Answer:
The range in which we can expect to find the middle 68% of most pregnancies is [245 days , 279 days].
Step-by-step explanation:
We are given that the lengths of pregnancies in a small rural village are normally distributed with a mean of 262 days and a standard deviation of 17 days.
Let X = lengths of pregnancies in a small rural village
SO, X ~ Normal([tex]\mu=262,\sigma^{2} = 17^{2}[/tex])
Here, [tex]\mu[/tex] = population mean = 262 days
[tex]\sigma[/tex] = standard deviation = 17 days
Now, the 68-95-99.7 rule states that;
68% of the data values lies within one standard deviation points.95% of the data values lies within two standard deviation points.99.7% of the data values lies within three standard deviation points.So, middle 68% of most pregnancies is represented through the range of within one standard deviation points, that is;
[ [tex]\mu -\sigma[/tex] , [tex]\mu + \sigma[/tex] ] = [262 - 17 , 262 + 17]
= [245 days , 279 days]
Hence, the range in which we can expect to find the middle 68% of most pregnancies is [245 days , 279 days].
Potatoes cost Janice $1.10 per pound, and she has $6.00 that she could possibly spend on potatoes or other items. If she feels that the first pound of potatoes is worth $1.50, the second pound is worth $1.14, the third pound is worth $1.05, and all subsequent pounds are worth $0.30, how many pounds of potatoes will she purchase? What if she only had $3.00 to spend?
Answer:
she will purchase 8 pounds of potatoes.
But if she had on $3.0
She will purchase only two pounds
Step-by-step explanation:
first pound of potatoes is worth $1.50, the second pound is worth $1.14, the third pound is worth $1.05, and all subsequent pounds are worth $0.30.
1.5+1.14+1.05+0.30=$ 3.99
6-3.99= 2.01
If the rest cost 0.5
Then there are 4 pounds of 0.5 in 2.01.
So total she will purchase 8 pounds of potatoes.
But if she had on $3.0
She will purchase only two pounds, as it will only purchase1.5+1.14+= 2.64 worth of potatoes.
With $6 Janice can get a total of 10.7 pounds, but with $3 she will be able to afford the first two pounds only
Given Data
Amount at hand = $6
First Pound cost = $1.50
Second Pound cost = $1.14
Third Pound cost = $1.05
Total = 1.5+1.14+1.05 = 3.69
Balance = 6-3.69 = $2.31
Hence the number of pound we can get with $2.31 for subsequent purchase will be
= 2.31/0.3
= 7.7 pounds
This means that the total number of pounds will be
= 3+7.7
=10.7pounds
subsequent Pound cost = $0.3
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What’s the correct answer for this?
Answer:
The answer is option 2.
Step-by-step explanation:
Given that the formula for length of arc is :
[tex]arc = \frac{θ}{360} \times 2 \times \pi \times r[/tex]
Answer:
B
Step-by-step explanation:
It could also be
[tex] \frac{\pi \times r \times m}{180} [/tex]
The AARP (American Association of Retired People) report that at least 60% of retired people under the age of 65 would return to work on a full-time basis if a suitable job were available. A sample of 500 retirees under the age of 65 showed that 315 would return to work. Can we conclude that more than 60% would return to work? Test at the 2% level of significance.
Answer:
Step-by-step explanation:
Proportion of retired people under the age of 65 would return to work on a full-time basis if a suitable job were available = 60/100 = 0.6 = P
Null hypothesis: P ≤ 0.6
Alternative: P > 0.6
First, to calculate the hypothesis test, lets workout the standard deviation
SD = √[ P x ( 1 - P ) / n ]
where P = 0.6, 1 - P = 0.4, n = 500
SD = √[ (0.6 x 0.4) / 500]
SD = √ (0.24 / 500)
SD = √0.00048
SD = 0.022
To calculate for the test statistic, we have:
z = (p - P) / σ where p = 315/500 = 0.63, P = 0.6, σ = 0.022
z = (0.63 - 0.6) / 0.022
z = 0.03/0.022
z = 1.36
At the 2% level of significance, the p value is less than 98% confidence level, thus we reject the null hypothesis and conclude that more than 60% would return to work.
Suppose the price of a pair of shoes, including 14% vat is R240. The price of the pair of shoes is reduced by 15% on sale. What is the reduced price of the shoes including VAT?
Answer:
$204
Step-by-step explanation:
First, you have to find the price without VAT. To calculate it you have to divide the price including VAT by one plus the rate:
Price including VAT= $240
VAT= 14%
240/(1+0.14)= 240/1.14= 210.52
Now, you have to calculate the 15% of 210.52 and subtract that amount from the price:
210.52*0.15=31.57
210.52-31.57=178.95
Then, you have to calculate the 14% VAT by multiplying the price for 1 plus the rate:
178.95*(1+0.14)= 178.95*1.14= 204
According to this, the reduced price of the shoes including VAT is $204.
heeeeeeeellllllllllppppppppppppp plz i nned help
Answer:
Step-by-step explanation:
This is a ratio problem. there are 20 Blonde Teachers, and 12 brown haired teachers. The fraction is 12/20. We can simplify this fraction to 3/5. for every 3 brown haired Teacher, there is 5 other blonde hair teachers.
What’s the correct answer for this?
Answer: choice B
Step-by-step explanation:
Events A and B are independent if the equation P(B)=P(B|A) or P(A∩B) = P(A) · P(B) holds true.
in this example
p(A)=1/6 {5}
p(B)=1/2 {1,3,5}
P(B|A)=1
so
P(B)≠P(B|A)
=>A and B are dependent
The temperature fell from 0 Degrees Fahrenheit to 15 and one-half Degrees Fahrenheit below 0 in 5 and three-fourths hours. Wen tried to find the change in temperature per hour. Her work is shown below. Negative 15 and one-half divided by 5 and three-fourths = negative StartFraction 31 over 2 EndFraction divided by StartFraction 23 over 4 EndFraction = negative StartFraction 31 over 2 EndFraction times StartFraction 23 over 4 EndFraction = Negative StartFraction 713 over 8 EndFraction
Answer:
The correct answer will be:
[tex]-\dfrac{62}{23}[/tex]
Step-by-step explanation:
It is given that :
Initial temperature, [tex]T_1 = 0^\circ F[/tex]
Final temperature,
[tex]T_2 = -15\dfrac{1}{2}^\circ F\\\Rightarrow T_2 = -\dfrac{15\times 2+1}{2} ^\circ F\\\Rightarrow T_2 = -\dfrac{31}{2} ^\circ F[/tex]
Time taken :
[tex]5\dfrac{3}{4}\ hrs = \dfrac{5 \times 4+3}{4}\ hrs = \dfrac{23}{4}\ hrs[/tex]
Change in temperature per hour:
[tex]\dfrac{\text{Difference of temperature}}{\text{Total Time Taken}}\\\Rightarrow \dfrac{T_2-T_1}{\text{Total Time Taken}}[/tex]
Putting the values of temperatures and time:
[tex]\dfrac{\dfrac{-31}{2}-0}{\dfrac{23}{4}}\\\Rightarrow \dfrac{\dfrac{-31}{2}}{\dfrac{23}{4}}\\\Rightarrow \dfrac{-31 \times 4}{2 \times 23}} \text{---- Error done by Wen at this step}\\\Rightarrow \dfrac{-31 \times 2}{23}}\\\Rightarrow \dfrac{-62}{23}}[/tex]
The error done by Wen was during calculating the values of fraction.
So, the correct answer is :[tex]\frac{-62}{23}}[/tex] instead of [tex]\frac{-713}{8}[/tex]
Answer:
C. Wen did not take the reciprocal of the divisor
Step-by-step explanation:
The sugar content of the syrup in canned peaches is normally distributed. Suppose that the variance is thought to be σ2=18 (milligrams)2. A random sample of n=10 cans yields a sample standard deviation of s=4.8 milligrams. Part 1 (a) Test the hypothesis H0:σ2=18 versus H1:σ2≠18 using α=0.05 Find χ02 .
Answer:
[tex]\chi^2 =\frac{10-1}{18} 23.04 =11.52[/tex]
The degrees of freedom are:
[tex] df =n-1=10-1=9[/tex]
Now we can calculate the critical value taking in count the alternative hypotheis we have two values:
[tex]\chi^2_{\alpha/2}= 2.70[/tex]
[tex]\chi^2_{1-\alpha/2}= 19.02[/tex]
Since the calculated value is between the two critical values we FAIL to reject the null hypothesis and we can't conclude that the true variance is different from 18
Step-by-step explanation:
Information given
[tex]n=10[/tex] represent the sample size
[tex]\alpha=0.05[/tex] represent the confidence level
[tex]s^2 =4.8^2= 23.04 [/tex] represent the sample variance obtained
[tex]\sigma^2_0 =18[/tex] represent the value to verify
System of hypothesis
We want to verify if the true variance is different from 18, so the system of hypothesis would be:
Null Hypothesis: [tex]\sigma^2 = 18[/tex]
Alternative hypothesis: [tex]\sigma^2 \neq 18[/tex]
The statistic would be given by:
[tex]\chi^2 =\frac{n-1}{\sigma^2_0} s^2[/tex]
And replacing we got:
[tex]\chi^2 =\frac{10-1}{18} 23.04 =11.52[/tex]
The degrees of freedom are:
[tex] df =n-1=10-1=9[/tex]
Now we can calculate the critical value taking in count the alternative hypotheis we have two values:
[tex]\chi^2_{\alpha/2}= 2.70[/tex]
[tex]\chi^2_{1-\alpha/2}= 19.02[/tex]
Since the calculated value is between the two critical values we FAIL to reject the null hypothesis and we can't conclude that the true variance is different from 18
which fraction is equivalent to 5/12+1/4
Answer:
2/3
Step-by-step explanation:
5/12+1/4
Make the fractions have the same denominator.
5/12 + 1×3/4×3
5/12 + 3/12
Add the fractions.
(5+3)/12
8/12
Simplify.
2/3
Answer:
[tex] = \frac{2}{3} \\ [/tex]
Step-by-step explanation:
[tex] \frac{5}{12} + \frac{1}{4} \\ \frac{5 + 1 \times 3}{12} \\ \frac{5 + 3}{12} \\ = \frac{8}{12} \\ = \frac{2}{3} [/tex]
Please answer this correctly
Answer: A ≈ 3.25 in squared
Step-by-step explanation:
So we can work backward using the perimeter of a circle formula to find the radius.
Perimeter of a whole circle: C = 2* 3.14 * r
Since we are only given a quarter of the perimeter we need to multiply the 3.57 by 4 to find C.
C = 14.28
Now we can solve for r using algebra.
[tex]14.28 = 2 *3.14*r[/tex]
Multiply 2 and 3.14
[tex]14.28 = 6.28*r[/tex]
Isolate r through division.
[tex]2.2739 =r\\[/tex]
Now that we have the radius of the circle we can plug that into the area of a circle.
Formula: [tex]A = 3.14* r^2[/tex]
Plug in and solve.
[tex]A = 3.14 * 2.2738^2[/tex]
A = 16.2355 in squared
This however is the area of the whole circle it is just looking for a quarter.
So we can divide the area of the whole circle by 4 to find the final answer.
16.2355 / 4 = 3.2471
Now round to the nearest hundreds which is the second number after the decimal.
A ≈ 3.25
Please answer this question !! 20 points and brainliest !! Thank u !!
Answer:
Step-by-step explanation:
The system of equations (a) is the correct one for this situation. We must now solve this system for x and y, which are the unit costs.
Multiplying the 2nd equation by (-3/2) yields -12x - 18y = -168.
Now eliminate x by combining the two equations
12x + 5 y = 116
-12x -18y = - 168
-------------------------
-13y = -52, or y = 4.
If y = 4 then 12x + 5y = 116, or
12x +5(4) = 116, or
12x + 20 = 116, or
12x = 96, or x = 8
Roller pens are $8/dozen and ball point pens are $4 per dozen.
I think it’s A:
Cause it is what it is
Wendy has only nickels and dimes in her pocket. The number of dimes is 8 less than triple the number of nickels. Let n represent the number of nickels. Write an expression for the number of dimes.
Answer:
The number of dimes is 3n - 8
Step-by-step explanation:
Here, we are interested in writing an expression for the number of dimes.
We proceed as follows;
Now, there are n nickels with the number of dimes been 8 less than 3 times the number of nickels
That would be;
3(n) -8 = 3n -8
Annie was told that her math test score was 3 standard deviations below the mean. If test scores were approximately normal with μ=99 and σ=4, what was Annie's score? Do not include units in your answer. For example, if you found that the score was 99 points, you would enter 99.
Answer:
[tex]X \sim N(99,4)[/tex]
Where [tex]\mu=99[/tex] and [tex]\sigma=4[/tex]
We want to find the Annie's score takign in count that the score is 3 deviations below the mean, so then we can find the value with this formula:
[tex] X = \mu -3\sigma[/tex]
And replacing we got:
[tex] X = 99 -3*4 = 87[/tex]
So then the Annie's score would be 87
Step-by-step explanation:
Let X the random variable that represent the test scores of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(99,4)[/tex]
Where [tex]\mu=99[/tex] and [tex]\sigma=4[/tex]
We want to find the Annie's score takign in count that the score is 3 deviations below the mean, so then we can find the value with this formula:
[tex] X = \mu -3\sigma[/tex]
And replacing we got:
[tex] X = 99 -3*4 = 87[/tex]
So then the Annie's score would be 87
Answer: 87
Step-by-step explanation:
We can work backwards using the z-score formula to find the x-value. The problem gives us the values for z, μ and σ. So, let's substitute these numbers back into the formula:
z−3−1287=x−μσ=x−994=x−99=x
We can think of this conceptually as well. We know that the z-score is −3, which tells us that x is three standard deviations to the left of the mean, 99. So we can think of the distance between 99 and the x-value as (3)(4)=12. So Annie's score is 99−12=87.
The number of customers waiting for gift-wrap service at a department store is an rv X with possible values 0, 1, 2, 3, 4 and corresponding probabilities 0.1, 0.2, 0.3, 0.25, 0.15. A randomly selected customer will have 1, 2, or 3 packages for wrapping with probabilities 0.55, 0.35, and 0.1, respectively. Let Y = the total number of packages to be wrapped for the customers waiting in line (assume that the number of packages submitted by one customer is independent of the number submitted by any other customer).
(a) Determine P(X = 3, Y = 3), i.e., p(3,3). (Round your answer to four decimal places.)
(b) Determine p(4,11). (Round your answer to four decimal places.) p(4,11) = ?
Answer:
(a)p(3,3)=0.0416
(b)p(4,11)=0.0002
Step-by-step explanation:
Number of Customers, X
[tex]\left\begin{array}{|c|ccccc|}X&0&1&2&3&4\\P(X)&0.1&0.2&0.3&0.25&0.15.\end{array}\right[/tex]
Y = the total number of packages to be wrapped for the customers waiting in line
[tex]\left\begin{array}{|c|ccccc|}Y&1&2&3\\P(Y)&0.55&0.35&0.1\end{array}\right|[/tex]
a. P(X = 3, Y = 3)
p(3,3) means that there are 3 customers with one gift each
The probability of this event happening:
[tex]0.25 \times 0.55^3=0.0416[/tex]
p(3,3)=0.0416
b. p(4,11)
For 4 people to have a total package of 11, there must be 3 customers with 3 packages each and 1 customer with 2 packages,
The probability of this happening is:
[tex]p(4,11)=0.15\times^4C_1\times0.1^3\times0.35\\p(4,11)=0.0002[/tex]
Lines f and g are perpendicular. The equation of line f is y = 1/4x + 2. What is
the equation of line g?
Answer:
y = -4x + (any number)
Step-by-step explanation:
you want to use the negative reciprocal of the slope for line f
so m = -4, since you didn't say any points it has to go through then the
equation of the line is y = -4x + (any number)
Answer:
-4× - 4 That is the answer
A sports company wants to package a ball with a 1.5-inch radius in sets of two. They have two options: a cylinder or a square prism. 2 balls are inside of a cylinder and 2 balls are inside of a square prism. The cylinder has a height of 6 inches and a radius of 1.5 inches. The square prism has base lengths of 3 inches and the prism has a height of 6 inches. The company wants to use the package that has the least amount of wasted space. The company should choose
Answer:
C. the cylinder because it has approximately 11.6 in.3 less wasted space than the prism.
The company should choose the cylinder because it has approximately 11.6 in.3 less wasted space than the prism.
We have given that,
A sports company wants to package a ball with a 1.5-inch radius in sets of two.
They have two options a cylinder or a square prism.
2 balls are inside of a cylinder and 2 balls are inside of a square prism. The cylinder has a height of 6 inches and a radius of 1.5 inches.
What is the square prism?A square prism is basically a cuboid, that has square bases. It has four rectangular faces and two square-shaped ends. In Geometry, we have studied various three-dimensional shapes called solid shapes or solids.
The square prism has base lengths of 3 inches and the prism has a height of 6 inches.
The company wants to use the package that has the least amount of wasted space.
Therefore, The company should choose the cylinder because it has approximately 11.6 in.3 less wasted space than the prism.
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A store that sells skis buys them from a manufacturer at a wholesale price of $87 The store's markup rate is 50% A. What price does the store charge its customers for the skis? B. What percent of the original price is the final price? C. What is the percent INCREASE from the original price to the final price?
Answer:
A. $130.50
B. 150%
C. 50%
Step-by-step explanation:
First, find 50% of 87 and add it to find the new price.
87 x 0.5 = 43.5
87 + 43.5 = 130.5
Then, find the percent by doing 130.5 ÷ 87, which is 150%
Calculate the percent increase by doing (130.5-87) ÷ 87, which is 50%
Answer:
see below
Step-by-step explanation:
First find the markup
87* 50%
87*.50
43.50
Add this to the wholesale price
87+43.50 =130.50
This is what they charge the customer
This is 150% percent of the wholesale price
100% + 50% = 150%
To find the percent increase
Take the new price and subtract the wholesale price
Then divide by the wholesale price
(130.50-87)/87 =50 % increase
Identify the table of values which represents the function y=x+2
Answer:
Option 1.
Step-by-step explanation:
y = x + 2
Put x as 1, 2 and 3 to find y.
y = (1) + 2
y = 3
y = (2) + 2
y = 4
y = (3) + 2
y = 5
When x = 1, y = 3.
When x = 2, y = 4.
When x = 3, y =5.
The values that represent the function is the first table.
Work out the value of x
Answer:
67.5
Step-by-step explanation:
right angle =90 degree
total = 360 degree
360-90=270
270÷4=67.5
What’s the correct answer for this question?
Answer:
B
Step-by-step explanation:
I put the answer in an atachement
Answer:
B.
Step-by-step explanation:
In the attached file
This data set represents the number of
children in 8 families
4, 2, 1, 2, 4, 2, 6, 3
The mean of this data set is 3. What is the
Mean Absolute Deviation (MAD)?
A. 1.25
B. 8
C. 3.3
D. 3
Answer:
1.25
Step-by-step explanation:
Lets sum up the values
24
and then divide by their number (8)
we get 3 on average
now let's see how much each data point varies from that average (this is always a positive value)
1, 1, 2, 1, 1, 1, 3, 0
Let's sum and divide by 8 again
10/8 = 1.25
(would really, reallly appreciate the brainliest)
Find the value when x= 2 and y =3.
2x⁰ y-²
1/9
2/9
-18
Answer:
36
Step-by-step explanation:
Answer:
The answer is 2/9
Step-by-step explanation:
2(2)^0 (3)^-2 = 2(1)/(3)^2 = 2/9
Which of the following expressions represents the distance between -4/3 and 1/3?
Answer:
none of the above
Step-by-step explanation:
Take the absolute value of the second point and subtract the first point
| 1/3 -( -4/3)|
Since this answer does not match any of the above choices
Answer:
None of the above.
Step-by-step explanation:
The distance between -4/3 and 1/3 is
|1/3-(-4/3)| =|1/3 + 4/3|
This is not in the option so;
None of the above.
Use the function rule f(x)=x^2
Find f(2.5)__
Answer:
6.25
Step-by-step explanation:
f(x)=x^2
f(2.5) = (2.5)^2
=6.25
Answer: 6.25
Step-by-step explanation: Notice that f is a function of x.
So we want to find f(2.5).
We find f(2.5) by plugging 2.5 in for x
everywhere that x appears in the function.
So we have f(2.5) = (2.5)².
(2.5)² is 6.25.
So our answer is 6.25.
5 Samir wants to work out the cost of the tiles needed to replace a roof.
The roof has 4 identical faces.
Each face is a triangle.
Each triangle has a base length of 7.6 m and a height of 4.8 m.
Samir has this information,
roof tiles
1 pack of tiles covers 13.8 m² (including overlaps)
each pack costs £716.10
Answer:
Step-by-step explanation:
Since the roof has 4 identical faces and each face is a triangle, we would determine the area of each face of the triangle by applying the formula,
Area of triangle = 1/2 × base × height
Area of each triangular face = 1/2 × 7.6 × 4.8 = 18.24m²
Area of the 4 faces = 4 × 18.24 = 72.96 m²
Since 1 pack of tiles covers 13.8 m², the number of packs needed to cover 72.96 m² is
72.96/13.8 = 5.29 packs
Since samir can only buy whole packs of these tiles, the number of packs of tiles that he needs to buy would be 6.
If 1 pack costs £716.1, then the total cost of the tiles needed for the 4 faces of this roof is
6 × 716.1 = £4296.6
HELPPP The number of members in the Triathlon Club was 36 in 2001 and has increased by 20% each year. Which exponential growth model shows the club's membership in terms of t, the number of years since 2001? s since 2001?
A. y=20(36)t
B. y=36(0.2)t
C. y=36(1.2)t
D. y=36(20)t
Answer:
C. y=36(1.2)t
Step-by-step explanation:
The number of members in the Club in t years after 2001 is given by an equation in the following format:
[tex]y = y(0)(1+r)^{t}[/tex]
In which y(0) is the number of members in 2001 and r is the growth rate, as a decimal.
The number of members in the Triathlon Club was 36 in 2001 and has increased by 20% each year.
This means that [tex]y(0) = 36, r = 0.2[/tex]
So
[tex]y = y(0)(1+r)^{t}[/tex]
[tex]y = 36(1+0.2)^{t}[/tex]
[tex]y = 36(1.2)^{t}[/tex]
The correct answer is C.
At a certain gas station 40% of the customers request regular gas, 35% request unleaded gas, and 25% request premium gas. Of those customers requesting regular gas, only 30% fill their tanks all the way up, while the remaining 70% only fill up part of their tank. Of those customers requesting unleaded gas, 60% fill their tanks all the way up, while of those requesting premium, 50% fill their tanks all the way up. If the next customer does not fill the tank all the way up (only fills it up part of the way), what is the probability that they requested regular gas?
Answer:
51.38% probability that they requested regular gas
Step-by-step explanation:
Bayes Theorem:
Two events, A and B.
[tex]P(B|A) = \frac{P(B)*P(A|B)}{P(A)}[/tex]
In which P(B|A) is the probability of B happening when A has happened and P(A|B) is the probability of A happening when B has happened.
In this question, we have that:
Event A: Not filling the tank
Event B: Regular gas
40% of the customers request regular gas
This means that [tex]P(B) = 0.4[/tex]
Of those customers requesting regular gas, 70% only fill up part of their tank.
This means that [tex]P(A|B) = 0.7[/tex]
Probability of not filling the tank:
70% of 40%(regular gas)
100 - 60 = 40% of 35%(unleaded gas).
100 - 50 = 50% of 25%(premium gas).
So
[tex]P(A) = 0.7*0.4 + 0.4*0.35 + 0.5*0.25 = 0.545[/tex]
What is the probability that they requested regular gas?
[tex]P(B|A) = \frac{0.4*0.7}{0.545}[/tex] = 0.5138
51.38% probability that they requested regular gas
Solve the equation for the indicated variable.
660w
C=
for w
h2
W=
(Simplify your answer.)
The first step is to multiply both sides by h^2. Afterwards, divide both sides by 660 to fully isolate w.
c = 660w/(h^2)
ch^2 = 660w
660w = ch^2
w = (ch^2)/660 is the answer
A university found that 20% of its students withdraw without completing the introductory statistics course. Assume that 20 students registered for the course. (a) Compute the probability that 2 or fewer will withdraw. If required, round your answer to four decimal places. (b) Compute the probability that exactly 4 will withdraw. If required, round your answer to four decimal places. (c) Compute the probability that more than 3 will withdraw. If required, round your answer to four decimal places. (d) Compute the expected number of withdrawals.
Answer:
a) 0.206 = 20.6% probability that 2 or fewer will withdraw.
b) 0.2182 = 21.82% probability that exactly 4 will withdraw.
c) 0.5886 = 58.86% probability that more than 3 will withdraw.
d) The expected number of withdrawals is 4.
Step-by-step explanation:
For each student, there are only two possible outcomes. Either they withdraw without completing the introductory statistics course, or they do not. Each student is independent of each other. So we use the binomial probability distribution to solve this question.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
20% of its students withdraw without completing the introductory statistics course.
This means that [tex]p = 0.2[/tex]
Assume that 20 students registered for the course.
This means that [tex]n = 20[/tex]
(a) Compute the probability that 2 or fewer will withdraw.
[tex]P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)[/tex]
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{20,0}.(0.2)^{0}.(0.8)^{20} = 0.0115[/tex]
[tex]P(X = 1) = C_{20,1}.(0.2)^{1}.(0.8)^{19} = 0.0576[/tex]
[tex]P(X = 2) = C_{20,2}.(0.2)^{2}.(0.8)^{18} = 0.1369[/tex]
[tex]P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) = 0.0115 + 0.0576 + 0.1369 = 0.206[/tex]
0.206 = 20.6% probability that 2 or fewer will withdraw.
(b) Compute the probability that exactly 4 will withdraw.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 4) = C_{20,4}.(0.2)^{4}.(0.8)^{16} = 0.2182[/tex]
0.2182 = 21.82% probability that exactly 4 will withdraw.
(c) Compute the probability that more than 3 will withdraw.
Either less than 3 withdraw, or more than 3 withdraw. The sum of the probabilities of these events is 1. So
[tex]P(X \leq 3) + P(X > 3) = 1[/tex]
We want P(X > 3). So
[tex]P(X > 3) = 1 - P(X \leq 3)[/tex]
In which
[tex]P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)[/tex]
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 0) = C_{20,0}.(0.2)^{0}.(0.8)^{20} = 0.0115[/tex]
[tex]P(X = 1) = C_{20,1}.(0.2)^{1}.(0.8)^{19} = 0.0576[/tex]
[tex]P(X = 2) = C_{20,2}.(0.2)^{2}.(0.8)^{18} = 0.1369[/tex]
[tex]P(X = 3) = C_{20,3}.(0.2)^{3}.(0.8)^{17} = 0.2054[/tex]
[tex]P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 0.0115 + 0.0576 + 0.1369 + 0.2054 = 0.4114[/tex]
[tex]P(X > 3) = 1 - P(X \leq 3) = 1 - 0.4114 = 0.5886[/tex]
0.5886 = 58.86% probability that more than 3 will withdraw.
(d) Compute the expected number of withdrawals.
The expected value of the binomial distribution is:
E(X) = np
In this question:
E(X) = 20*0.2 = 4
The expected number of withdrawals is 4.
Recently, FHA mortgages, which are insured by the federal government, accounted for 28% of all home-purchase mortgages that were approved. A random sample of 150 mortgage applications was selected. What is the probability that 48 or more from this sample were insured by the FHA?
Answer:
15.87% probability that 48 or more from this sample were insured by the FHA
Step-by-step explanation:
I am going to use the normal approximation to the binomial to solve this question.
Binomial probability distribution
Probability of exactly x sucesses on n repeated trials, with p probability.
Can be approximated to a normal distribution, using the expected value and the standard deviation.
The expected value of the binomial distribution is:
[tex]E(X) = np[/tex]
The standard deviation of the binomial distribution is:
[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]
Normal probability distribution
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score 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. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
When we are approximating a binomial distribution to a normal one, we have that [tex]\mu = E(X)[/tex], [tex]\sigma = \sqrt{V(X)}[/tex].
In this problem, we have that:
[tex]p = 0.28, n = 150[/tex]
So
[tex]\mu = E(X) = np = 150*0.28 = 42[/tex]
[tex]\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{150*0.28*0.72} = 5.5[/tex]
What is the probability that 48 or more from this sample were insured by the FHA?
Using continuity correction, this is [tex]P(X \geq 48 - 0.5) = P(X \geq 47.5)[/tex], which is 1 subtracted by the pvalue of Z when X = 47.5. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{47.5 - 42}{5.5}[/tex]
[tex]Z = 1[/tex]
[tex]Z = 1[/tex] has a pvalue of 0.8413.
1 - 0.8413 = 0.1587
15.87% probability that 48 or more from this sample were insured by the FHA