Given a test that is normally distributed with a mean of 100 and a standard deviation of 12, find:

(a) the probability that a single score drawn at random will be greater than 110

(b) the probability that a sample of 25 scores will have a mean greater than 105

(c) the probability that a sample of 64 scores will have a mean greater than 105

(d) the probability that the mean of a sample of 16 scores will be either less than 95 or greater than 105

Answers

Answer 1

Answer:

(a)  0.2033

(b)  0.0188

(c)  0.0004

(d)  0.095

Step-by-step explanation:

(a) the probability that a score at random is greater than 110 is obtained with a normal distribution of mean 100 and standard deviation 12 can be estimated using the z-table for Z = (110 -  100)/12 = 0.83

So P (X > 110) = P (Z > 0.83) = 0.2033

(b) Probability that a sample of 25 scores will have a mean greater than 105:

we use a standard distribution with the same mean (100) but the standard distribution reduced by a factor of [tex]\sqrt{25} = 5[/tex]. That is a standard deviation of 12/5 = 2.4. which gives a Z-value of (105-100) / 2.4 = 2.08

P (X> 105) = P (Z > 2.08) = 0.0188

(c) Probability that a sample of 64 scores will have a mean greater than 105:

we use a standard distribution with the same mean (100) but the standard deviation reduced by a factor of [tex]\sqrt{64} = 8[/tex]. That is a standard deviation of 12/8 = 1.5. which gives a Z-value of (105-100) / 1.5 = 3.33

P (X> 105) = P (Z > 3.33) = 0.0004

(d) the probability that the mean of a sample of 16 scores will be either less than 95 or greater than 105

This will be the addition of the two probabilities. We use a standard distribution with the same mean (100) but the standard distribution reduced by a factor of [tex]\sqrt{16} = 4[/tex]. That is a standard deviation of 12/4 = 3. which gives us two different Z values to study:

(105-100) / 3 = 1.67

and for X= 95 ==> Z = (95 - 100)/3 = - 1.67

P (X > 105) = P (Z > 1.67) = 0.0475

P (X < 95) = P (Z < -1.67) = 0.0475

which add up to: 0.095.

Answer 2

Using the normal distribution and the central limit theorem, it is found that there is a:

a) 0.2033 = 20.33% probability that a single score drawn at random will be greater than 110.

b) 0.0188 = 1.88% probability that a sample of 25 scores will have a mean greater than 105.

c) 0.0004 = 0.04% probability that a sample of 64 scores will have a mean greater than 105.

d) 0.095 = 9.5% probability that the mean of a sample of 16 scores will be either less than 95 or greater than 105.

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:

Mean of 100, hence [tex]\mu = 100[/tex].Standard deviation of 12, hence [tex]\sigma = 12[/tex].

Item a:

This probability is 1 subtracted by the p-value of Z when X = 110, hence:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

[tex]Z = \frac{110 - 100}{12}[/tex]

[tex]Z = 0.83[/tex]

[tex]Z = 0.83[/tex] has a p-value of 0.7967.

1 - 0.7967 = 0.2033

0.2033 = 20.33% probability that a single score drawn at random will be greater than 110.

Item b:

Sample of 25, hence [tex]n = 25, s = \frac{12}{\sqrt{25}} = 2.4[/tex].

This probability is 1 subtracted by the p-value of Z when X = 105, hence:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

By the Central Limit Theorem

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{105 - 100}{2.4}[/tex]

[tex]Z = 2.08[/tex]

[tex]Z = 2.08[/tex] has a p-value of 0.9812.

1 - 0.9812 = 0.0188

0.0188 = 1.88% probability that a sample of 25 scores will have a mean greater than 105.

Item c:

Sample of 64, hence [tex]n = 64, s = \frac{12}{\sqrt{64}} = 1.5[/tex].

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{105 - 100}{1.5}[/tex]

[tex]Z = 3.33[/tex]

[tex]Z = 3.33[/tex] has a p-value of 0.9996.

1 - 0.9996 = 0.0004

0.0004 = 0.04% probability that a sample of 64 scores will have a mean greater than 105.

Item d:

Sample of 16, hence [tex]n = 16, s = \frac{12}{\sqrt{16}} = 3[/tex].

Both 105 and 95 are the same distance of the mean, so we find one probability, and multiply by 2.

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{105 - 100}{3}[/tex]

[tex]Z = 1.67[/tex]

[tex]Z = 1.67[/tex] has a p-value of 0.9525.

1 - 0.9525 = 0.0475.

2 x 0.0475 = 0.095

0.095 = 9.5% probability that the mean of a sample of 16 scores will be either less than 95 or greater than 105.

A similar problem is given at https://brainly.com/question/24663213


Related Questions

if l || m,find the value of x, (8x + 20) (11x-31)

Answers

Answer:

x=17

Step-by-step explanation:

8x+20=11x-31

We simplify the equation to the form, which is simple to understand  

8x+20=11x-31

We move all terms containing x to the left and all other terms to the right.  

+8x-11x=-31-20

We simplify left and right side of the equation.  

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8x + 20 = 11x - 31

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Answer:

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:)

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Answers

Answer: D. A polygon is concave if no diagonal contains points outside the polygon.

Explanation:

Choice A is true since "equiangular" means "equal angles"

Choice B is true since all sides are congruent, and all angles are congruent for a regular polygon

Choice C is true. The term "lateral" means "side".

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The answer is "[tex]y= -x+7[/tex]"

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If [tex]\bold{\theta = 135^{\circ}}[/tex]

point: [tex]p=(2,5)[/tex]

Formula:

[tex]\to m= \tan \theta \\\\\to (y-y_1)= m(x-x_1)[/tex]

[tex]\to m= \tan \ 135^{\circ} = -1\\\\\to x_1= 2\\\\\to y_1=5[/tex]

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[tex]\to y= -x+7[/tex]

help me please i dont know what this is i cant think

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Answer:

ahhhhh swear that's scary honestly


Identify the following data set:
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Step-by-step explanation:

Answer:

Population

Step-by-step explanation:

I just took the quiz

Find the value of x. Round to the nearest degree.
13.
22
14
Not drawn to scale
a. 47
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c. 40
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Answers

Answer:

32

Step-by-step explanation:

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Answer:

X = 34

Step-by-step explanation:

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Answers

Answer:
43 cows and 18 ducks

Explanation:
Let c represent the number of cows in the farmhouse.
Let d represent the number of ducks in the farmhouse.

We know that:
- there are a total of 61 animals in the farmhouse
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- one duck has 2 legs
- there are a total of 208 legs

Using the information above, we can create two equations:

c+d=61
4c+2d=208

1) multiply the first equation by 2
If we do this, both equations will have 2d and we can subtract them to solve for c.

c+d=61
2c+2d=122

2) Subtract the first equation from the second

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Therefore, the number of cows in the farmhouse is equal to 43.

2) plug 43 into the first equation to solve for d
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43+d-43=61-43
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Therefore, the number of ducks in the farmhouse is equal to 18.

I hope this helps!

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Answers

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Step by Step Explanation:

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NOT SURE BUT HOPE IT IS CORRECT AND HELPS YOU

Suppose that we have a sample space S = {E1, E2, E3, E4, E5, E6, E7}, where E1, E2, . . . , E7 denote the sample points. The following probability assignments apply:

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Answers

Answer:

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Step-by-step explanation:

Given that:

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Also;

A = {E1, E4, E6}

B = {E2, E4, E7}

C = {E2, E3, E5, E7}

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P(C) = 0.20 + 0.20 + 0.15 + 0.05 = 0.6

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P(A ∩ B) = 0.25

The required probability for the intersection of A & B = 0.25

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Answers

Answer:

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Step-by-step explanation:

Answer:

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Step-by-step explanation:

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Answers

Answer:

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Step-by-step explanation:

The value of the given linear expression 3(12 - 8) + 5 is 17.

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Therefore, the variable x⁰ = 1.

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3(12 - 8) + 5

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Answers

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Step-by-step explanation:

Sam needs 7/8 cup mashed bananas and 5/6 cup mashed strawberries for a recipe. He wants to find whether he needs more bananas or more strawberries. How can he write 7/8 and 5/6 as a pair of fractions with the smallest common denominator?

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Answer:

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The amount of air pressure, (PSI) in the spare tire of a certain vehicle (Type A) brought for inspection are normally distributed with PSI of µ = 30 and σ = 4, and such spares tires with PSI below 25 are considered under-inflated.
The amount of air pressure, (PSI) in the spare tire of a certain vehicle (Type B) brought for inspection are normally distributed with PSI of µ = 27.7 and σ = 5.4, and such spare tires with PSI below 25 are considered under-inflated.
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Answer:

f

Step-by-step explanation:

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parallel?
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Enter the number that belongs in the green box.

Answers

Answer:

4

Step-by-step explanation:

Parallel lines have the same slope and since the slope of the first line is 4 the second lines slope is also 4

Answer:

Step-by-step explanation:

Slopes of parallel lines are the same.

m = 4

Pls help me out I really need help

Answers

Answer: -0.8

Step-by-step explanation:

A bag contains 150 marbles. Some are blue, and the rest are white. There are 21 blue marbles for every 4 white marbles. . How many blue marbles are in the bag

Answers

If goal to get white marble then 16 if goal to get blue marbles then its 84

BRAINLIEST APPERICATED :D
There are 84 blue marbles in the bag

If Michael can read 400 pages in 8 hours, how many pages can he read in 1 hour?

Answers

Step-by-step explanation:

He will read 50 pages in an hour

What Quadrant is 4,-8 located in

Answers

Answer:

4

Step-by-step explanation:

The price of one share of a stock fell 4 dollars each day for 8 days. How much value did one share of the stock lose after 8 days?

Answers

Answer:

After 8 days, the stock lost 32 dollars.

Step-by-step explanation:

If 4 dollars fell every day for 8 days, it would be 32 dollars because 4*8 is 32.

If b^2=a then b is what of a?

Answers

1:: If
a
is divisible by a square of a prime number, then we cannot conclude from |2
a
|
b
2
that |
a
|
b
.

Any time
a
is divisible by a square of a prime number, =⋅
a
=
k

p
n
where ≥2
n

2
and
p
does not divide
k
, you can see that
a
divides (⋅−1)2
(
k

p
n

1
)
2
, but
a
does not divide ⋅−1
k

p
n

1
.

Part 2: If
a
is not divisible by a square of a prime number, then we can conclude from |2
a
|
b
2
that |
a
|
b
.

On the other hand, if
a
is not divisible by a square of a prime number, then =12⋯
a
=
p
1
p
2

p
n
where
p
i
is prime for all
p
. Then, assuming
a
divides 2
b
2
, you can conclude that
p
i
divides 2
b
2
, and therefore
p
i
divides
b
(from prime factorization) for all
i
. From this, you can conclude that
a
divides
b
.

Answer:

[tex]\huge\boxed{b = \sqrt{a}}[/tex]

Step-by-step explanation:

If we have an equation [tex]b^2 = a[/tex] and we want to find what b is in relation to a, we can change the equation so that we have b on one side and whatever is on the other side is what b is.

[tex]b^2 = a[/tex]

To isolate b, we can take the square root of both sides as taking the square root of something squared results in the base.

[tex]\sqrt{b^2} = \sqrt{a}[/tex]

[tex]b = \sqrt{a}[/tex]

So b is the square root of a.

Hope this helped!

A car moves at a speed of 30m/s to the west for 3hr. What is the displacement of the car in km?

Answers

(30m/s divided by 2)x 10800 ( convert 3hr to seconds by x with 60squared)
= 162000 m ( convert m to km by dividing with 1000)
= 162 km


Plz mark as brainleast if this answers helped ya.

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Answers

Answer:

4 1/4

Step-by-step explanation:

the additive inverse is a number that when you add it with the number you already had, makes 0

-2 is the additive inverse of 2

so 4 1/4 is the inverse of -4 1/4

HELP PLS A carnival charges $2.50 per ride after an entrance fee.You paid a total of $22.50 after 6 rides.Find the total cost for 15 rides.

Answers

The cost of the entrance would be 7.5$, because 2.50$ x 6 equals 15.

Let’s multiply 6 x 2 to get twelve, and if six rides are 15$, that would be 30$ for twelve rides. But we still have three more.

2.50 x 3 would be 7.5$, and if we add that to 30$, we would get 37.5$.

With entry fee though, we would add 37.5$ to
7.5$, and so it would be 45$ on the dot.

Your answer would be 37.5$ without entry fee, and 45$ with entry fee.

If this helped.. please do me a favor and mark this as the Brainliest! :))
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