Answer:
Cohen's D
Step-by-step explanation:
Cohen's D is a statistic that measures effect size. It shows standardised difference between 2 means.
Effect size is defined as how large the effect of a something is or its magnitude.
Cohen's D works effectively when the sample is >50 (that is for large samples). However a correction factor can be used to make results from small samples more accurate
The formular for Cohen's D is:
D = (mean1 - mean2) ÷ (√({standard deviation1}^2 + {standard deviation 2}^2)/2)
This is the most appropriate method in the given scenario
What best explains whether a triangle with side links 5 cm 13 cm and 12 cm is a right triangle
Step-by-step explanation:
Pythagoras Theorem
If the sum of the squares of the smaller two sides is equal to the square if the third side then it is a right triangle
[tex] {a}^{2} + {b}^{2} = {c}^{2} [/tex]
So, (5)^2 + (12)^2
is 25 + 144 = 169
Which is equal to (13)^2 which is also 169
The sides of the given triangle follows pythagoras theorem, therefore it is a right triangle
Hope it helps:)
Answer:
Pythagorean theorem
Step-by-step explanation:
We can explain it using the Pythagorean theorem. Right triangles always have a hypotenuse which is the longest side. That means 13 must be the hypotenuse of the triangle. The Pythagorean theorem is a^2+b^2=c^2
We already know all the values since every side is given so we just fill it in.
5^2+12^2=13^2
25+144=169
169=169
It is a right triangle
Volume 1 (3)3 = 367
SSCE/JME-TYPE OF
2
The area of an equilateral triangle of side 8 cm is
A. 16V3 cm? B. 32/3 cm
B.
48 cm
cm?
D.
36V3 cm
A
parallelogram
of area 425 cmhas a height o
Answer:
[tex]A.\ 16\sqrt3\ cm^2[/tex] is the correct answer.
Step-by-step explanation:
Given that:
Side of an equilateral triangle = 8 cm
To find:
Area of the triangle will be:
[tex]A.\ 16\sqrt3\ cm^2[/tex]
[tex]B.\ \dfrac{32}{3} cm^2[/tex]
[tex]C.\ 48\ cm^2[/tex]
[tex]D.\ 36\sqrt3\ cm^2[/tex]
Solution:
First of all, let us have a look at the formula for area of an equilateral triangle:
[tex]A =\dfrac{\sqrt3}{4}a^2[/tex]
Where [tex]a[/tex] is the side of equilateral triangle and an equilateral triangle is a closed 3 sided structure in 2 dimensions which has all 3 sides equal to each other.
Here, we are given that side, [tex]a=8\ cm[/tex]
Putting the value in formula:
[tex]A =\dfrac{\sqrt3}{4}\times 8^2\\\Rightarrow A =\dfrac{\sqrt3}{4}\times 64\\\Rightarrow A =\sqrt3\times 16\\OR\\\Rightarrow \bold{A =16\sqrt3\ cm^2}[/tex]
Hence, [tex]A.\ 16\sqrt3\ cm^2[/tex] is the correct answer.
Question 1: A triangle has sides with lengths 5, 6, and 7. Is the triangle right, acute, or obtuse?
A)Right
B)Obtuse
C)Can't be determined
D) Acute
Question 2: A 15-foot statue casts a 20-foot shadow. How tall is a person who casts a 4-foot-long shadow?
A)0.33 feet
B)3.75 feet
C)3 feet
D)5 feet
Question 3: A triangle has sides with lengths 17, 12, and 9. Is the triangle right, acute, or obtuse?
A)Acute
B)Right
C)Can't be determined
D)Obtuse
Question 4: Two friends are standing at opposite corners of a rectangular courtyard. The dimensions of the courtyard are 12 ft. by 25 ft. How far apart are the friends?
A)21.34 ft.
B)21.93 ft.
C)27.73 ft.
D)19.21 ft.
Answer:
Question 1 = D) Acute
Question 2 = C)3 feet
Question 3 = D) Obtuse
Question 4 = C)27.73 ft.
Step-by-step explanation:
Question 1: A triangle has sides with lengths 5, 6, and 7. Is the triangle right, acute, or obtuse?
In order to be able to accurately classify that a triangle with 3 given sides is either a right , acute or obtuse angle, we use the Pythagoras Theorem
Where:
If a² + b² = c² = Right angle triangle
If a² +b² > c² = Acute triangle.
If a² +b² < c² = Obtuse triangle.
It is important to note that the length ‘‘c′′ is always the longest.
Therefore, for the above question, we have lengths
5 = a, 6 = b and c = 7
a² + b² = c²
5² + 6² = 7²
25 + 36 = 49
61 = 49
61 ≠ 49, Hence 61 > 49
Therefore, this is an Acute Triangle
Question 2: A 15-foot statue casts a 20-foot shadow. How tall is a person who casts a 4-foot-long shadow?
This is question that deals with proportion.
The formula to solve for this:
Height of the statue/ Length of the shadow of the person = Height of the person/ Length of the shadow of the person
Height of the statue = 15 feet
Length of the shadow of the person = 20 feet
Height of the person = unknown
Length of the shadow of the person = 4
15/ 20 = Height of the person/4
Cross Multiply
15 × 4 = 20 × Height of the person
Height of the person = 15 × 4/20
= 60/20
Height of the person = 3 feet
Therefore, the person is 3 feet tall.
Question 3: A triangle has sides with lengths 17, 12, and 9. Is the triangle right, acute, or obtuse?
In order to be able to accurately classify that a triangle with 3 given sides is either a right , acute or obtuse angle, we use the Pythagoras Theorem
Where:
If a² + b² = c² = Right angle triangle
If a² +b² > c² = Acute triangle.
If a² +b² < c² = Obtuse triangle.
It is important to note that the length ‘‘c′′ is always the longest.
Therefore, for the above question, we have lengths 17, 12, 9
9 = a, 12 = b and c = 17
a² + b² = c²
9² + 12² = 17²
81 + 144 = 289
225 = 289
225 ≠ 289
225 < 289
Hence, This is an Obtuse Triangle.
Question 4: Two friends are standing at opposite corners of a rectangular courtyard. The dimensions of the courtyard are 12 ft. by 25 ft. How far apart are the friends?
To calculate how far apart the two friends are we use the formula
Distance = √ ( Length² + Breadth²)
We are given dimensions: 12ft by 25ft
Length = 12ft
Breadth = 25ft
Distance = √(12ft)² + (25ft)²
Distance = √144ft²+ 625ft²
Distance = √769ft²
Distance = 27.730849248ft
Approximately ≈27.73ft
Therefore, the friends are 27.73ft apart.
Justine and Meagan played a trivia game. Justine answered a question incorrectly and lost 7 points. Then Meagan answered correctly and got the opposite score. Which is the correct way to represent that “the opposite of Justine’s score was equal to Meagan’s score
Answer:
[tex]m = -j[/tex], or in this case, [tex]m=-(-7)[/tex]
Step-by-step explanation:
Assuming that Justine's score is represented by [tex]j[/tex] and Meagan's score is represented by [tex]m[/tex], we know that [tex]j[/tex] will always be the opposite of [tex]m[/tex].
To represent opposite, we put a negative sign before the variable.
This makes the current number, even if it's negative, the opposite value.
Let's test it out.
Since Justine's score is -7, substituting it into the equation makes it [tex]m=-(-7)[/tex]. We know that two negatives make a positive, so [tex]m=7[/tex].
Now let's assume Justine's score is 7. Plugging it into the equation, we get [tex]m=-(7)[/tex]. That's the same thing as [tex]m = -1(7)[/tex], and -1 times 7 is -7.
Hope this helped!
By using words per sentence and characters per words, we can create formulas for the Flesch-Kincaid grade level readability. The readability statistics for random pages from a Harry Potter novel are given below.
Words/Sentence 15.7 9 16.3 14.5 9.7 7.4 14 16.1 13.9 12.5 17.2 11.5
Characters/Word 41 4.2 4.2 4.3 4.3 4.2 4.5 4.5 4.3 4 4.4 4.3
F-K Grade Level 5.2 3.7 6.1 4.9 4.4 3.2 5.6 6.9 5.7 4.1 6.7 4.4
What is the correct analysis statement for the coefficient of determination between words per sentence (x-variable) and characters per word (y-variable)?
a. 1.5% of the variance in characters per word is accounted for by words per sentence.
b. 8% of the variance in characters per word is accounted for by words per sentence.
c. 10% of the variance in characters per word is accounted for by words per sentence.
d. 31% of the variance in characters per word is accounted for by words per sentence.
e. None of these
Answer:
d. 31% of the variance in characters per word is accounted for by words per sentence.
Step-by-step explanation:
The coefficient of determination is the statistical measurement that determines the impact in difference of one variable which is explained by the difference in another variable. It is denoted by [tex]R^{2}[/tex] or R-squared. It helps in predicting the outcome of the event.
What is 45x62 Please help.
Answer:
45
62x
______
90
2700+
_________
2790
Step-by-step explanation:
Can your help me please?
Answer:
(-5, 0) and (0,4)
Step-by-step explanation:
Given equation: -4x + 5y = 20
Sub. in the values
When x = -5 and y = 0 (-5,0),
[tex] - 4( - 5) + 5(0) = 20 [/tex]
When x = 0 and y = 4 (0,4),
[tex] - 4(0) + 5(4) = 20[/tex]
That's how I would do it, not sure if your school has another method. Hope this helps :)
Answer: x = -5 and y = 4
Step-by-step explanation: its the first option do you need me to exlain how cuz its multiple choice
I need help with this math problem
Answer:
1). [tex]\frac{2}{x^{2}-x-12 }=\frac{2}{(x+3)(x-4)}[/tex]
2). [tex]\frac{1}{x^{2}-16 }=\frac{1}{(x-4)(x+4)}[/tex]
Step-by-step explanation:
In this question we have to write the fractions in the factored form.
Rational expressions are [tex]\frac{2}{x^{2}-x-12 }[/tex] and [tex]\frac{1}{x^{2}-16 }[/tex].
1). [tex]\frac{2}{x^{2}-x-12 }[/tex]
Factored form of the denominator (x² - x - 12) = x² - 4x + 3x - 12
= x(x - 4) + 3(x - 4)
= (x + 3)(x - 4)
Therefore. [tex]\frac{2}{x^{2}-x-12 }=\frac{2}{(x+3)(x-4)}[/tex]
2). [tex]\frac{1}{x^{2}-16 }[/tex]
Factored form of the denominator (x² - 16) = (x - 4)(x + 4)
[Since (a²- b²) = (a - b)(a + b)]
Therefore, [tex]\frac{1}{x^{2}-16 }=\frac{1}{(x-4)(x+4)}[/tex]
Which system of linear inequalities has the point (3, –2) in its solution set?
Answer:
see below
Step-by-step explanation:
We want where both inequalities are true
y > -3
-2 >-3 this is true
y ≥ 2/3x -4
-2≥ 2/3*3 -4
-2 ≥ 2 -4
-2≥ -2
This is true
This is is the graph
Answer:
[tex]\boxed{\sf Option \ 3}[/tex]
Step-by-step explanation:
[tex]\sf The \ values \ must \ be \ true \ for \ both \ inequalities.[/tex]
[tex]x = 3\\y = -2[/tex]
[tex]y>-3\\-2>-3\\ \sf True[/tex]
[tex]y\geq \frac{2}{3}x-4 \\ -2\geq \frac{2}{3}(3)-4\\2\geq 2-4\\-2\geq-2 \\ \sf True[/tex]
Given that f(x) = x + 4 and g(x) = x + 7, find (g - 4(x).
Answer: The value of [tex](g - f)(x)=4 .[/tex]
Step-by-step explanation:
Given functions : [tex]f(x) = x + 4[/tex] and [tex]g(x) = x + 7[/tex]
To find : [tex](g - f)(x)[/tex]
Difference between two functions: [tex](u-v)(x)=u(x)-v(x)[/tex]
Then, [tex](g-f)(x)=g(x)-f(x)[/tex]
[tex]=(x+7)-(x+4)=x+7-x-4\\\\=7-4=3[/tex]
Hence, the value of [tex](g - f)(x)=4 .[/tex]
Simplify the expression. Write the answer using scientific notation.
(5x107)(6x104)
A) 1.1 x 1012
B) 3.0x 1029
C) 3.0 x 1012
D) 1.1 x 1029
Answer:
3* 10 ^12
Step-by-step explanation:
(5x10^7)(6x10^4)
Multiply the numbers together
5*6 =30
Add the exponents
10^7 * 10 ^ 4 = 10 ^(7+4) = 10 ^11
30 * 10 ^11
But this is not scientific notation since 30 >10
Move the decimal one place to the left and add 1 to the exponent
3* 10 ^12
Answer:
3* 10 ^12
Step-by-step explanation:
Johnny and Steven ate a 12-piece pizza. If Johnny ate 3/4 of the pizza, how many pieces did Steven eat? *
Answer:
Steven ate 3 pieces
Step-by-step explanation:
If Johnny ate 3/4 , then Steven at 1 - 3/4 or 1/4
12 * 1/4 = 3
Steven ate 3 pieces
Answer:
3 slices of pizza
Step-by-step explanation:
There are 12 total slices of pizza. In order to find how much Johnny ate, we must multiply 12 by 3/4.
12/1 × 3/4 OR 12 × 0.75 = 9
Johnny ate 9 slices of pizza.
Then, we have to subtract 9 from 12 to determine how many slices Steven ate.
12 - 9 = 3
Steven ate 3 slices of pizza.
A librarian needs to package up all of the children's books and move them to a different location in the library. There are 625 books, and she can fit 25 books in one box. How many boxes does she need in order to move all of the books? 5 B. 25 C. 125 D. 600 E. 650
Answer: B. 25
Step-by-step explanation:
Given: Total books = 625
Number of books can fit in one box = 25
Now, the number of boxes she need to move all of the books = (Total books) ÷ (Number of books can fit in one box )
= 625÷25
= 25
hence, she requires 25 boxes in order to move all of the books.
So, correct option is B. 25.
If the circumference of a circle is equal to 14, what is the diameter?
Answer:
7
Step-by-step explanation:
Please help!!! A calculator was used to perform a linear regression on the values in the table. The results are shown to the right of the table.What is the line of best fit?A.y = –0.984x + 13.5B.y = –2.9x + 13.5C.–0.984 = –2.9x + 13.5D.y = 13.5x – 2.9
Answer:
B.y = –2.9x + 13.5 i think dont word me for it
Step-by-step explanation:
Equation of line is y = –2.9x + 13.5 which is correct option(B).
What is linear equation?A linear equation is defined as an equation in which the highest power of the variable is always one.
Let y = a + bx, where a is the y-intercept and b is the slope.
[tex]\sum{x} = 1 + 2 + 3 + 4 + 5 = 15[/tex]
[tex]\sum{y} = 11 + 8 + 4+ 1 + 0 = 24[/tex]
[tex]\sum {xy} = 11 + 16 + 12 + 4 + 0 = 43[/tex]
[tex]\sum{x^2} = 1 + 4 + 9 + 16 + 25 = 55[/tex]
[tex]n = 5[/tex]
[tex]b= \dfrac{n\sum{xy}-\sum{x}\sum{y}}{n{\sum{x^2}-(\sum{x}})^2}[/tex]
[tex]b= \dfrac{ 5 (43) - (15) . (24)}{{5 (55) - (15})^2}[/tex]
[tex]b= -2.9[/tex]
[tex]a = \dfrac{25}{5} - (-2.9)\dfrac{15}{5}[/tex]
[tex]a=13.5[/tex]
Hence, equation of line is y = –2.9x + 13.5
Learn more about linear equation here:
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A rotating light is located 16 feet from a wall. The light completes one rotation every 2 seconds. Find the rate at which the light projected onto the wall is moving along the wall when the light's angle is 20 degrees from perpendicular to the wall.
Answer:
a
Step-by-step explanation:
answer is a on edg
The length of a rectangle is increasing at a rate of 9 cm/s and its width is increasing at a rate of 7 cm/s. When the length is 12 cm and the width is 5 cm, how fast is the area of the rectangle increasing?
Answer:
129 m/s^2
Step-by-step explanation:
The length of a rectangle is increasing at a rate of 9m/s
dL/dt = 9m/s
The width is increasing at a rate of 7m/s
dw/dt= 7m/s
The formular for solving the area of a rectangle is length × width
Therefore, to calculate how fast the rectangle is increasing we will apply the product rule
dA/dt= L × dw/dt + W × dl/dt
= 12×7 + 5×9
= 84+45
= 129m/s^2
Hence the area of the rectangle is increasing at 129m/s^2
do numbers ever stop
Nope, i dont think so
Answer:
no the numbers are infinite
Step-by-step explanation:
The manager of the video department at a department store plans to purchase a large number of DVDs of a recent movie. One supplier is selling boxes of 20 DVD movies for $240, and a second supplier is selling boxes of 14 DVD movies for $170. Only complete boxes of DVD movies can be purchased. Complete part a) and b) below. a)
a) If the manager can purchase boxes of DVD movies from either or both suppliers, determine the maximum number of DVD movies that can be purchased for $415. Indicate how many boxes of 20 and how many boxes of 14 will be purchased.
— box(es) of 20 and — box(es) of 14
b) How much will the DVD movies cost?
They will cost $—
Answer:
1 box of 20 and 1 box of 14
They will cost $410
Step-by-step explanation:
1. Find how many boxes of 20 DVD movies can be bought
415 - 240 = 175
1 box of 20 DVD movies can be sold
2. Find how many boxes of 14 DVD movies can be bought from $175
175 - 170 = 5
1 box of 14 DVD movies can be bought
3. Find the cost
240 + 170 = 410
could I please have some help
Find the value of x.
Answer:
6x + 6 = 32
6x = 32 - 6
6x = 26
divide both sides by 6
6x/6 = 26/6
6x + 6 = 4.35
9x - 9 = 24
9x = 24 + 9
9x = 33
divide both sides by 9
9x/9 = 24/9
9x + 9 = 2.66
9x + 9 = 2.66
Answer: x=3
Step-by-step explanation:
[tex]\frac{32}{24} =\frac{4}{3} \\\\\frac{4}{3}=\frac{6x+6}{9x-9}\\ x=3[/tex]
In the expression 3x2 + y − 5, which of the following choices is the exponent in the term 3x2?
Answer:
2
Step-by-step explanation:
The term 3x² has 2 as an exponent, the correct option is C.
What are Exponents?Exponents are the base raised by power, it is written in the superscript of a number.
The expression is 3x² +y-5
The term 3x² has 2 as an exponent.
Therefore, the correct option is C.
The missing options are
A.3x2
B. y
A 2
C. -5
D. None of these choices are correct.
To know more about Exponents
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Vu is three times as old as Wu. In 25 years Wu will be twice as old as Vu. How old is Vu now?
Answer: Vu is 15 years old now.
Step-by-step explanation:
Let present age of WU be x.
Then, the present age of Vu = 3x
Also, After 25 years
Age of Wu = x+25
According to the question:
[tex](x+25)=2(3x)\\\\\Rightarrow\ x+25=6x\\\\\Rightarrow\5x=25\\\\\Rightarrow\ x=5[/tex]
Present age of Vu = 3(5) = 15
Hence, Vu is 15 years old now.
Answer:
j
Step-by-step explanation:j
For the equation ax+c =bx +d where a≠b and c≠d , what is x expressed in terms of a,b,c, and d?
Answer:
x = (d - c) / (a - b)
Step-by-step explanation:
Let's simply isolate and solve for the variable x.
ax + c = bx + d
ax - bx + c = d
x (a - b) = d - c
x = (d - c) / (a - b)
Thus, we have expressed x in terms of a,b,c, and d.
Cheers.
:)
The value of x in terms of a,b,c, and d will be x = (d - c) / (a - b).
What is an arithmetic operation?It is described as the process through which we add, subtract, multiply, and divide numerical values. It has the fundamental operators +, -, ×, and ÷.
The order in which arithmetic operations must be performed in an equation is referred to as PEMDAS. This rule states that operations must be performed as follows: parentheses, exponents, multiplication, or division, followed by addition or subtraction.
It is given that,
ax+c =bx +d
a≠b and c≠d
Rearrange the equation and solve it for the x in the following steps,
ax - bx + c = d
x (a - b) = d - c
x = (d - c) / (a - b)
Thus,the value of x in terms of a,b,c, and d will be x = (d - c) / (a - b).
Learn more about the arithmetic operation here:
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After basketball practice, Courtney spent extra time working on free throws. Overall, she made 3 of her 5 shots. What percent of her total shots did she make?
Answer:
I know the answer
Step-by-step explanation:
Courtney made 60% of her shots
Answer:
she made a 60%
Step-by-step explanation:
first you do this 3÷5
that gives you .60
inorder to get a percent multiply by 100
that gives you an end result of 60
add the percent sing and your done 60%
hope i helped
Help us plazz this is mathematics IGCSE fast as you can
Answer:
Step-by-step explanation:
y varies direcrtly with √(x+5) wich can be expressed mathematically as:
● y = k*√(x+5)
Let's calculate k khowing that y=4 and x=-1
● 4 = k*√(-1+5)
● 4 = k*√(4)
● 4 = k * 2
● k = 4/2
● k = 2
■■■■■■■■■■■■■■■■■■■■■■■■■■
Let's calculate y khowing that x = 11
● y = k*√(x+5)
● y = 2×√(11+5)
● y = 2× √(16)
● y = 2× 4
● y = 8
Answer:
The value of y is 8.
Step-by-step explanation:
Given that y is directly proportional to √(x+5) so the equation is y = k√(x+5) where k is constant. First, you have to find the value of k with given values :
[tex]y = k \sqrt{x + 5} [/tex]
[tex]let \: x = - 1,y = 4[/tex]
[tex]4 = k \sqrt{ - 1 + 5} [/tex]
[tex]4 = k \sqrt{4} [/tex]
[tex]4 = k(2)[/tex]
[tex]4 \div 2 = k[/tex]
[tex]k = 2[/tex]
So the equation is y = 2√(x+5). In order to find the value of y, you have to substitute x = 11 into the equation :
[tex]y = 2 \sqrt{x + 5} [/tex]
[tex]let \: x = 11[/tex]
[tex]y = 2 \sqrt{11 + 5} [/tex]
[tex]y = 2 \sqrt{16} [/tex]
[tex]y = 2(4)[/tex]
[tex]y = 8[/tex]
For a trip, Eli packed 3 shirts, 3 pairs of pants, and 2 pairs of shoes. How many different outfits can Eli make? A. 6 outfits B. 8 outfits C. 9 outfits D. 18 outfits Please include ALL work!
Answer:
18 outfits
Step-by-step explanation:
To determine the number of outfits available
3 shirts * 3 pairs of pants * 2 pairs of shoes
3*3*2
18
Hellllppp!!!! Please!Match the numbers with the correct label.
Answer:
(a = 1/7 (b = .2 (c = 3/9
Step-by-step explanation:
1/7 = .14
1/4 = .25
3/9 = .33
a & b are lower than 1/4 and c is higher
If f(x)=ax+b and f(2)=1 and f(-3)=11, what is the value of A?
Answer:
a = -2
Step-by-step explanation:
f(x)=ax+b
f(2)=1
f(-3)=11
f(2) = 1 means 2a+b =1
f(-3)=11 means -3a + b = 11
Subtracting the two equations
-(-3a +b =11) becomes 3a -b = -11 so we can add
2a+b =1
3a - b = -11
----------------------
5a = -10
Divide by 5
5a/5 = -10/5
a=-2
The heat evolved in calories per gram of a cement mixture is approximately normally distributed. The mean is thought to be 100, and the standard deviation is 2. You wish to test H0: μ = 100 versus H1: μ ≠ 100 with a sample of n = 9 specimens.
A. If the acceptance region is defined as 98.5 le x- 101.5, find the type I error probability alpha.
B. Find beta for the case where the true mean heat evolved is 103.
C. Find beta for the case where the true mean heat evolved is 105. This value of beta is smaller than the one found in part (b) above. Why?
Answer:
A.the type 1 error probability is [tex]\mathbf{\alpha = 0.0244 }[/tex]
B. β = 0.0122
C. β = 0.0000
Step-by-step explanation:
Given that:
Mean = 100
standard deviation = 2
sample size = 9
The null and the alternative hypothesis can be computed as follows:
[tex]\mathtt{H_o: \mu = 100}[/tex]
[tex]\mathtt{H_1: \mu \neq 100}[/tex]
A. If the acceptance region is defined as [tex]98.5 < \overline x > 101.5[/tex] , find the type I error probability [tex]\alpha[/tex] .
Assuming the critical region lies within [tex]\overline x < 98.5[/tex] or [tex]\overline x > 101.5[/tex], for a type 1 error to take place, then the sample average x will be within the critical region when the true mean heat evolved is [tex]\mu = 100[/tex]
∴
[tex]\mathtt{\alpha = P( type \ 1 \ error ) = P( reject \ H_o)}[/tex]
[tex]\mathtt{\alpha = P( \overline x < 98.5 ) + P( \overline x > 101.5 )}[/tex]
when [tex]\mu = 100[/tex]
[tex]\mathtt{\alpha = P \begin {pmatrix} \dfrac{\overline X - \mu}{\dfrac{\sigma}{\sqrt{n}}} < \dfrac{\overline 98.5 - 100}{\dfrac{2}{\sqrt{9}}} \end {pmatrix} + \begin {pmatrix}P(\dfrac{\overline X - \mu}{\dfrac{\sigma}{\sqrt{n}}} > \dfrac{101.5 - 100}{\dfrac{2}{\sqrt{9}}} \end {pmatrix} }[/tex]
[tex]\mathtt{\alpha = P ( Z < \dfrac{-1.5}{\dfrac{2}{3}} ) + P(Z > \dfrac{1.5}{\dfrac{2}{3}}) }[/tex]
[tex]\mathtt{\alpha = P ( Z <-2.25 ) + P(Z > 2.25) }[/tex]
[tex]\mathtt{\alpha = P ( Z <-2.25 ) +( 1- P(Z < 2.25) })[/tex]
From the standard normal distribution tables
[tex]\mathtt{\alpha = 0.0122+( 1- 0.9878) })[/tex]
[tex]\mathtt{\alpha = 0.0122+( 0.0122) })[/tex]
[tex]\mathbf{\alpha = 0.0244 }[/tex]
Thus, the type 1 error probability is [tex]\mathbf{\alpha = 0.0244 }[/tex]
B. Find beta for the case where the true mean heat evolved is 103.
The probability of type II error is represented by β. Type II error implies that we fail to reject null hypothesis [tex]\mathtt{H_o}[/tex]
Thus;
β = P( type II error) - P( fail to reject [tex]\mathtt{H_o}[/tex] )
∴
[tex]\mathtt{\beta = P(98.5 \leq \overline x \leq 101.5) }[/tex]
Given that [tex]\mu = 103[/tex]
[tex]\mathtt{\beta = P( \dfrac{98.5 -103}{\dfrac{2}{\sqrt{9}}} \leq \dfrac{\overline X - \mu}{\dfrac{\sigma}{n}} \leq \dfrac{101.5-103}{\dfrac{2}{\sqrt{9}}}) }[/tex]
[tex]\mathtt{\beta = P( \dfrac{-4.5}{\dfrac{2}{3}} \leq Z \leq \dfrac{-1.5}{\dfrac{2}{3}}) }[/tex]
[tex]\mathtt{\beta = P(-6.75 \leq Z \leq -2.25) }[/tex]
[tex]\mathtt{\beta = P(z< -2.25) - P(z < -6.75 )}[/tex]
From standard normal distribution table
β = 0.0122 - 0.0000
β = 0.0122
C. Find beta for the case where the true mean heat evolved is 105. This value of beta is smaller than the one found in part (b) above. Why?
[tex]\mathtt{\beta = P(98.5 \leq \overline x \leq 101.5) }[/tex]
Given that [tex]\mu = 105[/tex]
[tex]\mathtt{\beta = P( \dfrac{98.5 -105}{\dfrac{2}{\sqrt{9}}} \leq \dfrac{\overline X - \mu}{\dfrac{\sigma}{n}} \leq \dfrac{101.5-105}{\dfrac{2}{\sqrt{9}}}) }[/tex]
[tex]\mathtt{\beta = P( \dfrac{-6.5}{\dfrac{2}{3}} \leq Z \leq \dfrac{-3.5}{\dfrac{2}{3}}) }[/tex]
[tex]\mathtt{\beta = P(-9.75 \leq Z \leq -5.25) }[/tex]
[tex]\mathtt{\beta = P(z< -5.25) - P(z < -9.75 )}[/tex]
From standard normal distribution table
β = 0.0000 - 0.0000
β = 0.0000
The reason why the value of beta is smaller here is that since the difference between the value for the true mean and the hypothesized value increases, the probability of type II error decreases.
