Answer:
The equation of the tangent line is [tex]y = -\frac{5}{18}\cdot x +\frac{14}{9}[/tex].
Step-by-step explanation:
Firstly, we obtain the equation for the slope of the tangent line by implicit differentiation:
[tex]2\cdot x + 6\cdot y + 6\cdot x \cdot y' + 24\cdot y \cdot y' = 0[/tex]
[tex]2\cdot (x + 3\cdot y) + 6\cdot (x + 4\cdot y) \cdot y' = 0[/tex]
[tex]6\cdot (x + 4\cdot y) \cdot y' = -2\cdot (x+3\cdot y)[/tex]
[tex]y' = -\frac{1}{3}\cdot \left(\frac{x + 3\cdot y}{x + 4\cdot y} \right)[/tex] (1)
If we know that [tex](x,y) = (2, 1)[/tex], then the slope of the tangent line is:
[tex]y' = -\frac{1}{3}\cdot \left(\frac{2+3\cdot 1}{2 + 4\cdot 1} \right)[/tex]
[tex]y' =-\frac{5}{18}[/tex]
By definition of tangent line, we determine the intercept of the line ([tex]b[/tex]):
[tex]y = m\cdot x + b[/tex]
[tex]b = y - m\cdot x[/tex] (2)
If we know that [tex](x,y) = (2,1)[/tex] and [tex]m = -\frac{5}{18}[/tex], then the intercept of the tangent line is:
[tex]b = 1 - \left(-\frac{5}{18} \right)\cdot (2)[/tex]
[tex]b = \frac{14}{9}[/tex]
The equation of the tangent line is [tex]y = -\frac{5}{18}\cdot x +\frac{14}{9}[/tex].
change the following basis to Base 10 134 in base seven
Answer:
74 base 10.
Step-by-step explanation:
134 base 7 = 7^2 + 3*7 + 4
= 49 + 21 + 4
= 74 base 10
y + x + z =762500
z : x = 15/9 : 2
y : x = 1 : 3/4
Step-by-step explanation:
true
Which is a stretch of an exponential decay function?
f(x) =4/5(5/4)
f(x) = 4/5(4/5)
f(x) =5/4(4/5)
fx) = 5/4(5/4)
Answer:
f(x) = 4/5(5/4)Step-by-step explanation:
correct me if I am wrong
A chemical company makes two brands of antifreeze the first brand contains 65% pure antifreeze and the second brand contains 80% pure antifreeze in order to obtain 60 gallons of a mixture that contains 70% pure antifreeze how many gallons of each brand of antifreeze must be used
Answer:
30 gallons of each brand
Step-by-step explanation:
1 gallon of 60% + 1 gallon of 80% = 2 gallons 70% (60/2 = 30)
When 4(9y − 5) = 10(3y + 17) − 40 is solved, the result is:
A=20.
B=25.
C=-10.
D=10.
Answer:
B = 25
General Formulas and Concepts:
Pre-Algebra
Order of Operations: BPEMDAS
Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to RightDistributive Property
Equality Properties
Multiplication Property of Equality Division Property of Equality Addition Property of Equality Subtraction Property of EqualityAlgebra I
Terms/CoefficientsStep-by-step explanation:
Step 1: Define
Identify
4(9y - 5) = 10(3y + 17) - 40
Step 2: Solve for y
[Distributive Property] Distribute 4: 36y - 20 = 10(3y + 17) - 40[Distributive Property] Distribute 10: 36y - 20 = 30y + 170 - 40Combine like terms: 36y - 20 = 30y + 130[Subtraction Property of Equality] Subtract 30y on both sides: 6y - 20 = 130[Addition Property of Equality] Add 20 on both sides: 6y = 150[Division Property of Equality] Divide 6 on both sides: y = 25[tex]\huge\textsf{Hey there!}[/tex]
[tex]\large\textsf{4(9y - 5) = 10(3y + 17) - 40}\\\\\large\textsf{4(9y) + 4(-5) = 10(3y) + 10(17) - 10(40)}\\\\\large\textsf{36y - 20 = 30y + 170 - 40}\\\\\large\textsf{COMBINE the LIKE TERMS}\\\\\large\textsf{36y - 20 = (30 y)+ (170 - 40)}\\\\\large\textsf{36y - 20 = 30y + 130}\\\\\large\textsf{SUBTRACT 30y to BOTH SIDES}\\\\\large\textsf{36y - 20 - 30y = 30y + 130 - 30}}\\\\\large\textsf{Cancel out: 30y - 30y because that gives you 0}\\\\\large\textsf{Keep: 20 - 30y because helps solve for the y-value}[/tex]
[tex]\large\textsf{NEW EQUATION: 6y - 20 = 130}\\\\\large\textsf{ADD 20 to BOTH SIDES}\\\\\large\textsf{6y - 20 + 20 = 130 + 20}\\\\\large\textsf{Cancel out: -20 + 20 because that gives you 0}\\\\\large\textsf{Keep: 130 + 20 because that helps solve for the y-value}\\\\\large\textsf{130 + 20 = \bf 150}\\\\\large\textsf{NEW EQUATION: 6y = 150}\\\\\large\textsf{DIVIDE 6 to BOTH SIDES}\\\\\mathsf{\dfrac{6y}{6}= \dfrac{150}{6}}\\\\\large\textsf{Cancel: }\mathsf{\dfrac{6}{6}\large\textsf{ because that gives you 1}}[/tex]
[tex]\large\textsf{Keep: }\mathsf{\dfrac{150}{6}}\large\textsf{ because it helps solve for the y-value}\\\\\large\textsf{\bf y = }\mathsf{\dfrac{150}{6}}\\\\\large\textsf{OR }\\\\\mathsf{\dfrac{150}{6} }\large\textsf{ = \bf y}\\\\\\\large\textsf{SIMPLIFY ABOVE AND TOU YOU HAVE YOUR Y-VALUE}\uparrow\\\\\\\\\boxed{\boxed{\large\textsf{\huge\textsf{Answer: \bf y = 25} (Option B.)}}}\huge\checkmark\\\\\\\\\large\text{Good luck on your assignment and enjoy your day!}\\\\\\\\\\\frak{Amphitrite1040:)}[/tex]
How many boxes could you stack safely on a pallet if the pallet is 5 feet deep, five feet across, every box is 1 x 1 and the maximum safe stacking height is 5 boxes? *
Answer:
125 boxes
Step-by-step explanation:
5*5*5
Find the distance from the vertices of AABC to the corresponding vertices of the other three triangles, and enter them in the table. For AJKL-
you'll need to use the distance formula da (01 – 12) + (y1 - y2) . Verify your calculations using the tools available in GeoGebra.
Answer:
See explanation
Step-by-step explanation:
The question is incomplete, as the vertices of the triangle are not given.
A general explanation is as follows;
To calculate distance between two points, we use:
[tex]d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}[/tex]
Take for instance;
[tex]A = (1,4)[/tex]
[tex]B = (3,-2)[/tex]
Distance AB is:
[tex]AB = \sqrt{(1 - 3)^2 + (4 - -2)^2}[/tex]
[tex]AB = \sqrt{(- 2)^2 + (4 +2)^2}[/tex]
[tex]AB = \sqrt{(- 2)^2 + (6)^2}[/tex]
Evaluate the exponents
[tex]AB = \sqrt{4 + 36}[/tex]
[tex]AB = \sqrt{40}[/tex]
[tex]AB = 6.32[/tex]
Answer:
for edmentum
Step-by-step explanation:
Which of the following functions shows the reciprocal parent function, F(x) = 1, shifted left?
Answer:
G(x)=1/x+15
Step-by-step explanation:
.
Amy types at an average speed of 38 words per rinute. She has already typed 1,450 words of her final paper, which will be more than 4,000
words. Which inequality can be used to solve for x, the number of minutes it will take Amy to finish typing her paper?
ОА.
38x-1,450 > 76
OB.
38[X+1,450) > 4,000
Ос. .
38x> 4,000
OD.
38x + 1,450 > 4,000
Reset
Next
ntum. All rights reserved.
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74°F Mostly cloudy
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De here to search
Answer: D. 38x + 1,450 > 4,000
Step-by-step explanation:
It has to be greater than 4,000 so A makes no sense
The parentheses are in the wrong place completely changing the meaning for B
C disregards the info we have about how she's already typed 1,450 words
The answer has to be D
Two mechanics worked on a car. The first mechanic worked for 10 hours, and the second mechanic worked for 5 hours. Together they charged a total of $1125. What was the rate charged per hour by each mechanic if the sum of the two rates was $140 per hour?
Answer:
The first mechanic charged $ 85 an hour, and the second mechanic charged $ 55 an hour.
Step-by-step explanation:
Given that two mechanics worked on a car, and the first mechanic worked for 10 hours, and the second mechanic worked for 5 hours, and together they charged a total of $ 1125, to determine what was the rate charged per hour by each mechanic if the sum of the two rates was $ 140 per hour, the following calculation must be performed:
1125/15 = X
75 = X
80 x 10 + 60 x 5 = 800 + 300 = 1100
85 x 10 + 55 x 5 = 850 + 275 = 1125
Therefore, the first mechanic charged $ 85 an hour, and the second mechanic charged $ 55 an hour.
Solve the right triangle ABC, with C = 90.00◦ , a = 15.21 cm, b = 17.34 cm. Round to two decimal places.
Answer:
the hypotenuse side, c = 23.1 cmangle A = 41.26 ⁰angle B = 48.74 ⁰Step-by-step explanation:
Given;
first leg of the right triangle, a = 15.21 cm
second leg of the right triangle, b = 17.34 cm
Angle C = 90 ⁰
The missing parameters;
the hypotenuse side = cangle Aangle BUse Pythagoras theorem to calculate the missing side "c", which is the hypotenuse
c² = a² + b²
c² = (15.21)² + (17.34)²
c² = 532.02
c = √532.02
c = 23.1 cm
The missing angle A is calculated as;
[tex]tan(A) = \frac{a}{b} \\\\tan(A) = \frac{15.21}{17.34} \\\\tan(A) = 0.8772 \\\\A = tan^{-1} (0.8772)\\\\A = 41.26^0[/tex]
The missing angle is calculated as;
B = 90⁰ - A
B = 90⁰ - 41.26⁰
B = 48.74⁰
Is this a function? Yes or no?
Answer:
NO
Step-by-step explanation:
NO
Dora drove 80 km in 40 minutes then 120 km in 1 hour and then the final 200km took him 1 hour 20 minutes. what is his average speed for the whole journey
9514 1404 393
Answer:
133 1/3 km/hour
Step-by-step explanation:
Average speed is computed as ...
average speed = (total distance)/(total time)
= (80 km +120 km +200 km)/(2/3 + 1 + 1 1/3 hour) = 400 km/(3 hour)
= 133 1/3 km/hour
Suppose triangle ABC is reflected over the x-axis. If the distance between point A and A’ is 14, what is the distance between the x-axis and A’.
1. 7
2. -7
3. 3.5
4. There is not enough information given.
Given:
The triangle ABC is reflected over the x-axis.
The distance between point A and A’ is 14.
To find:
The distance between the x-axis and A’.
Solution:
If a figure is reflected across the x-axis then the corresponding parts are mirror image of each other about the x-axis.
It means the distance between A and x-axis is same as the distance between x-axis and A'.
The distance between point A and A’ is 14.
Let d be the distance between the x-axis and A’. Then,
[tex]d+d=14[/tex]
[tex]2d=14[/tex]
[tex]d=\dfrac{14}{2}[/tex]
[tex]d=7[/tex]
Therefore, the correct option is 1.
find the slope of a line perpendicular to the line below. y=2x+4
Please help me in this question
Answer:
3/8
Step-by-step explanation:
the total number of possible results is 4×4=16.
out of these 16 only the results
1 2
1 3
1 4
2 2
2 3
3 2
are desired results. these are 6.
so the probability of a desired result is 6/16 = 3/8
Private nonprofit four-year colleges charge, on average, $26,208 per year in tuition and fees. The standard deviation is $7,040. Assume the distribution is normal. Let X be the cost for a randomly selected college. Round all answers to 4 decimal places where possible.
a. What is the distribution of X? X ~ N(
26208
Correct,
7040
Correct)
b. Find the probability that a randomly selected Private nonprofit four-year college will cost less than 22,924 per year.
c. Find the 60th percentile for this distribution. $
(Round to the nearest dollar.)
Answer:
#########
Step-by-step explanation:
The table shows the relationship between the number of faculty members and the number of students at a local school. What is the missing value?
Faculty
Students
1
17
2
34
3
51
4
?
17
68
85
102
Answer:
68
Step-by-step explanation:
I did it on my test
The missing value in the table is 68. The correct answer would be option (B).
What is the linear relationship?A linear relationship is a connection that takes the shape of a straight line on a graph between two distinct variables - x and y. When displaying a linear connection using an equation, the value of y is derived from the value of x, indicating their relationship.
The table shows the relationship between the number of faculty members and the number of students at a local school.
Faculty Students
1 17
2 34
3 51
4 ?
The relationship between the number of faculty members and the number of students at the local school is that for every faculty member, there are 17 students.
Therefore, if there are 4 faculty members, we can find the number of students by multiplying 4 by 17, which gives us 68.
Thus, the missing value in the table is B. 68.
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GIVING OUT BRAINLIEST IF GIVEN AN ANSWER WITH THOUROUGH EXPLANATION AND NOT JUST AN ANSWER! THANKS!
During the first part of a 6-hour trip, you travel 240 miles at an average speed of r miles per hour. For the next 72 miles of the trip, you
increase your speed by 10 miles per hour. What were
your two average speeds?
9514 1404 393
Answer:
50 mph for 4.8 hours60 mph for 1.2 hoursStep-by-step explanation:
One way to write the relationship between time, speed, and distance is ...
time = distance/speed
For the first part of the trip, the time is ...
t1 = 240/r
For the second part of the trip, the time is ...
t2 = 72/(r+10)
The total time is 6 hours, so we have ...
t1 +t2 = 6
240/r +72/(r+10) = 6
We can simplify this a bit by multiplying by (r)(r+10)/6 to get ...
40(r+10) +12(r) = r(r+10)
r² -42r -400 = 0 . . . . . . . . subtract the left side and collect terms
(r -50)(r +8) = 0 . . . . . . . . factor
r = 50 . . . . . the positive solution of interest.
The two average speeds were 50 mph and 60 mph.
The ratio of the number of cherry tomatoes in a tossed salad to people served is 7:15. If Waldo wants to serve 105 people, how many cherry tomatoes will Waldo use
Answer: 49 cherry tomatoes.
Step-by-step explanation:
7 x
— = — cross multiply and done.
15 105
Using the following image, solve for x.
Answer:
please provide an image.
Consider the probability that no less than 37 out of 295 cell phone calls will be disconnected. Choose the best description of the area under the normal curve that would be used to approximate binomial probability.
a. Area to the right of 36.5
b. Area to the right of 37.5
c. Area to the left of 36.5
d. Area to the left of 37.5
e. Area between 36.5 and 37.5
==========================================================
Explanation:
The phrasing "no less than" means the same as "at least".
Saying "at least 37" means 37 is the lowest we can go.
If x is the number of disconnected calls, then [tex]x \ge 37[/tex] and we want to find the probability of this happening (the max being 295).
We could use the binomial distribution to find the answer, but that would require adding 295-37+1 = 259 different values which could get tedious. So we could use the normal approximation to make things relatively straight forward.
Assuming this binomial meets the requirements of the normal approximation, then we'd look under the normal curve for the area to the right of 36.5; which is why the answer is choice A.
Why 36.5 and not 37? This has to do with the continuity correction factor when translating from a discrete distribution (binomial) to a continuous one (normal).
If we used 37, then we'd be missing out on the edge case. So we go a bit beyond 37 to capture 36.5 instead. It's like a fail safe to ensure we do account for that endpoint of 37. It's like adding a buffer or padding.
------------
Side notes:
Choice B would be the answer if we wanted to excluded 37 from the group, ie if we wanted to calculate [tex]P(x > 37)[/tex] instead of [tex]P(x \ge 37)[/tex]. So we're moving in the opposite direction of choice A to avoid that edge case. We go with "right" instead of "left" since this is what the inequality sign says.Use the appropriate substitutions to write down the first four nonzero terms of the Maclaurin series for the binomial (1+3x)^(-1/3)
Answer:
First term=1
Second term=-x
Third term=[tex]2x^2[/tex]
Fourth term =[tex]-\frac{28}{3!}x^3[/tex]
Step-by-step explanation:
We are given that function
[tex]f(x)=(1+3x)^{-1/3}[/tex]
We have to find the first four non zero terms of the Maclaurin series for the binomial.
Maclaurin series of function f(x) is given by
[tex]f(x)=f(0)+f'(0)x+\frac{1}{2!}f''(0)x^2+\frac{1}{3!}f'''(0)x^3+....[/tex]
[tex]f(0)=(1+3x)^{\frac{-1}{3}}=1[/tex]
[tex]f'(x)=-\frac{1}{3}(1+3x)^{-\frac{4}{3}}(3)=-(1+3x)^{-\frac{4}{3}}[/tex]
[tex]f'(0)=-1[/tex]
[tex]f''(x)=\frac{4}{3}\times 3 (1+3x)^{-\frac{7}{3}}[/tex]
[tex]f''(0)=4[/tex]
[tex]f'''(x)=-4\times \frac{7}{3}\times 3(1+3x)^{-\frac{10}{3}}[/tex]
[tex]f'''(0)=-28[/tex]
Substitute the values we get
[tex](1+3x)^{-\frac{1}{3}}=1-x+\frac{4}{2!}x^2+\frac{-28}{3!}x^3+...[/tex]
[tex](1+3x)^{-\frac{1}{3}}=1-x+2x^2+\frac{-28}{3!}x^3+...[/tex]
First term=1
Second term=-x
Third term=[tex]2x^2[/tex]
Fourth term =[tex]-\frac{28}{3!}x^3[/tex]
Allie rode her bike up a hill at an average speed of 12 feet/second. She then rode back down the hill at an average speed of 60 feet/second. The entire trip took her 2 minutes. What is the total distance she traveled. [Hint: use t = time traveling down the hill]
Answer:
The total distance Allie traveled was 0.81 miles.
Step-by-step explanation:
Since Allie rode her bike up a hill at an average speed of 12 feet / second, and she then rode back down the hill at an average speed of 60 feet / second, and the entire trip took her 2 minutes, to determine what is the total distance she traveled, the following calculation must be performed:
12 + 60 = 72
72 x 60 = 4320
1000 feet = 0.189394 miles
4320 feet = 0.8181818 miles
Therefore, the total distance Allie traveled was 0.81 miles.
Construct a frequency distribution and a relative frequency histogram for the accompanying data set using five classes. Which class has the greatest relative frequency and which has the least relative frequency?Complete the table below. Use the minimum data entry as the lower limit of the first class.Class Frequency, f Relative frequencyx-x x xx-x x xx-x x xx-x x xx-x x x sumf= X?(Type integers or decimals. Round to the nearest thousandth as needed.)DATA:Triglyceride levels of 26 patients (in milligrams per deciliter of blood)138 199 240 143 294 175 240 216 223180 138 266 161 175 402 172 459 147391 152 199 294 188 320 421 161
Answer:
[tex]\begin{array}{cc}{Class}& {Frequency} & 138 - 202 & 14 & 203 - 267 & 5 & 268 - 332 & 3 & 333 - 397 & 1 & 398 - 462 & 3 \ \end{array}[/tex]
The class with the greatest is 138- 202 and the class with the least relative frequency is 333 - 397
Step-by-step explanation:
Solving (a): The frequency distribution
Given that:
[tex]Lowest = 138[/tex] --- i.e. the lowest class value
[tex]Class = 5[/tex] --- Number of classes
From the given dataset is:
[tex]Highest = 459[/tex]
So, the range is:
[tex]Range = Highest - Lowest[/tex]
[tex]Range = 459 - 138[/tex]
[tex]Range = 321[/tex]
Divide by the number of class (5) to get the class width
[tex]Width = 321 \div 5[/tex]
[tex]Width = 64.2[/tex]
Approximate
[tex]Width = 64[/tex]
So, we have a class width of 64 in each class;
The frequency table is as follows:
[tex]\begin{array}{cc}{Class}& {Frequency} & 138 - 202 & 14 & 203 - 267 & 5 & 268 - 332 & 3 & 333 - 397 & 1 & 398 - 462 & 3 \ \end{array}[/tex]
Solving (b) The relative frequency histogram
First, we calculate the relative frequency by dividing the frequency of each class by the total frequency
So, we have:
[tex]\begin{array}{ccc}{Class}& {Frequency} & {Relative\ Frequency} & 138 - 202 & 14 & 0.53 & 203 - 267 & 5 & 0.19 & 268 - 332 & 3 & 0.12 & 333 - 397 & 1 & 0.04 & 398 - 462 & 3 & 0.12 \ \end{array}[/tex]
See attachment for histogram
The class with the greatest is 138- 202 and the class with the least relative frequency is 333 - 397
Practice Question
1) VAT (value-added tax) is paid on things that you buy.
The table on the right shows the 2019 VAT rates.
This is how much VAT is charged on certain items
as a percentage of the item's cost.
VẬT (%)
20
5
0.
Items
Chocolate and crisps
Gas and electric
Fruit and vegetables
Currena
Before VAT is added, Simon pays 12p per unit of
electricity plus a fixed charge of £87 per year.
How much does Simon pay in VAT if he uses 3000 units of electricity in one year?
er hour
Shane and Space
Simon will pay £18 in VAT for using 3000 units of electricity in one year.
The VAT rate for gas and electric is 5%.
Therefore, Simon will pay VAT on his electricity usage.
Let's calculate Simon's annual electricity cost without VAT:
Cost per unit of electricity = 12p
= £0.12
Number of units used in one year = 3000
Electricity cost without VAT = Cost per unit × Number of units
= £0.12 × 3000
= £360
Now, let's calculate the VAT amount:
VAT rate = 5% = 0.05
VAT amount = Electricity cost without VAT × VAT rate
= £360 × 0.05
= £18
Therefore, Simon will pay £18 in VAT for using 3000 units of electricity in one year.
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Compute the total cost per year of the following pair of expenses. Then complete the sentence: On an annual basis, the first set of expenses is _______% of the second set of expenses.
Vern buys seven lottery tickets each week at a cost of $3 each and spends $900 per year on his textbooks.
Answer:
So 912$ is 58% of 1584 $
Step-by-step explanation:
Suppose that the walking step lengths of adult males are normally distributed with a mean of 2.5 feet and a standard deviation of 0.4 feet. A sample of 45 men’s step lengths is taken. Step 1 of 2 : Find the probability that an individual man’s step length is less than 1.9 feet. Round your answer to 4 decimal places, if necessary.
Answer:
0.0668 = 6.68% probability that an individual man’s step length is less than 1.9 feet.
Step-by-step explanation:
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set 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]
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 p-value, we get the probability that the value of the measure is greater than X.
Normally distributed with a mean of 2.5 feet and a standard deviation of 0.4 feet.
This means that [tex]\mu = 2.5, \sigma = 0.4[/tex]
Find the probability that an individual man’s step length is less than 1.9 feet.
This is the p-value of Z when X = 1.9. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{1.9 - 2.5}{0.4}[/tex]
[tex]Z = -1.5[/tex]
[tex]Z = -1.5[/tex] has a p-value of 0.0668
0.0668 = 6.68% probability that an individual man’s step length is less than 1.9 feet.
rotation 90° clockwise about the origin
Answer:
Take the picture you uploaded.
Click the rotate 'button' once.
Change the x to y and y to x on the graph. (axis labels)
Done
J (0, -1)
K(-4,-3)
I ( -4,-1)
Suppose 42% of the population has myopia. If a random sample of size 442 is selected, what is the probability that the proportion of persons with myopia will differ from the population proportion by less than 3%
Answer:
0.7994 = 79.94% probability that the proportion of persons with myopia will differ from the population proportion by less than 3%.
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal Probability Distribution
Problems of normal distributions can be solved using the z-score formula.
In a set 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]
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 p-value, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean [tex]\mu = p[/tex] and standard deviation [tex]s = \sqrt{\frac{p(1-p)}{n}}[/tex]
Suppose 42% of the population has myopia.
This means that [tex]p = 0.42[/tex]
Random sample of size 442 is selected
This means that [tex]n = 442[/tex]
Mean and standard deviation:
[tex]\mu = p = 0.42[/tex]
[tex]s = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.42*0.58}{442}} = 0.0235[/tex]
What is the probability that the proportion of persons with myopia will differ from the population proportion by less than 3%?
Proportion between 0.42 + 0.03 = 0.45 and 0.42 - 0.03 = 0.39, which is the p-value of Z when X = 0.45 subtracted by the p-value of Z when X = 0.39.
X = 0.45
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.45 - 0.42}{0.0235}[/tex]
[tex]Z = 1.28[/tex]
[tex]Z = 1.28[/tex] has a p-value of 0.8997
X = 0.39
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.39 - 0.42}{0.0235}[/tex]
[tex]Z = -1.28[/tex]
[tex]Z = -1.28[/tex] has a p-value of 0.1003
0.8997 - 0.1003 = 0.7994
0.7994 = 79.94% probability that the proportion of persons with myopia will differ from the population proportion by less than 3%.
