide both sides by 2:

[tex]6x^2+13x-6=0[/tex]

Compaer [tex]6x^2+13x-6=0[/tex] to [tex]ax^2+bx+c=0[/tex] to determine the values for [tex]a=6,b=13,c=-6[/tex].

[tex]a=6[/tex]

[tex]b=13[/tex]

[tex]c=-6[/tex]

We are going to use the quadratic formula to solve for our critical numbers, x.

[tex]x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}[/tex]

[tex]x=\frac{-13 \pm \sqrt{13^2-4(6)(-6)}}{2(6)}[/tex]

[tex]x=\frac{-13 \pm \sqrt{169+144}}{12}[/tex]

[tex]x=\frac{-13 \pm \sqrt{313}}{12}[/tex]

Let's separate the choices:

[tex]x=\frac{-13+\sqrt{313}}{12} \text{ or } \frac{-13-\sqrt{313}}{12}[/tex]

Let's approximate both of these:

[tex]x=0.3909838 \text{ or } -2.5576505[/tex].

This is a cubic function with leading coefficient 4 and 4 is positive so we know the left and right behavior of the function. The left hand side goes to negative infinity while the right hand side goes to positive infinity. So the maximum is going to occur at the earlier x while the minimum will occur at the later x.

The relative maximum is at approximately -2.5576505.

So the relative minimum is at approximate 0.3909838.

We could also verify this with more calculus of course.

Let's find the second derivative.

[tex]f(x)=4x^3+13x^2-12x-56[/tex]

[tex]f'(x)=12x^2+26x-12[/tex]

[tex]f''(x)=24x+26[/tex]

So if f''(a) is positive then we have a minimum at x=a.

If f''(a) is negative then we have a maximum at x=a.

Rounding to nearest tenths here:  x=-2.6 and x=.4

Let's see what f'' gives us at both of these x's.

[tex]24(-2.6)+25[/tex]

[tex]-37.5[/tex]  

So we have a maximum at x=-2.6.

[tex]24(.4)+25[/tex]

[tex]9.6+25[/tex]

[tex]34.6[/tex]

So we have a minimum at x=.4.

Now let's find the corresponding y-value for our relative minimum point since that would complete your question.

We are going to use the equation that relates x and y.

I'm going to use 0.3909838 instead of .4 just so we can be closer to the correct y value.

[tex]y=4(0.3909838)^3+13(0.3909838)^2-12(0.3909838)-56[/tex]

I'm shoving this into a calculator:

[tex]y=-58.4654411[/tex]

So the approximate relative minimum is (0.4,-58.5).

If you graph [tex]y=4x^3+13x^2-12x-56[/tex] you should see the graph taking a dip at this point.

Answer:

Step-by-step explanation:

Put the numbers in the formula and do the arithmetic. For repetitive calculations, it is convenient to define a function in a graphing calculator or spreadsheet.

Answer:

= 1/2[sin2Acos7A + cos2Asin7A + sin2Acos7A - cos2Asin7A]

we cut out

cos2Asin7A - cos2Asin7A

then we have

=1/2[sin2Acos7A + sin2Acos7A ]

= 1/2*2[ sin2Acos7A ]

cut out 2 we get

#sin2Acos7A

Answer:

Step-by-step explanation:

It is convenient to let a spreadsheet or graphing calculator do the repetitive evaluation of a function like this. That simplifies the work and reduces errors.

The function is shown in the attachment written in Horner form, which is convenient for evaluation by hand or using a calculator.

Answer:

I'm pretty sure its 1

Step-by-step explanation:

because if y = 1. 1 by the power of 3 is 1, and y by the power 2 is 1. 1- 10 is -9 . -9 times 1 equals -9, and -9 equals -9 therefore it's a true statement

Step-by-step explanation:

y³ (y² − 10) = -9y

Move everything to one side:

y³ (y² − 10) + 9y = 0

Factor out common term:

y (y² (y² − 10) + 9) = 0

Distribute:

y (y⁴ − 10 y² + 9) = 0

Factor:

y (y² − 1) (y² − 9) = 0

Solve:

y = 0, ±1, ±3

Since y > 0, the two possible values for y are 1 and 3.

Answer:

f(x)=3-4 In (x-2)=graph 3

f(x)=3-In x=graph 1

f(x)=In (x+1)=graph 4

f(x)= 2In (x+3)= graph 2

Step-by-step explanation:

Use a graph tool to visualize the functions.Attached are the graphed functions respectively.

Answer:

f(x) = 3 - 4㏑(x - 2) ⇒ graph 3

f(x) = 3 - ㏑(x) ⇒ graph 1

f(x) = ㏑(x + 1) ⇒ graph 4

f(x) = 2㏑(x + 3) ⇒ graph 2

Step-by-step explanation:

* Lets look to the graphs and solve the problem

- We will use some points on each graph and substitute in the function

  to find the graph of each function

- Remember: ㏑(1) = 0 and ㏑(0) is undefined

- Lets solve the problem

# f(x) = 3 - 4㏑(x - 2)

- Let x - 2 = 1 because ㏑(1) = 0, then f(x) will equal 3

∵ x - 2 = 1 ⇒ add 2 for both sides

∴ x = 3

- Substitute the value of x in f(x)

∴ f(x) = 3 - 4㏑(3 - 2)

∴ f(x) = 3 - 4㏑(1) ⇒ ㏑(1) = 0

∴ f(x) = 3

∴ Point (3 , 3) lies on the graph

- Look to the graphs and find which one has point (3 , 3)

∵ Graph 3 has the point (3 , 3)

∴ f(x) = 3 - 4㏑(x - 2) ⇒ graph 3

# f(x) = 3 - ㏑(x)

- Let x = 1 because ㏑(1) = 0, then f(x) will equal 3

- Substitute the value of x in f(x)

∴ f(x) = 3 - ㏑(1) ⇒ ㏑(1) = 0

∴ f(x) = 3

∴ Point (1 , 3) lies on the graph

- Look to the graphs and find which one has point (1 , 3)

∵ Graph 1 has the point (1 , 3)

∴ f(x) = 3 - ㏑(x) ⇒ graph 1

# f(x) = ㏑(x + 1)

- Let x = 0 because ㏑(1) = 0, then f(x) will equal 0

- Substitute the value of x in f(x)

∴ f(x) = ㏑(0 + 1) = ㏑(1) ⇒ ㏑(1) = 0

∴ f(x) = 0

∴ Point (0 , 0) lies on the graph

- Look to the graphs and find which one has point (0 , 0)

∵ Graph 4 has the point (0 , 0)

∴ f(x) = ㏑(x + 1) ⇒ graph 4

# f(x) = 2㏑(x + 3)

- Let x + 3 = 1 because ㏑(1) = 0, then f(x) will equal 0

∵ x + 3 = 1 ⇒ subtract 3 from both sides

∴ x = -2

- Substitute the value of x in f(x)

∴ f(x) = 2㏑(-2 + 3) = 2㏑(1) ⇒ ㏑(1) = 0

∴ f(x) = 0

∴ Point (-2 , 0) lies on the graph

- Look to the graphs and find which one has point (-2 , 0)

∵ Graph 2 has the point (-2 , 0)

∴ f(x) = 2㏑(x + 3) ⇒ graph 2

Answer: A.2x+30=90

wwww

Answer:

Exterior

Step-by-step explanation:

In any triangle an exterior angle is equal to the sum of the two opposite interior angles.

Answer:

C.

Step-by-step explanation:

The only information you really need in order to determine if this is a right triangle are the slopes of segments AB and BC.  If the slopes of these segments are opposite reciprocals of one another, then the lines are perpendicular, and the angle is a right angle (making the triangle a right triangle!).  Point A has coordinates (-5, 5), B(-3, 2), C(-6, 0).

The slope of segment AB:

[tex]m=\frac{2-5}{-3-(-5)}=-\frac{3}{2}[/tex]

The slope of segment BC:

[tex]m=\frac{0-2}{-6-(-3)}=\frac{2}{3}[/tex]

As you can see, the slopes are opposite reciprocals of one another so angle ABC is a right angle, and triangle ABC is a right triangle.  Choice C is the one you want.

Answer:

Yes, two sides are perpendicular and the side lengths fit the Pythagorean theorm

Step-by-step explanation:

Because it's the answer

Answer:

a

Step-by-step explanation:

Given

x² + 20x + 100 = 36 ( subtract 36 from both sides )

x² + 20x + 64 = 0 ← in standard form

Consider the factors of the constant term ( + 64) which sum to give the coefficient of the x- term ( + 20)

The factors are + 16 and + 4, since

16 × 4 = + 64 and 16 + 4 = + 20, hence

(x + 16)(x + 4) = 0 ← in factored form

Equate each factor to zero and solve for x

x + 16 = 0 ⇒ x = - 16

x + 4 = 0 ⇒ x = - 4

Answer: She will have $700 left over.

Step-by-step explanation: Since we know that a rectangle has two sides with the measurement, we can add the sides. 37+37+25+25=124. The fencing in 124 feet in total. Multiply the 124 feet by the price per foot. 124 x 75 =9,300. Subtract the price from your total amount of money. 10,000 - 9,300 = $700. She will have $700 left over.

Answer:

there would be $700 left over

Step-by-step explanation:

Answer:

900 miles will the plane be from its port when the boat is 54 miles away from its port

Step-by-step explanation:

Given data

boat away = 6 miles

plane away = 100 miles

to find out

How many miles will plane be from port when boat 54 miles away from its port

solution

first we consider x distance while plane travel and boat travel 54 miles

from question given the equation will be

boat away from its port / plane away from its port = boat away from its port/  plane away from its port

54 / x = 6/100

solve this equation

6 × (x) = 54 × 100

x = 54 × 100 / 6

x = 900

so 900 miles will the plane be from its port when the boat is 54 miles away from its port

Answer:

Option A is the correct choice.

Step-by-step explanation:

We have been given a histogram and we are asked to choose the correct statement about our given histogram.

Upon looking at our given histogram, we can see that our given data set is skewed to right. This means that means that the mean of the given data will be greater than median as our given data set has a long tail towards right or our data set is positively skewed.

Therefore, option A is the correct choice.

Answer:

  3/2

Step-by-step explanation:

Every coordinate of E'F'G'H' is 3/2 times that of EFGH, so the image is 3/2 times the size of the original.

___

For example, E'x/Ex = -3/-2 = 3/2; E'y/Ey = (-3/2)/-1 = 3/2.

The scale factor between quadrilateral EFGH and its image E'F'G'H' can be found by calculating the ratio of the lengths of corresponding sides. In this case, it is approximately 0.707.

In this task, we are asked to find the scale factor between a quadrilateral and its image. The scale factor can be found by dividing the lengths of corresponding sides in the image by the respective side length in the original figure.

First, let's calculate the distance between the vertices E and F in the original figure using the Euclidean distance formula: sqrt[(x2-x1)^2 + (y2-y1)^2] = sqrt[(1+2)^2 + (2+1)^2] = sqrt[9+9] = sqrt[18] = approximately 4.2426.

Then, let's do the same for vertices E' and F' in the image: sqrt[(3/2+3)^2 + (3+3/2)^2] = sqrt[(9/4 + 9/4)+(9/4 + 9/4)] = sqrt[(9/2)+(9/2)] = sqrt[9] = 3.

Your scale factor, then, is the length of EF' divided by the length of EF: 3 / 4.2426, which roughly equals 0.707.

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Answer:

  A.  $14,500

Step-by-step explanation:

The van depreciates ($16000 -1000 = $15000 in 5 years, so $3000 per year. It will be assumed to depreciate half that amount in half a year, so will be worth $1500 less than $16000 at the end of the first calendar year. The book value will be $14,500.

Final answer:

The book value of the minivan at the end of Year 1 is $14,500 after accounting for 6 months of straight-line depreciation of $1,500 from the original cost of $16,000.

Therefore, the correct answer is A. $14,500.

Explanation:

The student's question is related to calculating the book value of a minivan at the end of year 1 using the straight-line depreciation method. To find the book value, we need to first calculate the annual depreciation expense and then subtract it from the original cost of the minivan.

First, we calculate the annual depreciation expense:

Purchase price of minivan: $16,000

Residual value: $1,000

Useful life: 5 years

So, the annual depreciation expense is
(

$16,000

-

$1,000

) /

5 years

= $3,000 per year.

Since the minivan was bought on July 3, we need to account for a partial year of depreciation for year 1. Assuming the end of the year is December 31, that's 6 months (July through December) of depreciation in the first year. Therefore, it would be
$3,000 / 2 = $1,500 for 6 months.

To find the book value at the end of Year 1, we subtract the depreciation for the first 6 months from the purchase price:
$16,000 - $1,500 = $14,500.

Answer:

c

Step-by-step explanation:

C is your answer. Since this is a parallelogram, is means that there are two sets of sides with the same length. Because the measurement of angle K is 82 the angle directly opposite would have the same measurement. That's why angle M is also 82. When you add all the angles of a quadrilateral it adds up to 360 degrees. multiply 82 by 2 to get 164 and subtract that from 360 to get 196. you then have to divide that by 2 and you will get 98 which is the measurement for both angles L and N

Answer:

The correct answer is option C.

m<L = 98°, m<M = 82° and m<N = 98°

Step-by-step explanation:

From the figure we can see a parallelogram KLMN

Properties of parallelogram

1)Opposite sides are equal and parallel.

2) Opposite angles are equal.

3) Adjacent angles are supplementary.

To find the correct option

It is given that, m<K = 82°

By using properties of parallelogram we get

m<L = 98°, m<M = 82° and m<N = 98°

Therefore the correct answer is option C

Answer:

  3 mph

Step-by-step explanation:

Let c represent the current of the river in miles per hour. Then the ratio of speed downstream to speed upstream is ...

  (21 +c)/(21 -c) = 2.4/1.8

  1.8(21 +c) = 2.4(21 -c) . . . . . . multiply by 1.8(21-c)

  37.8 + 1.8c = 50.4 -2.4c . . . . eliminate parentheses

  4.2c = 12.6 . . . . . . . . . . . . . . . add 2.4c-37.8

  c = 3 . . . . . . . . . . . . . . . . . . . .divide by 4.2

The current of the river is 3 miles per hour.

Answer:

Y

Step-by-step explanation:

The answer is no, the result of the joint test for the null hypothesis that both [tex]\( \beta_1 = 0 \) and \( \beta_2 = 0 \)[/tex] is not necessarily implied by the results of two separate tests for each coefficient.

To understand why, let's consider the two scenarios:

1. Separate Tests: When we conduct two separate tests for [tex]\( \beta_1 = 0 \) and \( \beta_2 = 0 \)[/tex], we are looking at the significance of each predictor independently. We might find that neither [tex]\( \beta_1 \) nor \( \beta_2 \)[/tex] is significantly different from zero on its own. However, this does not account for the potential multicollinearity between [tex]\( X_1 \) and \( X_2 \)[/tex]. Multicollinearity can result in high variance of the coefficient estimates, leading to insignificant t-tests even if the predictors have a joint effect on the response variable.

2. Joint Test (F-test): The joint test, typically conducted using an F-test, assesses whether both [tex]\( \beta_1 \) and \( \beta_2 \)[/tex] are simultaneously equal to zero. This test takes into account the correlation between [tex]\( X_1 \) and \( X_2 \)[/tex] and evaluates the combined effect of both variables on the response variable. It is possible that while neither variable alone is significant, together they might have a significant effect.

The F-test for the joint hypothesis is based on the reduction in the sum of squared residuals when including [tex]\( X_1 \) and \( X_2 \)[/tex] in the model compared to a model with only the intercept (reduced model). The test statistic is calculated as:

[tex]\[ F = \frac{(\text{SSR}_{\text{reduced}} - \text{SSR}_{\text{full}}) / k}{\text{SSR}_{\text{full}} / (n - p - 1)} \][/tex]

where:

- [tex]\( \text{SSR}_{\text{reduced}} \)[/tex] is the sum of squared residuals from the reduced model.

- [tex]\( \text{SSR}_{\text{full}} \)[/tex] is the sum of squared residuals from the full model.

- [tex]\( k \)[/tex]is the number of restrictions (in this case, 2, since we are testing two coefficients).

- [tex]\( n \)[/tex] is the number of observations.

-  [tex]\( p \)[/tex] is the number of predictors in the full model (not including the intercept).

The degrees of freedom for the numerator are k and for the denominator are [tex]\( n - p - 1 \)[/tex].

In summary, the results from separate t-tests for [tex]\( \beta_1 \) and \( \beta_2 \)[/tex] do not necessarily inform us about the joint significance of these coefficients. It is entirely possible for the separate tests to show non-significance while the joint F-test shows significance, indicating that the predictors have a joint effect on the dependent variable even if their individual effects are not significant. Conversely, it is also possible for the separate tests to show significance for one or both coefficients, while the joint test does not show significance, suggesting that the combined effect of the predictors is not significant.