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The amount of time required for an oil and filter change on an automobile is normally distributed with a mean of 45 minutes and a standard deviation of 10 minutes. 1)If a random sample of 16 cars is selected, a) what is the probability that the sample mean will be between 39 and 48 minutes? b) 95% of all sample means will fall between what two values? c) 90% of the sample means will be greater than what value? 2)What sample size is needed in order for the standard deviation of the mean to be one fifth of the population standard deviation?
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Answer:
1a
[tex]P(39 < X < 48 ) = 0.8767[/tex]
1b
95% of all sample means will fall between [tex] 40.1 < \mu < 49.9 [/tex]
1c
[tex]\= x = 41. 795[/tex]
2
[tex]n = 25[/tex]
Step-by-step explanation:
From the question we are told that
The mean is [tex]n = 45[/tex]
The population standard deviation is [tex]\sigma = 10[/tex]
The sample size is n = 16
Generally the standard error of the mean is mathematically represented as
[tex]\sigma_{x} = \frac{ \sigma}{\sqrt{n} }[/tex]
=> [tex]\sigma_{x} = \frac{ 10 }{\sqrt{16 } }[/tex]
=> [tex]\sigma_{x} = 2.5[/tex]
Generally the probability that the sample mean will be between 39 and 48 minutes is
[tex]P(39 < X < 48 ) = P( \frac{ 39 - 45}{ 2.5} < \frac{X - \mu }{\sigma } < \frac{ 48 - 45}{ 2.5} )[/tex]
=> [tex]P(39 < X < 48 ) = P(-2.4 < Z< 1.2 )[/tex]
=> [tex]P(39 < X < 48 ) = P( Z< 1.2 ) - P(Z < -2.4)[/tex]
From the z table the area under the normal curve to the left corresponding to 1.2 and -2.4 is
=> [tex]P( Z< 1.2 ) = 0.88493[/tex]
and
[tex]P( Z< - 2.4 ) = 0.0081975[/tex]
So
[tex]P(39 < X < 48 ) = 0.88493 -0.0081975[/tex]
=> [tex]P(39 < X < 48 ) = 0.8767[/tex]
From the question we are told the confidence level is 95% , hence the level of significance is
[tex]\alpha = (100 - 95 ) \%[/tex]
=> [tex]\alpha = 0.05[/tex]
Generally from the normal distribution table the critical value of is
[tex]Z_{\frac{\alpha }{2} } = 1.96[/tex]
Generally the margin of error is mathematically represented as
[tex]E = Z_{\frac{\alpha }{2} } * \frac{\sigma }{\sqrt{n} }[/tex]
=> [tex]E = 1.96 * 2.5 [/tex]
=> [tex]E =4.9 [/tex]
Generally the 95% of all sample means will fall between
[tex]\mu -E < and \mu +E[/tex]
=> [tex]45 -4.9\ and \ 45 + 4.9[/tex]
Generally the value which 90% of sample means is greater than is mathematically represented
[tex]P( \= X > \= x ) = 0.90[/tex]
=> [tex]P( \= X > \= x ) = P( \frac{\= X - \mu }{ \sigma_x} > \frac{\= x -45 }{ 2.5} ) = 0.90[/tex]
=> [tex]P( \= X > \= x ) = P( Z > z ) = 0.90[/tex]
Generally from the z-table the critical value of 0.90 is
[tex]z = -1.282[/tex]
[tex]\frac{\= x -45 }{ 2.5} = -1.282[/tex]
=> [tex]\= x = 41. 795[/tex]
Considering question 2
Generally we are told that the standard deviation of the mean to be one fifth of the population standard deviation, this is mathematically represented as
[tex]s = \frac{1}{5} \sigma[/tex]
Generally the standard deviation of the sample mean is mathematically represented as
[tex]s = \frac{\sigma }{ \sqrt{n} }[/tex]
=> [tex]\frac{1}{5} \sigma = \frac{\sigma }{ \sqrt{n} }[/tex]
=> [tex]n = 5^2[/tex]
=> [tex]n = 25[/tex]
Using the normal distribution and the central limit theorem, it is found that:
1)
a) 0.8767 = 87.67% probability that the sample mean will be between 39 and 48 minutes.
b) 95% of all sample means will fall between 40.1 and 49.9 minutes.
c) 90% of the sample means will be greater than 41.8 minutes.
2) A sample size of 25 is needed.
In a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
It measures how many standard deviations the measure is from the mean. After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X. By the Central Limit Theorem, the sampling distribution of sample means of size n has standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].In this problem:
The mean is of 45 minutes, hence [tex]\mu = 45[/tex].The standard deviation is of 10 minutes, hence [tex]\sigma = 10[/tex].Sample of 16 cars, hence [tex]n = 16, s = \frac{10}{\sqrt{16}} = 2.5[/tex].Item 1, question a:
The probability is the p-value of Z when X = 48 subtracted by the p-value of Z when X = 39, hence:
X = 48:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{48 - 45}{2.5}[/tex]
[tex]Z = 1.2[/tex]
[tex]Z = 1.2[/tex] has a p-value of 0.8849.
X = 39:
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{39 - 45}{2.5}[/tex]
[tex]Z = -2.4[/tex]
[tex]Z = -2.4[/tex] has a p-value of 0.0082.
0.8849 - 0.0082 = 0.8767
0.8767 = 87.67% probability that the sample mean will be between 39 and 48 minutes.
Item b:
The critical value for a 95% confidence interval is |z| = 1.96, hence:
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]-1.96 = \frac{X - 45}{2.5}[/tex]
[tex]X - 45 = -1.96(2.5)[/tex]
[tex]X = 40.1[/tex]
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]1.96 = \frac{X - 45}{2.5}[/tex]
[tex]X - 45 = 1.96(2.5)[/tex]
[tex]X = 49.9[/tex]
95% of all sample means will fall between 40.1 and 49.9 minutes.
Item c:
Greater than the 10th percentile, which is X when Z has a p-value of 0.1, so X when Z = -1.28.
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]-1.28 = \frac{X - 45}{2.5}[/tex]
[tex]X - 45 = -1.28(2.5)[/tex]
[tex]X = 41.8[/tex]
90% of the sample means will be greater than 41.8 minutes.
Item 2:
This is n for which [tex]s = \frac{10}{5} = 2[/tex], hence:
[tex]s = \frac{\sigma}{\sqrt{n}}[/tex]
[tex]2 = \frac{10}{\sqrt{n}}[/tex]
[tex]2\sqrt{n} = 10[/tex]
Simplifying both sides by 2:
[tex]\sqrt{n} = 5[/tex]
[tex](\sqrt{n})^2 = 5^2[/tex]
[tex]n = 25[/tex]
A sample size of 25 is needed.
A similar problem is given at https://brainly.com/question/14789302
given the endpoint (-9,3) and (1,8) partition the segment into the ratio 2:3
Answers
Answer:
[tex]P = (-5,5)[/tex]
Step-by-step explanation:
Let the segments be A and B
[tex]A = (-9,3)[/tex]
[tex]B = (1,8)[/tex]
[tex]m:n = 2:3[/tex]
Required
Determine the endpoints after it is divided into ratios.
Let the point be represented with P
The formula to use is:
[tex]P = (\frac{mx_2 + nx_1}{m + n},\frac{my_2 + ny_1}{m + n})[/tex]
Where
[tex]A = (-9,3)[/tex] ----- [tex](x_1,y_1)[/tex]
[tex]B = (1,8)[/tex] ------ [tex](x_2,y_2)[/tex]
So:
[tex]P = (\frac{mx_2 + nx_1}{m + n},\frac{my_2 + ny_1}{m + n})[/tex]
[tex]P = (\frac{2 * 1 + 3 * -9}{2 + 3},\frac{2 * 8 + 3 * 3}{2 + 3})[/tex]
[tex]P = (\frac{-25}{5},\frac{25}{5})[/tex]
[tex]P = (-5,5)[/tex]
The perimeter of a rectangle is 138
meters and the area is 1080 square
meters. Find the length and the width of
the rectangle.
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y = 3
y = –2
y = 2
y = 0
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y = –10x − 8
Answers
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Step-by-step explanation: because the formula is mx+b=y and the -8 is the b
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a. (10, 24, 26)
b. (5, 12, 24)
C. (12, 18, 30)
d. (1, 2,9)
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The answer is B because it has to have a right angle in it
Please help answer 2 and 3 the best you possibly can
Answers
Answer:
2. x = 3
3. x = 5
Answer:
Below.
Step-by-step explanation:
(2). 6x + 51 + 111 = 180
6x = 180 - 162 = 18
x = 3.
(3). 6x = 30
x = 5.
resolver
4×3
____
7+5
Answers
Answer:
THEY ARE EQUAL =
Step-by-step explanation:
The unshaded regions are quarter circles. Which choice best approximates the area of the shaded region? Use ≈3.14.
Answers
Answer:
123.6cm^2
Step-by-step explanation:
24^2=576
area of all four quarter circles combined: 144pi
576-122pi
f(a) = a^4- 6a² - 10a - 4 at a=-2
Answers
Step-by-step explanation:
I'm sorry I don't know the answer i know your mad because I'm writing this
Solve: 2y - 3.1 when y = 3.41
Answers
Answer:
3.72
Step-by-step explanation
2x3.42=6.82
6.82-3.1=3.72
Answer:
First, substitute y by the value
2(3.41)-3.1
Multiply the bracket by 2
It will be 0.58651026393, so you’ll round it up to make it shorter to make it easy for calculation
0.59-3.1=-2.51
-2.51 is the answer to your question
Step-by-step explanation:
I hope this helped, please give me brainliest
What is the equation in slope-intercept form of the line that passes through the points (0,-4) and (2,0)
Answers
Answer:
m=2
Step-by-step explanation:
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Which is better 32 miles or 12.7 kilometers and why
Answers
Answer:
32 miles
Step-by-step explanation:
32 miles is equal to 51.499008 Kilometers
12.7 kilometers is equal to 7.891414 miles
At the local harvest festival Marty bought 2 pumpkin pie slices and 2 cups of hot chocolate $10.00. Patty bought 4 slices of pie and 5 cups of hot chocolate for $22.50. How much is a slice of pumpkin pie and how much is a cup of hot chocolate?
Answers
Answer:
1 cup of hot chocolate is $2.5
1 slice of pumpkin pie is $2.5
Step-by-step explanation:
Represent Pumpkin Pie with P and Hot Chocolate with C:
Marty:
[tex]2P + 2H = 10[/tex]
Patty:
[tex]4P + 5H = 22.5[/tex]
So, we have the following set of equation:
[tex]2P + 2H = 10[/tex] -------- (1)
[tex]4P + 5H = 22.5[/tex] -------- (2)
Multiply the first equation by 2
[tex]2(2P + 2H = 10)[/tex]
[tex]4P + 4H = 20[/tex] -------- (3)
Subtract (3) from (2)
[tex]4P + 5H = 22.5[/tex]
[tex]4P + 4H = 20[/tex]
--------------------------
[tex]5H - 4H = 22.5 - 20[/tex]
[tex]H = 2.5[/tex]
Substitute 2.5 for H in [tex]2P + 2H = 10[/tex]
[tex]2P + 2 * 2.5 = 10[/tex]
[tex]2P + 5 = 10[/tex]
[tex]2P = 10 - 5[/tex]
[tex]2P = 5[/tex]
Divide through by 2
[tex]P = \frac{5}{2}[/tex]
[tex]P=2.5[/tex]
Hence, 1 cup of hot chocolate is $2.5 and 1 slice of pumpkin pie is $2.5
The sum of two fractions is 1. If one fraction is 3/8 greater than the other, what are the two fractions?
Answers
Answer:
Well it could be a number of fractions. It could be 1/8 and 4/8, or 2/8 and 3/8.
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Answers
Answer:
increases
Step-by-step explanation:
on on the side, it looks like you need to enter stuff into the table. All you need to do is copy the numbers from the table and transfer them to the empty one.
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Write your answer in simplest form.
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Answer:
hey so the answer is m=-4
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Answers
Answer:
16.1
Step-by-step explanation:
Can someone please help me with this?!!
Answers
Answer:
x=19
Step-by-step explanation:
x+8=2x-11
-11+11=0
8+11=19
x-x=0
2x-x=x
x=19
Trevor made this number pattern 2,9,16,23,30,37 if the pattern continues what will be the eight number be
Answers
Answer:
44
Step-by-step explanation:
Rewrite the equation in Standard Form: 2=y-2x
Answers
Answer:
Standard Form: 2x - y = -2
By which rule are these triangles congruent?
Answers
Answer:
AAS (Angle-Angle-Side)
Step-by-step explanation:
Topic: congruence and similarity
If you like to venture further, feel free to check out my insta (learntionary). I'll be constantly posting math tips and notes! Thanks!
Answer:
Angle Angle Side
Step-by-step explanation:
You can see you have two angles next to each other and one side next to that so just see which proof it fits
Hope This Helps
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Answers
Answer:
The answer for the question is 11 and 12 I just got a test back and it was correct
Step-by-step explanation:
Answer:
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Step-by-step explanation:
127.86
+78.95
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Answers
Answer:
229.9
Step-by-step explanation:
Answer:
229.9
Step-by-step explanation:
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Parallelograms pre test on edg 2020
!ANSWERS !
1.c
2.c
3.a
4.b
5.c
6.c
7.d
8.d
9.d
10.c
You’re welcome ;)
Answers
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A.-3,3
B.3
C.no solution
D.-3
Answers
Answer: b
Step-by-step explanation:Step 1: Simplify both sides of the equation.
6x=x2+9
Step 2: Subtract x^2+9 from both sides.
6x−(x2+9)=x2+9−(x2+9)
−x2+6x−9=0
Step 3: Factor left side of equation.
(−x+3)(x−3)=0
Step 4: Set factors equal to 0.
−x+3=0 or x−3=0
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Answers
Answer:
Simplify the solution: 13^1d^3
Step-by-step explanation:
Any expression raised to the power of one equal itself.
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Click to review the online content. Then answer the question(s) below, using complete sentences. Scroll down to view additional questions. Online Content: Site 1 When graphing inequalities, where can the solutions be found?
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Answer:
what do u need help with