Answer:
The probability that the sample proportion of 80 LL70 batteries is within 0.03 of the population proportion is 0.44
Step-by-step explanation:
Sample proportion being within margin, or margin of error (ME) around the mean can be found using the formula
ME=[tex]\frac{z*\sqrt{p*(1-p)}}{\sqrt{N} }[/tex] where
z is the corresponding statistic of the probability that the sample proportion is within the 0.03 of the population proportionp is the claimed proportion (86% or 0.86) N is the sample size (80)Then 0.03=[tex]\frac{z*\sqrt{0.86*0.14}}{\sqrt{80} }[/tex] from this we get:
z≈0.773 and the p(z)≈0.439
Therefore, the probability that the sample proportion is within 0.03 of the population proportion is 0.44
Sahir took 1 h 22 min to do his english homework 45 min to do his maths homework, and 10 min to do his urdu. How long did he take to complete all his homework?
Answer:
2 hr and 17 minutes
Step-by-step explanation:
add 45 to 10+= 55
55+ 1 hr 22 = 1 hour and 77 minutes which converts to:
2 hour and 17 minutes
good luck!
The lenght of the library is 13 inches.The lenght of the actual library is 78 feet. Which scale leon use to create the scale drawing of the school library?
So the scale used was
1 inch = 6 feet
Step-by-step explanation:
Given
Length of library on drawing = 13 inches
Actual length of library = 78 feet
In order to find the scale we divide the actual length by the length of library on drawing.
So,
Scale = [tex]Scale = \frac{Actual\ length}{Length\ on\ drawing}\\=\frac{78}{13}\\=6[/tex]
So the scale used was
1 inch = 6 feet
Keywords: Scale, Maps
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A poll that does not attempt to generate a random sample, but instead invites people to volunteer to participate is called:________
Answer:
Self selection sampling.
Step-by-step explanation:
A poll that does not attempt to generate a random sample, but instead invites people to volunteer to participate is called - self selection sampling.
Self-selection sampling is a sampling method where researchers allow the people or individuals, to choose to take part in research on their own accord.
Describe where the function has a vertical asymptote and how you found your answer. Remember that an asymptote is represented by an equation of a line and not just a single value.
Answer:
x=-4 is a vertical asymptote
Step-by-step explanation:
A vertical asymptote of the graph of a rational function f(x) is a line x=a, such that one of of these statements is fulfilled
[tex]\displaystyle \lim _{x\to a^{+}}f(x)=\pm \infty[/tex]
[tex]\displaystyle \lim _{x\to a^{+}}f(x)=\pm \infty[/tex]
Our function is
[tex]\frac{x^2+7x+10}{x^2+9x+20}[/tex]
To find the candidate values of a, we set the denominator to zero
[tex]x^2+9x+20=0[/tex]
Factoring
[tex](x+4)(x+5)=0[/tex]
Which gives us two possible vertical asymptotes: x=-4 or x=-5
We now must confirm if one of the two conditions are true for each value of a
[tex]\displaystyle \lim _{x\to -4^{-}}\frac{x^2+7x+10}{x^2+9x+20}[/tex]
The numerator can be factored as
[tex]x^2+7x+10=(x+2)(x+5)[/tex]
So our limit is
[tex]\displaystyle \lim _{x\to -4^{-}}\frac{(x+2)(x+5)}{(x+4)(x+5)}[/tex]
Simplifying:
[tex]=\displaystyle \lim _{x\to -4^{-}}\frac{(x+2)}{(x+4)}=+\infty[/tex]
We can see x=-4 is a vertical asymptote
Checking with x=-5, and using the simplified limit:
[tex]\displaystyle \lim _{x\to -5^{-}}\frac{(x+2)}{(x+4)}=3[/tex]
[tex]\displaystyle \lim _{x\to -5^{+}}\frac{(x+2)}{(x+4)}=3[/tex]
The limit exists and is 3, so x=-5 is NOT a vertical asymptote
The only vertical asymptote of the function is x=4
A computer can perform c calculations in s seconds. How many minutes will it take the computer to perform k calculations?
A. 60ks/c
B. ks/c
C. ks/(60c)
D. 60c/(ks)
E. k/(60cs)
Answer:
C. ks/(60c)
Step-by-step explanation:
Units analysis is helpful here.
A. (60 sec/min)(k calc)(s sec)/(c calc) = 60ks/c sec²/min ≠ min
B. (k calc)(s sec)/(c calc) = ks/c sec ≠ min
C. (k calc)(s sec)/((60 sec/min)(c calc)) = ks/(60c) min . . . correct units
D. (60 sec/min)(c calc)/((k calc)(s sec)) = 60c/(ks) /min ≠ min
E. (k calc)/((60 sec/min)(c calc)(s sec)) = k/(60cs) min/sec² ≠ min
The only choice in which the units work out correctly is choice C.
Your father will be going to your grandparents house for dinner.He drives 40 minutws at a constant soeed of 30 miles per houer.He reaches the highway,quickly speeds up,and drives for another 30 minutes at a constant speed of 60 miles per hour.How far did you and your father travel altogether?
40 minutes is 2/3 of an hour.
30 miles per hour x 2/3 hour = 20 miles.
30 minutes = 1/2 hour.
60 miles per hour x 1/2 hour = 30 miles.
Total = 20 + 30 = 50 miles.
Can someone answer ?
Answer:
X is less sign but is greater than sign -2
Step-by-step explanation:
Mr. Croft is getting ready for Thanksgiving and needs some help. He needs some advice on cooking his turkey. He bought a 16 1/2 pound turkey. The cookbook says to cook the turkey for 20 minutes per pound and 25 minutes per pound if the turkey is stuffed. He plans on stuffing his turkey. If he has to have dinner ready for 12:45 pm what is the exact time he needs to put the turkey in the oven? Oh yeah, the cookbook says to let the turkey cool for 10 minutes after coming out the oven.
Answer: 5:42:2 am
Step-by-step explanation:
He bought a 16 1/2 pound turkey. Converting to improper fraction, it becomes 33/2 pounds of turkey.
The cookbook says to cook the turkey for 20 minutes per pound and 25 minutes per pound if the turkey is stuffed. Since he wants to stuff the Turkey, he would have to cook for 25 minutes per pound. The total amount of time that he would have to cook the stuffed turkey will be 25 × 33/2 = 412.5 minutes
Let us converted to hours
If 60 minutes = 1 hour,
412.5 minutes = 412.5/60 = 6.875
He would spend 6.875 = 55/8 hours
in cooking the stuffed turkey.
The cookbook also says to let the turkey cool for 10 minutes after coming out the oven. Let us convert the 10 minutes to hours
60 minutes = 1 hour
10 minutes = 10/60 = 1/6
Total time it will take to get the Turkey ready for dinner will be
1/6 + 55/8 = 169/24 = 7.042 hours which means 7 hours, 2 minutes and 58 seconds
If he has to have dinner ready for 12:45 pm, he needs to put the turkey in the oven at 5:42:2 am at most
Find the slope of each line defined below and compare their values.
Answer: slope=-1, slope is -1/6, line a has a greater slope
Step-by-step explanation:
slope of line a is "m" in y=mx+b form
so slope=-1
slope for line b
first find two point, slope is change in y divided by change in x
1 down for every 6 units forward, so slope is -1/6
line a has a greater slope
Jordan enters 3.4 x6.8 into his calculator. He writes the digits 2312 from the display and forgets the decimal point. Where should Jordan put her decimal point? Explain
Answer:
23.12
Step-by-step explanation:
Jordan has entered 3.4 x 6.8 on his calculator.
3.4 x 6.8 = (34/10) x (68/10) = (34 x 68) / (100) = 23.12
Or an easier way would be, observing the two numbers.
In 3.4 , there is one digit after decimal point.
In 6.8 there is one digit after decimal point.
So in their product there should be two digits after the decimal point.
So the answer is 23.12
Montel bought a spool of string for making kites. It contained 10 3/20 meters of string. He used 6 9/10 meters of string for kites. How many meters of string does Montel have left?
Answer: he has 65/20 meters of string left
Step-by-step explanation:
Montel bought a spool of string for making kites. It contained 10 3/20 meters of string. Let us first convert this length of string to improper fraction, it becomes 203/20 meters
He used 6 9/10 meters of string for kites. Changing 6 9/10 to improper fraction, it becomes 69/10 meters
To determine how many meters of string that Montel have left, we will subtract the length of string used from the total length of string. It becomes
203/20 - 69/10 = 203 - 138/20 = 65/20 meters of string left
The experimental probability of rain in a certain town is 20 percent. In the next 45 days, how many days can one expect it to rain?
Answer:
9 days
Step-by-step explanation:
Experimental probability is the number of times an event occurs divided by the total number of trials of the event.
We know the experimental probability, based on past data, that rain will fall 20% chance.
So, one can expect, that in the next 45 days, rain will fall 20% of the days, according to our experimental probability.
First, we find decimal equivalent of 20%. We divide by 100:
20% = 20/100 = 0.2
Now we multiply this with the number of days:
0.2 * 45 = 9
Thus,
One can expect it to rain 9 days
A parabola opens upward and has no vertical stretch. The complex roots of the quadratic function are 6 + 4i and 6 – 4i. Determine the function rule.
Answer:
Step-by-step explanation:
Describing the function rule means that you are going to write the equation of the parabola using that roots. If x = 6 + 4i, then the factor for that is
(x - 6 - 4i).
If x = 6 - 4i, then the factor for that is
(x - 6 + 4i).
FOILing that together gives you a long string of x- and i-terms with a constant or 2 thrown in:
[tex]x^2-6x+4ix-6x+36-24i-4ix+24i-16i^2[/tex]
What's nice here is that 4ix and -4ix cancel each other out; likewise 24i and -24i. Once that is all canceled away, we are left with
[tex]x^2-12x+36-16i^2[/tex]
The i-squared is what makes this complex. i-squared = -1, so
[tex]x^2-12x+36-16(-1)[/tex] and
[tex]x^2-12x+36+16[/tex] and
[tex]x^2-12x+52=y[/tex]
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What series of transformations would carry trapezoid ABCD onto itself?
(x + 0, y − 4), 90° clockwise rotation, reflection over the y-axis
(x + 0, y − 4), 180° rotation, reflection over the y-axis
(x + 6, y + 0), 90° clockwise rotation, reflection over the x‐axis
(x + 6, y + 0), 180° rotation, reflection over the x‐axis
Answer:
(x + 6, y + 0), 180° rotation, reflection over the x‐axis
Step-by-step explanation:
The answer can be found out simply , a trapezoid has its horizontal sides usually parallel meanwhile the vertical sides are not parallel.
The horizontal parallel sides are on the x-axis.
Reflection over y- axis would leave the trapezoid in a vertical position such that the trapezoid ABCD won't be carried on the transformed trapezoid as shown in figure.
So option 1 and 2 are removed.
Now, a 90 degree rotation would leave the trapezoid in a vertical position again so its not suitable again.
In,The final option (x + 6, y + 0), 180° rotation, reflection over the x‐axis, x+6 would allow the parallel sides to increase in value hence the trapezoid would increase in size,
180 degree rotation would leave the trapezoid in an opposite position and reflection over x-axis would bring it below the Original trapezoid. Hence, transformed trapezoid A`B`C`D` would carry original trapezoid ABCD onto itself
Rob buys 6 tickets to the basketball game. He pays$8.50 for parking. His total cost is $40.54. What is the cost of each ticket? Plzzzzzzzz tell me what it is
Answer:
Step-by-step explanation:
26.04
please help, i im having trouble
Answer:
Elimination
Step-by-step explanation:
Use the equation for variance below, along with the given data set, to answer the following questions.
Sigma squared = StartFraction (x 1 minus mu) Squared + (x 2 minus mu) squared + ellipsis + (x N minus mu) squared Over N EndFraction
What does the numerator evaluate to?
What does the denominator evaluate to?
The variance equals:
Answer:
The given equation is
[tex]\sigma^{2}=\frac{(x_{1}- \mu)^{2}+(x_{2}- \mu)^{2}+...+(x_{n}- \mu)^{2} }{N}[/tex]
What does the numerator evaluate to?
You can observe that the numerator is formed by the sum between the difference of squares. The difference is between each data and the mean of the data, this difference compares the deviation of each data regarding the mean, that's why the variance measures the degree of deviation of a data set. Additionally, the squares are made to have each difference as a positive number.
What does the denominator evaluate to?
On the other hand, the denominator represents the total number of data. With this denominator, the deviation evaluated in the numerator can be distributed to each data.
At last, the variance equals the degree of deviation of each data in average.
Answer:
94, 5, 18.8
Step-by-step explanation:
second part: 4.34
edge
Rewrite the equation −5x2+3x+5y2+5y−3z2+4z+12=0 −5x2+3x+5y2+5y−3z2+4z+12=0 in cylindrical and spherical coordinates. NOTE: write any greek letters using similar standard characters - i.e., for θθ use t, for rhorho use r, for ϕϕ use f, etc.
Answer:
Cylindrical:
5r^2(sin(t)^2 - cos(t)^2) +r(3cos(t) + 5sin(t)) - 3z^2 + 4z + 12 = 0
Spherical:
5r^2sin(t)^2 (sin(f)^2 - cos(f)^2) - 3r^2 cos(t)^2 + r sin(t) (3cos(f) + 5sin(f)) + 4r cos(t) +12 = 0
Step-by-step explanation:
In cylindrical coordinates:
x = r cos(t)
y = r sin(t)
z = z
Let us reorganize the original equation
−5x^2+3x+5y^2+5y−3z^2+4z+12=0
5 (y^2-x^2) + 3x + 5y - 3z^2 + 4z + 12 = 0
Now, we can replace x and y:
5 (r^2 sin(t)^2 - r^2 cos(t)^2) + 3rcos(t) + 5r sin(t) - 3z^2 + 4z + 12 = 0
5r^2(sin(t)^2 - cos(t)^2) +r(3cos(t) + 5sin(t)) - 3z^2 + 4z + 12 = 0
In spherical coordinates:
x = r sin(t) cos(f)
y = r sin(t) sin(f)
z = r cos(t)
Let us reorganize the equation:
5 (y^2-x^2) - 3z^2 + 3x + 5y + 4z + 12 = 0
5r^2sin(t)^2 (sin(f)^2 - cos(f)^2) - 3r^2 cos(t)^2 + r sin(t) (3cos(f) + 5sin(f)) + 4r cos(t) +12 = 0
Marcy has $150 to buy packages of hot dogs and hamburgers for her booth at the carnival. At her local grocery store she found packages of hot dogs at cost six dollars and packages of hamburger is that cost $20. Right in equation that could marcy has $150 to buy packages of hot dogs and hamburgers for her booth at the carnival. At her local grocery store she found packages of hot dogs that cost six dollars in packages of hamburgers that cost $20. Right in equation that could be used to find the possible combination of hot dog and hamburger packages Marcy can by using her budget of exactly $150.
Answer:
Step-by-step explanation:
Marcy wants to buy packages of hot dogs and hamburgers for her booth at the carnival and the total cost must not exceed her budget of $150
she found packages of hot dogs that cost $6 per one and packages of hamburger is that cost $20 per one .
Let x = number of packages of hot dogs that she can buy.
Let y = number of packages of hamburgers that she can buy.
The cost of x packages of hot dogs = 6×x = $6x
The cost of y packages of hamburgers = 20×y = $20y
The equation will be
Since 6x + 20y must not exceed 150
The equation will be
6x + 20y lesser than or equal to 150
The owner of a souvenir shop wants to order some custom printed mugs and t-shirts what is the price per t-shirt for 75 T-shirts if the t-shirts cost 637.50 dollars
Answer:
The price per t-shirts is $8.5.
Step-by-step explanation:
Let us assume the price per t- shirt = $m
The cost of 75 t-shirts = $637.50
Now, the total cost for 75 t-shirts = 75 x ( cost of 1 t-shirt)
⇒ $637.50 = 75 x ( m)
⇒ m = $637.50 / 75
or, m = 8.5
Hence, the price per t-shirts is $8.5.
(3 points) There are 50 apple trees in an orchard. Each tree produces 800 apples. For each additional tree planted in the orchard, the output per tree drops by 10 apples. How many trees should be added to the existing orchard in order to maximize the total output of trees ?
Answer:
15 trees should be added to the existing orchard in order to maximize the total output of trees.
Step-by-step explanation:
Number of trees = 50
Number of apples each tree gives = 800
For each additional tree planted in the orchard, the output per tree drops by 10 apples. Let x be the number of trees added, then,
Number of apple is given by:
[tex]V(x) = (50+x)(800-10x) = -10x^2+300x + 40000 [/tex]
First, we differentiate V(x) with respect to x, to get,
[tex]\displaystyle\frac{d(V(x))}{dx} = \frac{d(-10x^2+300x + 40000)}{dx} = -20x + 300[/tex]
Equating the first derivative to zero, we get,
[tex]\displaystyle\frac{d(V(x))}{dx} = 0\\\\-20x + 300 = 0[/tex]
Solving, we get,
[tex]-20x + 300 = 0\\\\x=\displaystyle\frac{300}{20} = 15[/tex]
Again differentiation V(x), with respect to x, we get,
[tex]\displaystyle\frac{d^2(V(x))}{dx^2} = -20[/tex]
At x = 15
[tex]\displaystyle\frac{d^2(V(x))}{dx^2} < 0[/tex]
Thus, by double derivative test, the maxima occurs at x = 15 for V(x).
Thus, 15 trees should be added to the existing orchard in order to maximize the total output of trees.
Maximum output of apples =
[tex]V(15) = (50+15)(800-10(15)) = 42250[/tex]
Let the geometric sequence {an} be defined as follows: {12, 6, 3, 3 /2 , 3 /4 , 3/ 8 ,...} Find the sum of the entire sequence, S∞. A) 24 B) 240 C) 1,080 Eliminate D) 2,176,782,336
Answer:
A) 24
Step-by-step explanation:
The given geometric series is {12, 6, 3, 3 /2 , 3 /4 , 3/ 8 ,...}
Each term can be represented as a product of its previous term and \[\frac{1}{2}\]
The generic term of the series can be represented as the product of the first term 12 and \[\frac{1}{2}^{n-1}\] where n is the index of the term in the series.
The sum to infinity of such a series is given by the following formula:
\[\frac{term1}{1-ratio}\]
Substituting and calculating:
\[\frac{12}{1-\frac{1}{2}}\]
=\[\frac{12}{\frac{1}{2}}\]
=12*2 = 24
A family has a monthly budget of $2,400. How much money is spent on each category? 44% is spent on housing. 23% is spent on food. 6% is spent on clothing. 17% is spent on transportation. The rest is put into savings.
Answer:
housing, $1056food, $552transportation, $408savings, $240clothing, $144Step-by-step explanation:
Multiply the budget total by the percentage for each category:
housing: 44% × $2400 = $1056food: 23% × $2400 = $552clothing: 6% × $2400 = $144transportation: 17% × $2400 = $408savings: 10% × $2400 = $240__
Find "the rest" by subtracting the total of percentages from 100%, or by subtracting the total of budget categories from $2400. Here, everything (not including savings) adds to 90% × $2400 = $2160, so the amount to savings is 10%, or $240.
In a family's monthly budget of $2,400, $1,056 is spent on housing, $552 on food, $144 on clothing, $408 on transportation, and the remaining $240 is put into savings.
Explanation:To calculate how much money is spent on each category for a family with a monthly budget of $2,400, we need to multiply each percentage by the total budget. For example:
Housing: 44% of $2,400 is $1,056.00,Food: 23% of $2,400 is $552.00,Clothing: 6% of $2,400 is $144.00,Transportation: 17% of $2,400 is $408.00.
The rest of the money, which is 10% of the total budget (the remaining percentage when subtracting the other categories from 100%), is put into savings. Therefore, the amount put into savings is 10% of $2,400, which equals $240.00.
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A game spinner has eight equal sections: three sections numbered 1, one section numbered 2, and four sections numbered 3. The spinner is spun twice. What is the probability that the sum of the two spins will be five?
The probability that the sum of two spins on this spinner will be five is 7/32, which can be calculated by determining individual probabilities for each combination that gives a sum of 5 and then adding these probabilities together.
Explanation:The question is asking for the probability that the sum of two spins on a specific spinner will be equal to five. First, we need to look into the different ways we can achieve a sum of 5. These possible combinations include (1,4), (2,3), and (3,2), where the values in the parentheses represent the results of the first and second spins respectively.
Now we calculate the probability of each combination happening. The probability of getting a 1 on a spin is 3/8 because there are three sections numbered 1 out of eight total sections. The probability of getting a 2 on a spin is 1/8 because there's only one section numbered 2. Likewise, the probability of getting a 3 is 4/8 (simplified to 1/2) because there are four sections numbered 3.
The combined probability of each scenario would be (3/8*1/2) for (1,4), (1/8*1/2) for (2,3) and (1/2*1/8) for (3, 2). To find the total probability, we simply add these probabilities together. That gives us (3/16) + (1/32) + (1/32) = 7/32. Therefore, the probability that the sum of two spins will be five is 7/32.
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You have a bag with two coins. One will come up heads 40% of the time, and the other will come up heads 60%. You pick a coin randomly, flip it and get a head. What is the probability it will be heads on the next flip?
The probability it will be heads on the next flip is 0.24.
How to calculate probability?From the information given, a bag comes up heads 40% of the time, and the other will come up heads 60%.
The probability it will be heads on the next flip will be:
= 40% × 60%
= 0.4 × 0.6
= 0.24
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Shelia works 8 hours per day Monday, Wednesday, and Friday, and 6 hours a day on Tuesday and Thursday. She does not work on Saturday or Sunday. She earns $324 per week. How much is she paid in dollars per hour?
Answer:She earns $9 per hour
Step-by-step explanation:
Shelia works 8 hours per day Monday, Wednesday, and Friday. This means she works a total of 8×3 = 24 hours for Monday, Wednesday and Friday
She also works 6 hours a day on Tuesday and Thursday. This means that she works for a total of 6×2 = 12 hours on Tuesday and Thursday.
Total hours worked for the week will be sum of hours worked on Monday, Tuesday, Wednesday, Thursday and Friday. It becomes
24 + 12 = 36 hours
She earns $324 per week. The amount that she earns in a week will be
Total amount earned in a week/total number of hours worked in the week. It becomes
324/36 = 9
She earns $9 per hour
Among 46- to 51-year-olds, 28% say they have called a talk show while under the influence of peer pressurepeer pressure. Suppose sevenseven 46- to 51-year-olds are selected at random. (a) What is the probability that at least one has not called a talk showcalled a talk show while under the influence of peer pressurepeer pressure? (b) What is the probability that at least one has called nbspcalled a talk showa talk show while under the influence of peer pressurepeer pressure?
Answer:
0.8997, 0.9999
Step-by-step explanation:
Given that among 46- to 51-year-olds, 28% say they have called a talk show while under the influence of peer pressure.
i.e. X no of people who say they have called a talk show while under the influence of peer pressure is binomial with p = 0.28
Each person is independent of the other and there are only two outcomes
n =7
a) the probability that at least one has not called a talk showcalled a talk show while under the influence of peer pressure
=[tex]P(Y\geq 1)[/tex] where Y is binomial with p = 0.72
= [tex]1-(1-0.72)^7\\=0.9999[/tex]
b) the probability that at least one has called a talk showcalled a talk show while under the influence of peer pressurepeer pressurethe probability that at least one has not called a talk showcalled a talk show while under the influence of peer pressure
=[tex]P(x\geq 1) =1-P(0)\\= 1-(1-0.28)^7\\=0.8997[/tex]
yz = 7ln(x + z), (0, 0, 1) (a) the tangent plane Correct: Your answer is correct. (b) parametric equations of the normal line to the given surface at the specified point. (Enter your answer as a comma-separated list of equations. Let x, y, and z be in terms of t.)
Answer:
The equation of tangent plane is 7x - y + 7z - 7 = 0
Parametric equation of normal line
x = 7 t , y=-t , z=1+7 t
Step-by-step explanation:
Equation of tangent
fₓ (x₀ , y₀ , z₀) (x-x₀) + fy (x₀ , y₀ , z₀) (y-y₀) +fz(x₀ , y₀ , z₀)(z-z₀)=0 (1)
From taking derivation we get
fₓ (x₀ , y₀ , z₀) = 7
fy (x₀ , y₀ , z₀)= -1
fz(x₀ , y₀ , z₀) = 7
putting these value in equation 1
(7) (x-0) + (-1)(y-0) + 7(z-1)=0
7x - y + 7z - 7 = 0
The equation of tangent plane is 7x - y + 7z - 7 = 0
b) Parametric equation
x=x +f ₓ (P)t , y = y₀ +f y (P) t , z=z₀ +f z (P) t
x=0 +7 t , y =0+(-1) t , z=1+7 t
x=7 t ,y=-t , z= 1+7 t
Parametric equation of normal line
x = 7 t , y=-t , z=1+7 t
Test the hypothesis that the average content of containers of a particular lubricant is 10 liters if the contents of a random sample of 10 containers are 10.2, 9.7, 10.1, 10.3, 10.1, 9.8, 9.9, 10.4, 10.3, and 9.8 liters. Use a 0.01 level of significance and assume that the distribution of contents is normal.
Answer:
We fail to reject the null hypothesis that the average content of containers of the lubricant is 10 liters, this at the significance level of 0.01
Step-by-step explanation:
Let X be the random variable that represents the content of a container of the lubricant. We have observed n = 10 values, [tex]\bar{x}[/tex] = 10.06 and s = 0.2459. We assume that X is normally distributed.
We have the following null and alternative hypothesis
[tex]H_{0}: \mu = 10[/tex] vs [tex]H_{1}: \mu \neq 10[/tex] (two-tailed alternative)
We will use the test statistic
[tex]T = \frac{\bar{X}-10}{S/\sqrt{10}}[/tex] because we have a small sample size. And the observed value is
[tex]t = \frac{10.06-10}{0.2459/\sqrt{10}} = 0.7716[/tex]
if [tex]H_{0}[/tex] is true, then T has a t distribution with n-1 = 9 degrees of freedom.
The rejection region for a two-tailed alternative and a significance level of 0.01 is given by RR = {t | t < -3.2498 or t > 3.2498}, where 3.2498 is the value such that there is an area of 0.005 above this number and under the density of the t distribution with 9 df.
Because the observed value 0.7716 does not fall inside RR, we fail to reject the null hypothesis.
The hypothesis that the average content of containers of a particular lubricant is 10 liters is not acceptable.
How to classify the hypotheses?There are two hypotheses. First one is called null hypothesis and it is chosen such that it predicts nullity or no change in a thing. It is usually the hypothesis against which we do the test. The hypothesis which we put against null hypothesis is alternate hypothesis.
Null hypothesis is the one which researchers try to disprove.
When to use the z test and when to use the t test?If the sample taken is of size less than t test, then you can use the t test.
If the sample size is larger or equal to 30, you can use the z test.
It is because as the sample size grows more and more, the z test statistic approximates more and more the normal distribution.
For the considered case, we can use:
[tex]\mu[/tex] = average content of containers of a particular lubricant (of population)Null Hypothesis: [tex]H_0: \mu \neq 10[/tex]Alternative hypothesis: [tex]H_1: \mu = 10[/tex]It is because we want to show that the average content of containers of a particular lubricant is 10 liters. Thus, we took alternative hypothesis such that the is = 10
Now, since the sample size is n = 10 < 30, we will use t-test here.
We evaluate 't' as:
[tex]t = \dfrac{\overline{x} - \mu}{s/\sqrt{n}}[/tex] (sample standard deviation is used if population standard deviation is not available).
where, the symbols denote:
[tex]\overline{x}[/tex] = sample means = sample standard deviationn = sample size[tex]\mu[/tex] = hypothesized mean of populationFor this case, the sample is of size n = 10
Its observed values are: 10.2, 9.7, 10.1, 10.3, 10.1, 9.8, 9.9, 10.4, 10.3, 9.8
The mean is obtained as:
[tex]\overline{x} = \dfrac{\sum{x_i}}{n} = \dfrac{100.6}{10} = 10.06[/tex] (in litres)
The standard deviation of sample would be:
[tex]s = \dfrac{\sum(x_i - \overline{x})^2}{n} = \dfrac{\sum(x_i - 10.06)^2}{10} = 0.544[/tex]
The values for this case are:
[tex]\overline{x} = 10.06[/tex] [tex]s= 0.544[/tex][tex]n = 10[/tex][tex]\mu = 10[/tex]Thus, we get:
[tex]t = \dfrac{\overline{x} - \mu}{s/\sqrt{n}} = \dfrac{10 - 10.06}{0.544/\sqrt{10}} \approx 0.349[/tex]
The level of significance here is 0.01
At this degree of freedom and level of significance, the critical value of t-test statistic is [tex]t_{\alpha/2} = \pm3.2498[/tex] (two tailed)
Since [tex]t < |t_{\alpha/2}|[/tex] we may accept the null hypothesis, and thus, haven't got significant evidence to accept the alternative hypothesis.
(if the obtained value would be bigger than critical value, then we'd reject null hypothesis).
Therefore, the hypothesis that the average content of containers of a particular lubricant is 10 liters is not acceptable.
Learn more about z-test for single mean here:
https://brainly.com/question/25316952
In survey conducted by Quinnipiac University from October 25-31, 2011, 47% of a sample of 2,294 registered voters approved of the job Barack Obama was doing as president. What is the 99% confidence interval for the proportion of all registered voters who approved of the job Barack Obama was doing as president?
Answer:
Confidence interval = [ 0.4432, 0.4968]
Step-by-step explanation:
Registered voters approved of the job Barack Obama was doing as president,
p = 47% = 0.47
Total sample size, n = 2,294
Confidence level = 99%
now,
confidence interval for the proportion of all registered voters who approved of the job Barack Obama was doing as president
= p ± [tex]z\times\sqrt{\frac{p\times(1-p)}{n}}[/tex]
for 99% confidence level, z value is 2.58
= 0.47 ± [tex]2.58\times\sqrt{\frac{0.47\times(1-0.47)}{2294}}[/tex]
= 0.47 ± 0.0268
or
Confidence interval = [0.47 - 0.0268, 0.47 + 0.0268 ]
or
Confidence interval = [ 0.4432, 0.4968]