Jessica bought 4 gallons of paint. Of that, she used 3/4 to paint. So the ammount she used was
[tex]4\cdot(\frac{3}{4})=\frac{4\cdot3}{4}=3[/tex]So she used 3 gallons of paint.
A convenience store manager notices that sales of soft drinks are higher on hotter days, so he assembles the data in the table. (a) Make a scatter plot of the data. (b) Find and graph a linear regression equation that models the data. (c) Use the model to predict soft-drink sales if the temperature is 95°F.
ANSWER and EXPLANATION
a) First we have to make a scatter plot. We do this by plotting the calues of High Temperature on the x axis and Number of cans sold on the y axis:
b) We want to find and graph the linear regression equation that models the data.
The linear regression equation will be in the form:
y = a + bx
[tex]\begin{gathered} \text{where} \\ a\text{= }\frac{(\sum ^{}_{}y)(\sum ^{}_{}x^2)\text{ - (}\sum ^{}_{}x)(\sum ^{}_{}xy)}{n(\sum ^{}_{}x^2)\text{ }-\text{ (}\sum ^{}_{}x)^2} \\ \text{and b = }\frac{n(\sum ^{}_{}xy)\text{ - (}\sum ^{}_{}x)(\sum ^{}_{}y)}{n(\sum ^{}_{}x^2)\text{ }-\text{ (}\sum ^{}_{}x)^2} \end{gathered}[/tex]We have from the question that:
x = High Temperature
y = Number of cans added
So, we have to find xy and x^2. We will form a new table:
Now, we will find a and b:
[tex]\begin{gathered} a\text{ = }\frac{(4120)(39090)\text{ - (}554)(297220)}{8(39090)\text{ }-554^2} \\ a\text{ = }\frac{\text{ 161050800 - 164659880}}{312720\text{ - 306916}} \\ a\text{ = }\frac{-3609080}{5804} \\ a\text{ }\cong\text{-62}2 \end{gathered}[/tex][tex]\begin{gathered} b\text{ = }\frac{8(297220)\text{ - (554})(4120)}{5804} \\ b\text{ = }\frac{2377760\text{ - 2282480}}{5804} \\ b\text{ = }\frac{95280}{5804} \\ b\text{ }\cong\text{ 16} \end{gathered}[/tex]Therefore, the linear regression equation is:
y = -622 + 16x
Now, let us graph it using values of x (High Temperature):
That is the Linear Regression Graph.
c) To predict soft drink sales if the temperature is 95 degrees Farenheit, we will put the x value as 95 and find y. That is:
y = -622 + 16(95)
y = 898
The model predicts that 898 cans of soft drinks will be sold when the High Temperature is 95 degrees Farenheit.
Find the quotient32 divided by 517 what is quotient and what is remainder
Calculate the division as shown below
Therefore, the quotient is 16 and the remainder is 5
The answer is 16R5In which quadrant does 0 lie if the following statements are true:sin 0 > 0 and sec 0 < 0Quadrant IQuadrant IIQuadrant IIIQuadrant IV
Given the conditions in the question:
1. sin θ > 0, therefore, it must be positive. From that, we can conclude that y must be on the positive side, therefore, located at the top of the coordinate plane.
2. sec θ < 0, therefore, it must be negative. From that, we can conclude that x must be on the negative side, therefore, located at the left side of the coordinate plane.
Therefore, the quadrant that the θ belongs to is in the top and left of the coordinate plane and that is Quadrant II.
find the volume and total surface area of a right circular cone whose base diameter is 10 cm and whose altitude is 20 cm.
SOLUTION
Given the question in the question tab, the following are the solution steps to calculate the required measurements.
Step 1: write the given parameters
[tex]\begin{gathered} \text{diameter}=10\operatorname{cm},\text{altitude}=\text{height}=20\operatorname{cm} \\ r=\frac{d}{2}=\frac{10}{2}=5\operatorname{cm} \end{gathered}[/tex]Step 2: Calculate the volume of the right circular cone
[tex]\begin{gathered} V=\frac{\pi r^2h}{3} \\ V=\frac{\pi\times5\times5\times20}{3} \\ V=\frac{500\pi}{3}=523.5987756 \\ V\approx523.5988\operatorname{cm}^3 \end{gathered}[/tex]Step 3: Calculate the total surface area of the right circular cone
[tex]\begin{gathered} \text{TSA}=\pi r(r+\sqrt[]{h^2+r^2)} \\ \text{TSA}=\pi(5)(5+\sqrt[]{20^2+5^2)} \\ \text{TSA}=5\pi(5+\sqrt[]{400+25)} \\ \text{TSA}=5\pi(5+\sqrt[]{425})=5\pi(5+20.61552813) \\ \text{TSA}=5\pi(25.615528134) \\ \text{TSA}=402.3677749 \\ \text{TSA}\approx402.3678cm^2 \end{gathered}[/tex]Hence, the volume and the total surface area of the given right circular cone are approximately 523.5988cm³ and 402.3678cm² respectively
a diver takes a dive in the red sea. He initially descends 100 feet. Then rises 28 before descending another 33. What is his final position
Descending: subtraction
Rises: Addition
The diver descends 100ft: Position -100 (the 0 is the sea level)
Then rises 28: Add 28 to the previous position: Position -72
[tex]-100+28=-72[/tex]...before descending another 33: Subtract 33 to the previous position:
[tex]-72-33=-105[/tex]Then, the final position is -105ft (105 ft under the sea level)
On December 13, 2007, one South African rand was worth 0.15 U.S. dollars.(a) On that date, how many rand was 44.11 dollars worth?Round your answer to the nearest hundredth of a rand.rand(b) On that date, how many dollars was 168.18 rand worth?Round your answer to the nearest hundredth of a dollars. I need help with this math problem.
Explanation
Part A
Given that one South African rand was worth 0.15 U.S. dollars. 44.11 dollars will be worth
[tex]\frac{44.11}{0.15}=294.07[/tex]Answer: 294.07 rands
Part B
On that date, how many dollars was 168.18 rand worth?
[tex]168.18\times0.15=25.23[/tex]Answer: 25.23 dollars
involving two rolls of a dieESEAn ordinary (falr) die is a cube with the numbers 1 through 6 on the sides (represented by painted spots). Imagine that such a dle is rolled twice in successionand that the face values of the two rolls are added together. This sum is recorded as the outcome of a single trial of a random experiment.Compute the probability of each of the following events.Event A: The sum is greater than 8.Event B: The sum is an odd number.Write your answers as fractions.Ola(a) P(A) = 1х5?(b) P(B) = 0
Answer:
[tex]\begin{gathered} a)\text{ }\frac{5}{18} \\ \\ b)\text{ }\frac{1}{2} \end{gathered}[/tex]Explanation:
Here, we want to compute some probabilities
The first thing to do is to get the count of results in our sample space
In the sample space, the total possible results is 36
Now, let us get the probabilities
a) The event that the sum is greater than 8
We have to count possible results greater than 8 here
3, 6 (3 on the first die, 6 on the second)
6,3 (6 on the first die, 3 on the second)
6,4 (6 on the first die, 4 on the second)
4,6
4,5
5,4
5,5
5,6
6,5
6,6
The number of possible results greater than 8 is 10
Thus, we have the probability as the count of this divided by the total number of possible results
Mathematically, we have that as:
[tex]\frac{10}{36}\text{ = }\frac{5}{18}[/tex]b) The sum is an odd number
For the sum to be an odd number, we know that if we add a table of 6 rows for all the sums, the even sum on each line is 3
The total even sum is 6 * 3 = 18
The probability is thus:
[tex]\frac{18}{36}\text{ = }\frac{1}{2}[/tex]A rectangular vegetable garden will have a width that is 3 feet less than the length, and an area of 54 square feet. If x
represents the length, then the length can be found by solving the equation:
x(x-3) = 54
What is the length, x, of the garden?
The length is
feet.
The solution is?
To determine the length of the rectangular flower garden, we need to derive equations from the given measurements and relations. The given measurements are the area, and the relation of the width and the length. From these, we generate the equation needed. We do as follows:
Area = Length x Width
where length = x ft
width = x - 3 ft
area = 54 ft^2
54 ft^2 = x ft (x -3) ft
54 ft^2 = x^2 - 3x ft^2
Solving for the value of x, we will have two values which are
x = -6 ft ( NOTE: this value can't be the answer since we cannot have a negative value for the length)
x = 9 ft = length
What is the slope of this line? :(
Answer:
y=1/4x+1
Step-by-step explanation:
Answer:
m=1/4
Step-by-step explanation:
Got it correct
all you need is in the photo I DON'T WANT STEP BY STEP ANSWER FAST please fdsd
We have the following:
[tex]\begin{gathered} x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ a=5 \\ b=0 \\ c=-80 \end{gathered}[/tex]replacing:
[tex]\begin{gathered} x=\frac{-0\pm\sqrt[]{0^2-4\cdot5\cdot-80}}{2\cdot5}=\frac{\pm\sqrt[]{1600}}{10}=\frac{\pm40}{10}=\pm4 \\ x_1=4 \\ x_2=-4 \end{gathered}[/tex]is it true that all whole numbers are rational numbers ? why or why not
all whole numbers are rational numbers
because we can write 21 as 21/1 in rational form.
.
We can write any whole number (a) into the form of
[tex]\frac{a}{b}[/tex]where b = 1,
so all whole numbers can be written in form of rational numbers.
question: determine whether the function is one-to-one. If it is sketch the graph of its inverse.i already found out it is a one-to-one, i just don't know how to graph its inverse
The inverse function of f(x) = y is f(y) = x
This means, switch the coordinates of the points of the graph
If we choose a point on the given graph like (5, 3) it will be (3, 5) in the inverse function
Also, point (-5, -3) it will be (-3, -5) in the inverse function
You can plot them with point (0, 0) and draw the curve
Let me try to show it
It will be like that
4 ft 12 ft The pitch of the roof is
As shown : in the figure
The pitch of the roof is the angle between the roof and the horizontal line
As shown we have a right angle triangle
The opposite side to the angle = 4 ft
And the adjacent side to the angle = 12 ft
According to the given sides, we will calculate the angle using tan function
So, let the angle = x
So,
[tex]\begin{gathered} \tan x=\frac{opposite}{adjacent} \\ \\ \tan x=\frac{4}{12}=\frac{1}{3} \\ \\ x=\tan ^{-1}\frac{1}{3}\approx18.435^o \end{gathered}[/tex]So, the pitch angle of the roof = 18.435
instead of writing the angle , just we will write the slope = rise/run
So, the pitch of the roof = 1/3
Which table shows a proportional relationship between miles traveled and gas used?
Miles Traveled Gas Used
27.3 mi 1.5 gal
49.16 mi 3.8 gal
Miles Traveled Gas Used
120 mi 6.2 gal
180 mi 12.2 gal
Miles Traveled Gas Used
135 mi 7.4 gal
135.5 mi 7.9 gal
Miles Traveled Gas Used
270 mi 15 gal
135 mi 7.5 gal
Answer:
D
Step-by-step explanation:
270mi 15gal
135mi 7.5gal
135/270=0.5
7.5/15=0.5
or
135/7.5=18
270/15=18
You are a landscaper working on the design of a parking lot in a new shopping center. You are measuring the length of a grass median that will be exactly as long as four parking spots and their dividing lines. One of these spots is a handicapped spot, which is 1018 feet wide and next to the curb. The other three spots are 838 feet wide. There are four dividing lines between the spots, and each measures 18 foot. What is the length of the grass median, D?
SOLUTION
From the given information:
one of the spots is
[tex]10\frac{1}{8}ft[/tex]Other three spots are
[tex]8\frac{3}{8}ft\text{ wide}[/tex]There are four dividing line of
[tex]\frac{1}{8}\text{foot}[/tex]The total length of the grass median is:
[tex]10\frac{1}{8}+3(8\frac{3}{8})+4(\frac{1}{8})[/tex]Calculate the value
[tex]\begin{gathered} \frac{81}{8}+3(\frac{67}{8})+\frac{4}{8} \\ =\frac{81}{8}+\frac{201}{8}+\frac{4}{8} \\ =\frac{81+201+4}{8} \\ =\frac{286}{8} \end{gathered}[/tex]Reduce the fraction
[tex]\frac{286}{8}=35\frac{6}{8}=35\frac{3}{4}[/tex]Therefore the length of the grass median is
[tex]35\frac{3}{4}[/tex]How far is the girl from the monument that is 30 ft high? Round your answer to a nearest foot. Show your work.
Given:
The diagram is shown alongside.
The height of the monument is 30 ft high.
The angle of elevation is 63 degrees
The objective is to find the distance between the monument and where the girl is standing.
Since it forms a right angled triangle so we can apply trigonometric ratios:
Now,
[tex]\tan 63^{\circ}=\frac{perpendicular}{\text{base}}[/tex]Perpendicular = 30 ft
Base = ?
Substituting the values,
[tex]\begin{gathered} \tan 63^{\circ}=\frac{30}{\text{base}} \\ \text{Base}=\frac{30}{\tan63^{\circ}} \\ \text{base}=\frac{30}{1.962610} \\ \text{base}=15.285767422\text{ ft} \end{gathered}[/tex]Therefore, the girl is at a distance of 15 ft from the monument.
Lincoln made 3 quarts of iced tea and Jasmine made 5 quarts of iced tea using the same recipe. Part A: How many cups of iced tea did Lincoln and Jasmine make all together? cho mark
Part A
number of ice tea lincoln made = 3 quarts
number of ice tea jasmine made = 5 quarts
Altogether we have = 8 quarts
But, there are four cups in 1 quart
Therefore, 8 quarts would give 8 x 4 cups = 32 cups
In conclusion, jasmine and lincoln made 32 cups of ice tea altogether.
Part B
There are 16 cups in one gallon
Lincoln and jasmine made 32 cups of ice tea
Therefore the number of gallons of ice tea they made is
=32/16 = 2gallons
Also, 1/2 bottle = 1 gallon
Therefore, the 2 gallons would give
[tex]\begin{gathered} =\frac{2}{\frac{1}{2}}=\frac{2}{0.5}=4 \\ \end{gathered}[/tex]Therefore the 2 gallons would give 4 bottles of ice tea
Write these numbers from least to greatest: 0, -6.1, 4, 10/2
Answer:
10/2, 4, 0, -6.1
Step-by-step explanation:
It is the only answer that makes sense.
pls mark brainliest
Answer: -6.1, 0, 4 , 10/2
Step-by-step explanation:
-6.1 is the only negative number, so it is the least. Zero comes next. 10/2 is 5, so 4 comes before it. Therefore 4 is the 3rd installment in these ordered numbers.
10 ptsQuestion 3Find the measure of each interior angle. Round to the nearest hundredth.4x(6x - 90)(3x + 31)(7x+19)X=(4x)" =(6x - 90)(7x + 19)° =(3x + 31)º =
Alexandre, this is the solution:
Let's recall that the interior angles of a rhombus add up to 360 degrees.
Upon saying that, we have:
4x + 3x + 31 + 6x - 90 + 7x + 19 = 360
20x - 40 = 360
Adding 40 at both sides:
20x - 40 + 40 = 360 + 40
20x = 400
Dividing by 20 at both sides:
20x/20 = 400/20
x = 20
Now, we can calculate each of the angles and prove they add up to 360 degrees, as follows:
• 4x = 4 * 20 =, 80
,• 3x + 31 = 3 * 20 + 31 = 60 + 31 =, 91
,• 7x + 19 = 7 * 20 + 19 = 140 + 19 = ,159
,• 6x - 90 = 6 * 20 - 90 = 120 - 90 = ,30
80 + 91 + 159 + 30 = 360
Based on a recent study, the pH level of the arterial cord (one vessel in the umbilical cord) is normally distributed with mean 7.38 and standard deviation of 0.14. Find the percentageof preterm infants who have the following arterial cord pH levels.a. pH levels between 7.00 and 7.50.b. pH levels over 7.46A.The percentage of arterial cord pH levels that are between 7.00 and 7.50 is ____%.(Round to two decimal places as needed.)B.The percentage of arterial cord pH levels that are over 7.46 is ___%.(Round to two decimal places as needed.)
We have the pH level as a random normal variable with mean 7.38 and standard deviation of 0.14.
A) We have to calculate the percentage of infants that are expected to have pH levels between 7.00 and 7.50.
We can approximate this as the probability of selecting a random infant and it has a pH level within this interval.
Then, to calculate the percentage we will use the z-scores for each boundary of the interval:
[tex]z_1=\frac{X_1-\mu}{\sigma}=\frac{7-7.38}{0.14}=\frac{-0.38}{0.14}\approx-2.7143[/tex][tex]z_2=\frac{X_2-\mu}{\sigma}=\frac{7.5-7.38}{0.14}=\frac{0.12}{0.14}\approx0.8571[/tex]Then, we can use the standard normal distribution to look for the probabilities for each z-score and calculate the probability as:
[tex]\begin{gathered} P(7.00Given that the probability is 0.80099, we can express the percentage as:[tex]P=0.80099\cdot100\%=80.01\%[/tex]B) We now have to calculate the percentage that is above 7.46.
We start by calculating the z-score as:
[tex]z=\frac{X-\mu}{\sigma}=\frac{7.46-7.38}{0.14}=\frac{0.08}{0.14}\approx0.571428[/tex]Then, we can calculate the probability as:
[tex]P(X>7.46)=P(z>0.571428)\approx0.28385[/tex]This correspond to a percentage of 28.39%.
Answer:
A) 80.01%
B) 28.39%
point).
Questions
1. (1) How many observations are collected? (2) How many variables are collected? (3) write
I
down quantitative variables (4) write down qualitative variables
2. Describe the data visually
2.1.(1) Make a frequency tables and histogram for the “Hour” variable (bin limit = 1), and (2)
describe the shapes of the histogram
2.2.(1) Make a frequency tables and histogram for the “Type” variable, and (2) prepare a 2-D
pie chart and write down the title of chart
2.3.(1) Make a frequency tables and histogram for the “Weekday” variable, and (2) prepare a 2-
D pie chart and write down the title of chart
2.4.(1) Make a frequency tables and histogram for the “Location” variable, and (2) prepare a 2-D
pie chart and write down the title of chart
2.5.(1) Make a scatter plot of the data for the “Time” and “Hour variables, placing “Hour" on the
X-axis and "Time" on the Y-axis. Add titles and modify the default colors, fonts, etc., to make
the scatter plot easy to understand. (2) Describe the relationship (if any) between X and Y.
Weak? Strong? Negative? Positive? Linear? Nonlinear?
2.6.(1) Make a dot plot of the “Hour" variable for the “Deposit Type" variable. (2) Make a dot
plot of the "Hour" variable for the "Withdraw Type” variable. (3) Compare the shapes of both
charts
1). 68 observations
1.2) 8 variables
1.3) Quantitative Variables: Type, Time, Date, DayCode, Hour, and Amount
1.4) Qualitative Variables: Location, Weekday
Question 1) Let's examine that table to find out the number of collected observations. Counting each row, we have 68 observations. Each one informing the type, time, date, day code, weekday, Location, Hour, and Amount
1.2) We have then 8 variables namely (type, time, date, day code, weekday, Location, Hour, and Amount)
1.3) The Quantitative Variables are the ones whose entries are numerical, so examining then we can state that:
Type, Time, Date, DayCode, Hour, and Amount each and every one of them receives a numerical entry.
1.4) Qualitative or Categoricals variables are the ones whose entry is not a numerical one. So we can enlist the following ones as Qualitative:
Location
Weekday
Hence the answers are:
1). 68 observations
1.2) 8 variables
1.3) Quantitative Variables: Type, Time, Date, DayCode, Hour, and Amount
1.4) Qualitative Variables: Location, Weekday
If 200 is added to a positive integer I, the result is a square number. If 276 is added to to the same integer I, another square number is obtained. Find I.
Solution:
[tex]\begin{gathered} Let\text{ } \\ 200\text{ + I= x}^2----------\left(1\right) \\ 276+I\text{ =y}^2----------\left(11\right) \\ Subtract\text{ equation \lparen1\rparen from equation \lparen11\rparen} \\ 276+1-\left(200_+I\right?=y^2-x^2 \\ 76=\left(y-x\right?\left(y+x\right? \end{gathered}[/tex]Now y+x and y-x differ in 2x.
One of them is even, because their product is even, so the other must be even too.
76=2*2*19 and 19 is prime.
We can assume x,y>=0,
Thus, y+x=2.19, and y-x=2
from here y=20, x=18
Therefore,
[tex]\begin{gathered} 200+1=18^2 \\ 200+I=324 \\ I=324-200 \\ I=124 \end{gathered}[/tex]The answer is I = 124
A company charges $7 dor a t-shirt and ships any order for $22 a school principal ordered a number of t-shirts for the school store the total cost of the order was 1,520 how many t-shirts did the principal buy
the equation is
[tex]7x+22=1520[/tex]then solve for x
[tex]\begin{gathered} 7x+22-22=1520-22 \\ 7x=1498 \\ \frac{7x}{7}=\frac{1498}{7} \\ x=214 \end{gathered}[/tex]answer: the principal bought 214 t-shirts
How can I know how many students scored 5 in their test?
Based on the given table, consider that the value in the column frequency specifies the number of times that a certain score (first column) is repeated in a given data.
In this case, the value of the frequency for a specific score determines the number of students with such a score in their tests.
As you can notice, for the value of the frequency equal to 3, the corresponding value of the score is 5. It means that 3 student get 5 scores in their tests.
after a translation 8 units left
The given transformation is 8 units left.
The pre-image vertices are A(1,-4), B(1,-6), C(5,-6), and D(5,-4).
Using the transformation, we have:
[tex]\begin{gathered} A^{\prime}(1-8,-4)=A^{\prime}(-7,-4) \\ B^{\prime}(1-8,-6)=B^{\prime}(-7,-6) \\ C^{\prime}(5-8,-6)=C^{\prime}(-3,-6) \\ D^{\prime}(5-8,-4)=D^{\prime}(-3,-4) \end{gathered}[/tex]The image below shows the graph of the image
Evaluate h(x) at x = 6, x = 8, and x= 12. h(x)=1.31^×
Answer : h(6) = 5.054
h(8) = 8.673
h(12) = 25.542
Given that h(x) = 1.31^x
[tex]\begin{gathered} h(x)=1.31^x \\ \text{ find the value of h(6) when x = 6} \\ h(6)=1.31^6 \\ h(6)\text{ = 5.05}4 \\ \text{when x = 8} \\ h(8)=1.31^8 \\ h(8)\text{ = 8.67}3 \\ \text{when x = 12} \\ h(12)=1.31^{12} \\ h(12)\text{ = 25.54}2 \end{gathered}[/tex]Therefore,
h(6) = 5.054
h(8) = 8.673
h(12) = 25.542
Evaluate the expression shown below and write your answer as a fraction in
simplest form.
-3/8+(-9/10)
Answer:
-1 11/40
Step-by-step explanation:
-51/40 this is simplest form.
Step-by-step explanation:
Determine the input value for which the statementf(x) = g(x) is true.From the graph, the input value is approximatelyf(x) = 3 and g(x) = 3x-23 = {x-25= xThe x-value at which the two functions' values areequal is
You can see from the graph, f (x) is a constant value and g (x) = -5, when x = -2, g (x) = - 2, when x = 0 and g (x) = 1, when x = 2.
The annual rainfall in a town has a mean of 54.11 inches and a standard deviation of 12.59 inches. Last year there was rainfall of 48 inches. How many standard deviations away from the mean is that? Round your answer to two decimal places.
SOLUTION
Mean=54.11, standard deviation = 12.59
X=48
Using the z formula
[tex]z=\frac{x-\mu}{\sigma}[/tex]Substituting values gives
[tex]z=\frac{48-54.11}{12.59}[/tex]Solve for z
[tex]z=-0.4853[/tex]This shows that the result shows that the value x=48 is 0.4853 standard deviation to the left of the mean.
The Lyon Restaurant Survey provides food, decor, and service ratings for some of the top restaurants across the United States. For 186 restaurants located in Boston, the average price of a dinner, including one drink and tip, was 48.60 Dollars. You are leaving on a business trip to Boston and will eat dinner 23 of these restaurants, randomly selected. Your company will reimburse you for a maximum of 50 dollars per dinner. Business associates familiar with these restaurants have told you that the meal cost at one-third of these restaurants will exceed 50 dollars.
a. What is the probability that none of the meals will exceed the cost covered by your company?
b. What is the probability that at most 12 of the meals will exceed the cost covered by your company? What is the probability that between 4 and 8 of the meals will exceed the cost covered by your company?
c. Calculate the expected number of restaurants that will exceed the cost covered by your company.
d. Calculate the probability of the first question by using the binomial distribution approximation. Therefore, in this case we will consider the possibility of repetition in the randomly selected restaurants. Define p=r/N as the success probability.N is the size of the population. r is the number of elements considered as successes in the population.
e. Calculate the probability of the second question by using the binomial disribution approximation.
f. Calculate the probability of the third question by using the binomial disribution approximation.
g. Calculate the expected number of the fourth question by using the binomial disribution aproximation
Using the binomial distribution, the probabilities are given as follows:
a. None: 0%.
b.
At most 12: 0.9814 = 98.14%.Between 4 and 8: 0.6249 = 62.49%.c. The expected number of restaurants that will exceed the cost covered by your company is of 7.67.
Using the normal approximation, the probabilities are:
a. None: 0.0008 = 0.08%.
b.
At most 12: 98.38 = 98.38%.Between 4 and 8: 0.6121 = 61.21%.The difference in these probabilities is due to the small sample size.
Binomial distributionThe formula for the probability of x successes is:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
In which the parameters are given by:
n is the number of trials of the experiment.p is the probability of a success on a single trial of the experiment.Considering that you will eat dinner at 23 restaurants, and at around one-third of them the meal cost will exceed 50 dollars, the values of these parameters are given as follows:
n = 23, p = 1/3 = 0.3333.
The probability that none will exceed is P(X = 0), hence:
P(X = 0) = (1 - 0.3333)^23 = 0% (rounded).
The probability of at most 12 is:
P(X <= 12) = P(X = 0) + P(X = 1) + ... + P(X = 12).
Using a binomial distribution calculator with the given parameters, the probability is:
P(X <= 12) = 0.9814 = 98.14%.
The probability that between 4 and 8 dinners are paid is:
P(4 <= X <= 8) = P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8)
Using a calculator, or the mass function P(X = x) and adding each probability, the desired probability is:
P(4 <= X <= 8) = 0.0493 + 0.0937 + 0.1405 + 0.1707 + 0.1707 = 0.6249 = 62.49%.
Normal approximationThe first step for the normal approximation is finding the mean and the standard deviation, as follows:
Mean = expected number: [tex]\mu = np = 23 \times 0.3333 = 7.67[/tex]Standard deviation: [tex]\sigma = \sqrt{np(1-p) = \sqrt{23 \times 0.3333 \times 0.6667} = 2.26[/tex]The probability of none, using continuity correction, is P(X < 0.5), which is the p-value of Z when X = 0.5, hence:
(the p-value of Z is found using the z-score table).
[tex]Z = \frac{X - \mu}{\sigma}[/tex] (z-score formula)
Z = (0.5 - 7.67)/2.26
Z = -3.17
Z = -3.17 has a p-value of 0.0008.
Hence the probability is 0.0008 = 0.08%.
The probability of at most 12 is P(X <= 12.5), using continuity correction, which is the p-value of Z when X = 12.5, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
Z = (12.5 - 7.67)/2.26
Z = 2.14
Z = 2.14 has a p-value of 0.9838.
Hence the probability is of 98.38 = 98.38%.
The probability of between 4 and 8 dinners being paid is P(3.5 <= X <= 8.5), which is the p-value of Z when X = 8.5 subtracted by the p-value of Z when X = 3.5, hence:
X = 8.5:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
Z = (8.5 - 7.67)/2.26
Z = 0.37
Z = 0.37 has a p-value of 0.6443.
X = 3.5:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
Z = (3.5 - 7.67)/2.26
Z = -1.85
Z = -1.85 has a p-value of 0.0322.
Hence the probability is:
0.6443 - 0.0322 = 0.6121 = 61.21%.
More can be learned about the binomial distribution at https://brainly.com/question/24756209
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