SAT verbal scores are normally distributed with a mean of 430 and a standard deviation of 120 (based on data from the College Board ATP). (a) If a single student is randomly selected, find the probability that the sample mean is above 500. (b) If a sample of 35 students are selected randomly, find the probability that the sample mean is above 500. These two problems appear to be very similar. Which problem requires the application of the central limit theorem, and in what way does the solution process differ between the two problems?

Answers

Answer 1

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

a) 0.281 = 28.1% probability that the sample mean is above 500.

b) 0.0003 = 0.03% probability that the sample mean is above 500.

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 430, hence [tex]\mu = 430[/tex].The standard deviation is of 120, hence [tex]\sigma = 120[/tex].

Item a:

The probability is the p-value of Z when X = 500, hence:

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

[tex]Z = \frac{500 - 430}{120}[/tex]

[tex]Z = 0.58[/tex]

[tex]Z = 0.58[/tex] has a p-value of 0.719.

1 - 0.719 = 0.281

0.281 = 28.1% probability that the sample mean is above 500.

Item b:

Sample of 35, hence [tex]n = 35, s = \frac{120}{\sqrt{35}}[/tex]

Then:

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

By the Central Limit Theorem

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

[tex]Z = \frac{500 - 430}{\frac{120}{\sqrt{35}}}[/tex]

[tex]Z = 3.45[/tex]

[tex]Z = 3.45[/tex] has a p-value of 0.9997.

1 - 0.9997 = 0.0003

0.0003 = 0.03% probability that the sample mean is above 500.

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Related Questions

Find the height of the trapezoid.
7 m
Area = 51 m2
10 m

Answers

The answer is 6

h = 2 A/a+b = 2 51/10+7 = 6

TRUE OR FALSE

There are whole numbers that are not integers.

There are rational numbers that are not whole
numbers.

All rational numbers are integers.

All whole numbers are rational numbers.

Answers

False. All whole numbers are integers.True. For example, 1/2 is a rational number but it is not a whole number.False. For example, 2/7 is a rational number but not an integer.True. Each and every whole numbers are rational numbers.

ANSWERS:

[tex]1. False \\ 2. True \\ 3. False \\ 4. True \\ [/tex]

Hope you could get an idea from here.

Doubt clarification - use comment section.

pls help me
I m giving brainliest

Answers

Step-by-step explanation:

the general line equation format is

y = ax + b

a is the slope or gradient of the line. it is the ratio of "y coordinate change / x coordinate change".

b is the y-intercept, the y value when x = 0. we get this by putting one point into the equation and solve for b.

(i)

A (3, 5)

B (5, 9)

x changes by +2 (from 3 to 5).

y changes by +4 (from 5 to 9).

so, the slope (or gradient) is +4/+2 = 2

(ii)

the semi-finished equation is

y = 2x + b

let's use A to get b :

5 = 2×3 + b = 6 + b

b = -1

and the full equation is

y = 2x - 1

(iii)

for point (4, k) we get

k = 2×4 - 1 = 8 - 1 = 7

9
00
8-
7+
6+
5-
3+
H
-9-8-7-6-5-4-3-2
2 3 4 5 6 7 8 9
-2+
-3+.
-4+
-5+
-6+
-7+
op -

what is the slope of the graph

Answers

Answer:

-5+

Step-by-step explanation:

Slope of the line passing through the points (1, 4) and (3, 2) is -1.

What is Slope of Line?

The slope of the line is the ratio of the rise to the run, or rise divided by the run. It describes the steepness of line in the coordinate plane.

The slope intercept form of a line is y=mx+b, where m is slope and b is the y intercept.

The slope of line passing through two points (x₁, y₁) and (x₂, y₂) is

m=y₂-y₁/x₂-x₁

The line passing through the points (1, 4) and (3, 2)

Plug in the points in the slope formula.

Slope = 2-4/3-1

=-2/2

=-1

Hence, slope of the line passing through the points (1, 4) and (3, 2) is -1.

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Two positive numbers are in the ratio 2 : 3. Find the numbers if ​(a) their sum is 55 and ​(b) their product is 384.

Answers

say the numbers are "a" and "b", we know that they're in the ratio of 2:3, or namely that

[tex]\stackrel{\textit{in the ratio of 2 to 3}}{a~:~b\qquad \qquad 2:3}\qquad \qquad \implies \qquad \cfrac{a}{b}=\cfrac{2}{3}\implies \cfrac{3a}{2}=b \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{their sum is 55}}{a + b = 55}\implies a + \cfrac{3a}{2}=55\implies \stackrel{\textit{multiplying both sides by }\stackrel{LCD}{2}}{2\left( a + \cfrac{3a}{2} \right)=2(55)} \\\\\\ 2a+3a=110\implies 5a=110\implies a = \cfrac{110}{2}\implies \boxed{a = 22}[/tex]

[tex]\stackrel{\textit{we know that}}{\cfrac{3a}{2}=b}\implies \cfrac{3(22)}{2}=b\implies \boxed{33=b} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{their product is 384}}{ab=384}\implies a\left( \cfrac{3a}{2} \right)=384\implies \cfrac{3a^2}{2}=384\implies 3a^2=768 \\\\\\ a^2=\cfrac{768}{3}\implies a^2=256\implies a=\sqrt{256}\implies \boxed{a=16} \\\\\\ \stackrel{\textit{we know that}}{\cfrac{3a}{2}=b}\implies \cfrac{3(16)}{2}=b\implies \boxed{24=b}[/tex]

The positive numbers for both the parts are obtained as 22,33 and 16,24 respectively.

How to write a linear equation?

A linear equation for the given case can be written by assuming any variable as the unknown quantity. Then, as per the given data the required operations are done and it is equated to some value.

(a) As per the question, the numbers can be considered as 2x and 3x respectively.

Then, the equation for their sum can be written as follows,

3x + 2x = 55

⇒ 5x = 55

⇒ x = 11

The numbers are obtained as 2 × 11 = 22 and 3 × 11 = 33.

(b) Similarly, the equation for the product of numbers can be written as,

3x × 2x = 384

⇒ 6x² = 384

⇒ x² = 64

⇒ x = 8

The numbers are obtained as 2× 8 = 16 and 3 × 8 = 24.

The required number for both the cases are 22,33 and 16,24 respectively.

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Given x+2y=-6 solve for y

y= x-6/2
y= -x+6/2
y= -x-6/2

Answers

Answer:

y= -x-6/2

Step-by-step explanation:

Given,

x+2y=-6

or,2y=-x-6 (cross multiplication)

or,y= -x-6/2

A new health drink has 190% of the recommended daily allowance (RDA) for a certain vitamin. The RDA for this vitamin is 50 mg. How many milligrams of the vitamins are in the drink?

Answers

There is 95mg of vitamins in the drink

Percentages and proportion

If a new health drink has 190% of the recommended daily allowance (RDA) for certain vitamins and the RDA for this drink is 50mg.

The amount of milligrams of the vitamins in the drink is expressed as shown:

A = 190 % of 50mg

A = 1.9 * 50

A = 95mg

Hence there is 95mg of vitamins in the drink

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Answer:

There is 95mg of vitamins in the drink

Someone help please

Answers

It is 4pi/7, or b. Just want to help ;)

Please help, very urgent!!!! 50 points!!

Answers

Answer:

f = 14

Step-by-step explanation:

it is because skephalo and hot and poggers science and dnf

Mrs. Lovejoy and Christi spent the day baking cookies. When they divided them into 8 stacks, they had 14 cookies in each stack. How many cookies did they have? Solve and label the word problem. Please I need help.​

Answers

Answer:

they had 112 cookies

Step-by-step explanation:

14×8 = 112

PLEASE HELP ME IN THIS QUESTION THE ASSIGNMENT IS DUE IN 2 MINUTES

Answers

If the bottle can only hold half a liter than the volume would not be 1.5 liters

guys? can ypu pls help me with this? it's our exam day, thank you very much​

Answers

Answer:

Step-by-step explanation:

Please look for the question in the picture.

Answers

Answer:

Step-by-step explanation:

B

whether the negative sign is next to the fraction bar or next to the numerator, it doesnt matter.

A results in the last number being different from the one in the question.

answer is B

Find the value of x! GIVING BRAINILEST

Answers

Answer:

the value of x must be 71°

Step-by-step explanation:

180° - (38° + 33°) = 109°

x = 180° - 109°

x = 71°

Answer:

x = 71

Step-by-step explanation:

The exterior angle of a triangle is equal to the sum of the 2 opposite interior angles.

x is an exterior angle of the triangle , then

x = 38 + 33 = 71

What fraction is equivalent to 4/3?

Answers

8/6
16/12
40/30


if you divide the first one by 2 you will get 4/3

If you divide the second one by 4 you will get 4/3

The last one if you divide it by 10 you will get 4/3

Solve for S
Please help me

Answers

Answer:

s = 19°

Step-by-step explanation:

opposite angles = equal angles

so

106 = 6s - 8

114 = 6s

s = 114 : 6

s = 19°

--------------- check

106 = 6 * 19 - 8

106 = 106

the answer is good

LAST ATTEMPT IM MARKING AS BRAINLIEST!! (Finding missing sides- similar triangles Algebraic practice )

Answers

TV / BC = TU / DC
TV = 20^2 - 12^2 = 256 = 16

substitute:
16 / 4x + 3 = 12 / 9
16 / 4x + 3 = 4 / 3

cross multiply:
48 = 16x + 12
36 = 16x

2.25 = x

Answer:

x=3

Step-by-step explanation:

What u should always do is look at the letters of the triangles given:

It will tell u that:

[tex] \frac{tu}{dc} = \frac{uv}{cb} [/tex]

Now we once again just sub in the values:

[tex] \frac{12}{9} = \frac{20}{4x + 3} [/tex]

we can simplify the fraction on the right

[tex] \frac{4}{3} = \frac{20}{4x + 3} [/tex]

And now we cross multiply

3×20=4(4x+3)

60=16x+12

60-12=16x

48=16x

Therefore :

x=3

Solve the Inequality for x.
-5x + 20 < 5

A) x > 5

B) x < 3

C) x > 3

D) x < -5

Thanks!!! :D

Answers

Answer:

x>3

Step-by-step explanation:

-5x+20<5

-5x<5-20

-5x<-15

5x>15

x>15/5

x>3

Answer:

A) x > 3

Step-by-step explanation:

[tex]-5x + 20 < 5\\\rule{150}{0.5}\\-5x + 20 - 20 < 5 - 20 \\\\-5x < -15 \\\\\frac{-5x < -15}{-5}\\\\\boxed{x > 3}[/tex]

Find the value of x in the figure.

Answers

Answer:

64 degrees

Step-by-step explanation:

X equals 64 degrees because it shares a vertical angle pair with another angle which is 64 degrees. Since they are across from each other, x is 64 degrees.

Vertical angles are a pair of opposite angles formed by intersecting lines.

Statement of cash flows (partial)

Answers

Yeaa that is what the problem is

Jarad, the track star, ran the 88 yards in 10 seconds. How fast is this in miles per hour?

Answers

Answer:

0.18 i think

Step-by-step explanation:

In 2004 5,545 woman were participating in college lacrosse and in 2008 6,830 woman joined was is the rate of change

Answers

Answer:

321.25 per year

Step-by-step explanation:

In 2004 they had 5,545

In 2008 they had 6,830

Subtract them both and you get 4 years, 1285 people

Divide 1285 by 4 and get 321.25 per year.


helpp ☹️☹️ plsplspsospsl

Answers

Answer:

BRO BRING THE PHONE CLOSER TO THE SCREEN WE CANT SEE ANYTING

Step-by-step explanation:

If 3 paintings cost $30, then how much does 1 painting cost?

Answers

Answer:

10$

Step-by-step explanation:

because 10$ + 10$ + 10$ equals 30$ so one is 30$

what is the value of x that makes the equation 8 x -4 equal 96 true​

Answers

Answer:

x= 12.5 or 25/2

Step-by-step explanation:

8x - 4 (+4) = 96 (+4)

8x (/8) = 100 (/8)

x= 12.5 or 25/2

I am so bad at this.Do you know the answer?

Answers

Answer:

it's for sure option (B) cuz it's represented in the form of a straight line which means it's linear!

hope helpful~

Answer:

B

Step-by-step explanation:

I thinks its the best answer

Please help I need the answer quick this assignment is almost due!

Answers

the correct answer is x= -3

Answer:

x = -3

Step-by-step explanation:

The two angles shown are corresponding angles, meaning that they have the same value. Because of this, we can create an equation to solve for the value of x.

2x + 9 = 7x + 24

Remember, those two angles are similar, and that's why we can make the two values equal to each other. First, let's remove 7x from both sides.

2x - 7x + 9 = 7x - 7x + 24

-5x + 9 = 24

Then, we'll remove 9 from both sides.

-5x + 9 - 9 = 24 - 9

-5x = 15

Finally, we'll divide both sides by -5 to isolate x and figure out what it is equivalent to.

-5x/-5 = 15/-5

x = -3

And there it is... the value of x. Hopefully that's helpful! :)

What percent of 38 is 12

Answers

Answer:

31%

Step-by-step explanation:

hope this helps

After deducting grants based on need, the average cost to attend the University of Southern California (USC) is $27,175 (U.S. News & World Report, America’s Best Colleges, 2009 ed.). Assume the population standard deviation is $7,400. Suppose that a random sample of 60 USC students will be taken from this population.

a. Refer to Exhibit 2. What is the value of the standard deviation of the mean? (Note: keep two decimal places.)

b.What is the probability that the sample mean will be more than $27,175

c.What is the probability that the sample mean will be within $1,000 of population mean? (Note: keep two decimal places for the z value and four decimal places for the final probability value.)

d.What would be the probability in Part c if the sample size were increased to 100? (Note: keep two decimal places for the z value and four decimal places for the final probability value.)

Answers

The solutions are:

(a) Standard error is 957.04. (b) The probability is 50%. (c)The probability is 0.8491. (d) The probability is 0.8617.

To calculate the values, use the standard error formula for the sample mean:

(a)

The standard error is given by:

[tex]Standard\ error =\dfrac{ Population \ standard \ deviation}{\sqrt{sample\ size}}\\Standard\ error=\dfrac {\$ 7,400 }{ \sqrt60}\\Standard\ error =\ $957.04[/tex]

(b)

To find the probability that the sample mean will be more than $27,175, convert it into a z-score and then find the area under the normal distribution curve:

z = (sample mean - population mean) / standard error

z = ($27,175 - $27,175) / $957.04

z = 0

Since the z-score is 0, the area under the curve to the right of this point is 0.5 or 50%.

c. To find the probability that the sample mean will be within $1,000 of the population mean, we need to find the area between the range of $(27,175 - $1,000) and $(27,175 + $1,000).

Lower limit: $27,175 - $1,000 = $26,175

Upper limit: $27,175 + $1,000 = $28,175

Now, we calculate the z-scores for both limits:

lower = ($26,175 - $27,175) / $957.04 ≈ -1.045

upper = ($28,175 - $27,175) / $957.04 ≈ 1.045

Using a standard normal distribution table or calculator, the probability of a z-score being between -1.045 and 1.045 is approximately 0.8491.

(d)

If the sample size were increased to 100.

[tex]Standard error = \dfrac{ \$7,400}{ \sqrt{100}} \\Standard error = \$740[/tex]

Find the probability that the sample mean will be within $1,000 of the population mean, did in part c:

Lower limit: $27,175 - $1,000 = $26,175

Upper limit: $27,175 + $1,000 = $28,175

lower = ($26,175 - $27,175) / $740

lower ≈ -1.351

upper = ($28,175 - $27,175) / $740

upper ≈ 1.351

Using a standard normal distribution table or calculator, the probability of a z-score being between -1.351 and 1.351 is approximately 0.8617.

Hence,

Standard error is 957.04.The probability is 50%.The probability is 0.8491.The probability is 0.8617.

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Layla missed 38 questions on her multiple choice math final that had a total of 200 problems. What percentage of questions did Layla answer correctly?

Answers

Answer:

81%

Step-by-step explanation:

200-38= 162

162/200= 81/100

81%

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