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?
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.
To learn more about the normal distribution and the central limit theorem, you can take a look at https://brainly.com/question/24663213
are the following expressions equivalent 5(2z+3) and (5x2)+(z+3)
Answer: yes they are
Step-by-step explanation:
Divide 5.6 by 0.7 the quotient is what ?
X=[-1,+∞),Y=R,f(x)=y sao cho y^2-2y=x∀x∈X
Answer:
30
Step-by-step explanation:
sh well cm to cm et FL to
can’t figure it out, need help
Answer: 74+39=113. So the answer is 113
2x +6y = 2
х= 8 - Зу
What is X and Y?
Answer:
no solutions exist
Step-by-step explanation:
We can solve this by using system of equations and using substitution.
2(8-3y)+6y=2 Substitute x=8-3y into the first equation.
16-6y+6y=2 Distribute.
16=2 Simplify.
16 does not equal 2, so there are no solutions.
Beatrice has two coins. The first coin is fair and the second coin is biased. The biased coin comes up heads with probability 2/3 and tails with probability 1/3. Beatrice selects one of the two coins at random and flips the selected coin 4 times. The result is HHTH. What is the probability that the fair coin was selected?
Triangle in space with vertices P= (1, 0, 0), Q = (0,1,0), R=(0, 0, 2), find angle at P P=
Answer:
71.6°
Step-by-step explanation:
The angle can be found from the dot product of PQ and PR.
PQ·PR = |PQ|×|PR|×cos(α)
where α is the angle between the two segments.
cos(α) = (Q -P)·(R -P)/(|Q -P|×|R -P|)
= ((0, 1, 0) -(1, 0, 0))·((0, 0, 2) -(1, 0, 0))/(|Q -P|×|R -P|)
= (-1, 1, 0)·(-1, 0, 2)/(√(((-1)² +1²)((-1²) +2²)) = (1+0+0)/√10
α = arccos(1/√10) ≈ 71.6°
The angle at P is about 71.6°.
_____
Additional comment
The side lengths of the triangle are √2, √5, √5. As we have seen, the angle at P is bounded by the sides of length √2 and √5. The law of cosines can also be used to arrive at the angle between these sides.
Try changing these improper fractions to mixed numbers.
16/5?
( teach me how to solve this please!)
Answer:
3 1/5
Step-by-step explanation:
The improper fraction 16/5 is called improper because the top number(the numerator) is bigger than the bottom number(the denominator) we "fix" this by dividing and creating a mixed number (it is called mixed because there is a whole part and a fraction part):
As 16÷5 >>> doing this division gives you 3 wholes and 1/5 leftover. We write this as
3 1/5 (in handwriting the three is big and the 1/5 is a fraction, it is the mixed number they are asking you for)
Write the sphere in standard form. 4x2 + 4y2 + 4z2 − 24x + 16y = 1
Answer:
(x -3)^2 +(y +2)^2 +z^2 = 13.25
Step-by-step explanation:
Divide by 4 and complete the squares.
x^2 -6x +y^2 +4y +z^2 = 1/4
(x^2 -6x +9) +(y^2 +4y +4) +z^2 = 13 1/4
(x -3)^2 +(y +2)^2 +z^2 = 13.25
which of the following is NOT a rational number, -n its like on top of it, -4/7, 3/5 or 2/3
Step-by-step explanation:
i am very sorry but I don't know the answer
how do I do this word problem?
Answer:
5 weeks
Step-by-step explanation:
John deposits $5 per week and has $10, so his equation is 5x + 10.
Jasmine's equation is 3x + 20.
Make them equal to each other and solve for x:
5x + 10 = 3x + 20
Subtract 10 from each side
5x = 3x + 10
Subtract 3x from each side
2x = 10
Divide each side by 2
x = 5
Answer:
5 weeks
Step-by-step explanation:
John:$10,$15,$20,$25,$30,$35
Jasmine:$20,$23,$26,$29,$32,$35
5 weeks
Select All equivalent expressions to 4( x + 2)
Answer:
3x 3-x and 3+x
URGENNTTT PLEASE ILL GIVE 50
Take Triangle ABC, A(2,-1), B(1,4) and C(-2,2). Do
the series of mappings to find A'B'C'.
First translate
it 3 units to the right and 2 units up, next reflect
about the x-axis, and finally rotate 180 degrees.
Answer:
(3-2)
Step-by-step explanation:
What is the answer to this : Simplify 4 4128
Answer:
1032/1
Step-by-step explanation:
Find the GCD (or HCF) of numerator and denominator
GCD of 4128 and 4 is 4
Divide both the numerator and denominator by the GCD
4128 ÷ 4
4 ÷ 4
Reduced fraction:
1032
1
Therefore, 4128/4 simplified to lowest terms is 1032/1.
Find the total area of the shape shown below in square feet. Enter only the number.
16 feet
5 feet
12 feet
9 feet
7 feet
7 feet
The solution is
Answer:
129, because you split the shape into two find the area of them separately and then add them up
plz help me to answer
For the 20ft from doc. Set up triangle, with dr/dt on the hypotenuse = 2. 6 on the y, 20 on the x. Use x^2+ y^2= r^2. Differentiate and solve for dx/dt. 2x(dx/dt) + 2y(dy/dt)=2r(dr/dt); Since y value does not change, your y vector is equal to 0. Giving us 2x(dx/dt)=2r(dr/dt) ; dx/dt=(2r(dr/dt))/2x
Do the same thing for 10ft
No it is not surprising because as the angle between the dock and the boat decreases, more force is being used to pull up on the boat, not pull it inwards. Thus, the boats horizontal velocity will slow down as it beats the dock.
A random sample of 25 boxes of cereal have a mean of 372.5grams and a standard deviation of 15 grams. Does an average box of cereal contain more than 368 grams of cereal? Test hypothesis at the 5% significance level, outline all steps involved. (9mks)
Using the t-distribution, it is found that since the test statistic is less than the critical value for the right-tailed test, there is not enough evidence to conclude that an average box of cereal contain more than 368 grams of cereal.
At the null hypothesis, it is tested if the average box of cereal does not contain more than 368 grams, that is:
[tex]H_0: \mu \leq 368[/tex]
At the alternative hypothesis, it is tested if it contains, that is:
[tex]H_1: \mu > 368[/tex]
We have the standard deviation for the sample, hence, the t-distribution is used to solve this question.
The test statistic is given by:
[tex]t = \frac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}}[/tex]
The parameters are:
[tex]\overline{x}[/tex] is the sample mean. [tex]\mu[/tex] is the value tested at the null hypothesis. s is the standard deviation of the sample. n is the sample size.In this problem, the values of the parameters are: [tex]\overline{x} = 372.5, \mu = 368, s = 15, n = 25[/tex].
Hence, the value of the test statistic is:
[tex]t = \frac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}}[/tex]
[tex]t = \frac{372.5 - 368}{\frac{15}{\sqrt{25}}}[/tex]
[tex]t = 1.5[/tex]
The critical value for a right-tailed test, as we are testing if the mean is greater than a value, with a significance level of 0.05, with 25 -1 = 24 df, is of [tex]t^{\ast} = 1.71[/tex].
Since the test statistic is less than the critical value for the right-tailed test, there is not enough evidence to conclude that an average box of cereal contain more than 368 grams of cereal.
You can learn more about the use of the t-distribution to test an hypothesis at https://brainly.com/question/13873630
expand and simplify
5(3y+2)+5(5y+2)
Answer:
Step-by-step explanation:
15y + 10 + 25y + 10
Answer:
Solution here,
Step-by-step explanation:
For simplify
First multiply the terms inside the brackets with the term outside the bracket i.e 5.
5(3y+2)+5(5y+2)
15y+10+25y+10
35y+20
someone pleaseeee answer!!!
Answer:
7.79
Step-by-step explanation:
in 30, 60, 90 triangle, hypotenuse is equal to "2x", shortest length is equal to "x" and second longest is equal to "x × square root of 3". hope this helps
Birdseed costs $0.57 a pound and sunflower seeds cost $0.87 a pound. Angela Leinenbachs' pet store wishes to make a 30 pound mixture of birdseed and sunflower seeds that sells for $0.81 per pound. How many pounds of each type of seed should she use?
Step-by-step explanation:
x = pounds birdseed
y = pounds sunflower seeds
x + y = 30
0.57x + 0.87y = 30×0.81
with this last trick (to calculate the full cost of the whole 30 pounds based on the targeted individual price per pound) we bring both constraints into the play.
from the first equation we get
x = 30 - y
and we use that in the second equation :
0.57×(30-y) + 0.87y = 30×0.81 = $24.30
17.1 - 0.57y + 0.87y = 24.3
0.3y = 7.2
y = 24
x = 30 - y = 30 - 24 = 6
she uses 6 pounds of birdseed and 24 pounds of sunflower seeds.
What is the slope of the function?
–10
- 5
5
10
Answer:
?
Step-by-step explanation:
what is the function?
a slope going up means it is positive and a slope going down means it is negative
mom left $5.00 for his 7 children. how much did each child receive
a.0.17
b.0.70
c.17
d.71
please answer it correctly.
Answer:
.71
Step-by-step explanation:
Answer:
B. 0.70
Step-by-step explanation:
Just divide the total amount of money by the total amount of people.
5/7=0.71428...
if you round up correctly, you get 0.70; thus, it is B.
Also, his is for he, mom is a she, so use her instead of his...
f(x) = x – 3
g(x) = x^2 + 1
Find:
f(g(x)) =
PLEASE HELP WILL MARK BRAINLIEST!!
[tex]\huge \bf༆ Answer ༄[/tex]
Let's solve ~
[tex] \sf{f(x) = x - 3}[/tex][tex] \sf f(g(x)) = g(x) - 3[/tex][tex] \sf{f(g(x)) = ({x}^{2} + 1}) - 3[/tex][tex] \sf{f(g(x)) = {x}^{2} - 2}[/tex]you can further simplify to get ~
[tex] \sf{(x + \sqrt{2} )(x - \sqrt{2} )}[/tex]I need help with this!!!!
Answer:
246b
Step-by-step explanation: i did this
What is the distance between points A and B?
A. 1/3
B. 1/2
C. 1 1/3
D. 1 1/2
Answer:
A. 1/3
Step-by-step explanation:
From 1 to ___, ____, and 2 is half and half that would be 2 and a half. the only solution would be 1/3
Answer:
boom
Step-by-step explanation:
LAST ATTEMPT IM MARKING AS BRAINLIEST!! (Pythagorean theorem )
Step-by-step explanation:
The Pythagorean theorem is:
[tex] {c}^{2} = {a}^{2} + {b}^{2} [/tex]
where c is the hypotenuse.
The two wires in the sketch are the hypotenuse so we just use the formula
let the wires=x
So:
[tex] {x}^{2} = {12}^{2} + {6}^{2} [/tex]
[tex] {x}^{2} = 144 + 36[/tex]
[tex] {x}^{2} = 180[/tex]
[tex]x = \sqrt{180} [/tex]
[tex]x = 13.42[/tex] feet
Find the arc length for arcs of circles as follows:
radius: 12 inches
central angle: radians
- - -
Answer:
≈ 11.78 in
Step-by-step explanation:
The arc length is calculated as
arc = circumference of circle × fraction of circle
= 2πr × [tex]\frac{\frac{\pi }{4} }{2\pi }[/tex] ( cancel 2π on numerator/ denominator )
= 15 × [tex]\frac{\pi }{4}[/tex]
= [tex]\frac{15\pi }{4}[/tex]
≈ 11.78 in ( to 2 dec. places )
Lashonda finished a race in 5 minutes. How many seconds is this
Answer:
300 seconds
Step-by-step explanation:
There are 60 seconds in a minute. If there are 5 minutes, you have to multiply 5 by 60.
5 x 60 = 300
-hope it helps
Answer:
360 sec
Step-by-step explanation:
this is because 60sec is one mintute
so 60x5= 360
Please help me find the height of 302 after some seconds?
Step-by-step explanation:
Since the initial velocity of the object is 170 ft/s, the expression for h is given by
[tex]h = -16t^2 + 170t[/tex]
In order to find the time it takes for the object to reach the height of 302 ft, we to rewrite the equation above as
[tex]302 = -16t^2 +170t \Rightarrow 16t^2 - 170t + 302 = 0[/tex]
This is a quadratic equation whose roots are
[tex]t = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/tex]
where a = 16, b = -170 and c = 302. Using these values, we get
[tex]t = \dfrac{170 \pm \sqrt{(-170)^2 - 4(16)(302)}}{2(16)}[/tex]
[tex]\;\;\;=\dfrac{170 \pm \sqrt{28900 - 19328}}{32}[/tex]
[tex]\;\;\;= \dfrac{170 \pm 97.8}{32}[/tex]
[tex]\;\;\;= 2.3\:\text{s},\;\;8.4\:\text{s},[/tex]
This means the object will reach the height of 302 ft 2.3 seconds after launch and then at 8.4 seconds after launch (on its way down).
Answer:
Step-by-step explanation: