Final answer:
The volume of the pancreas is estimated using the Midpoint Rule by adding the cross-sectional areas and multiplying by the distance between sections, resulting in an estimated volume of 100.5 cubic centimeters.
Explanation:
The question asks us to use the Midpoint Rule to estimate the volume of the human pancreas given its length (12 cm) and cross-sectional areas at 1 cm intervals. To do this, we add up the areas of the cross-sections (excluding the first and last since they're 0) and multiply by the distance between the sections (1 cm). This approach approximates the volume using cylindrical segments where each segment's volume is its cross-sectional area times the height (1 cm).
The given cross-sectional areas are 7.9, 15.3, 18.1, 10.2, 10.7, 9.5, 8.5, 7.8, 5.6, 4.2, and 2.7 square cm. Adding these gives a total of 100.5 square cm. Since the distance between each section is 1 cm, multiplying 100.5 by 1 cm gives an estimated pancreatic volume of 100.5 cubic centimeters.
This method, while not perfectly accurate due to it being an approximation, provides a useful estimate of the volume based on the available data. It's a practical application of the Midpoint Rule in estimating volumes from cross-sectional data.
as a lifeguard, sara earns a base pay of $80 per day. if her day involves swim instruction, sara earns an additional 9t dollars, where t represents the number of hours she gives instruction and $9 is her hourly rate. The total amount Sara earns in a day can be expressed as 80 + 9t.
(a) What does 9t represent in this context?
(b) What are the terms and the coefficients in the expression 80 + 9t
(c) Rewrite the expression for 7 hours of swim instruction.
(d) how much will sara earn in all?
Answer:
Step-by-step explanation:
Sara's base pay is $80
(a) The 9t represents the additional amount of money that she earns if she gives instruction for t hours. 9 stands for $9 per hour
b)The term in the expression is t which stands for the number of hours that she instructs. The coefficient is 9 which stands for the hourly rate
c) The expression for 7 hours would be
We will substitute t into 80 + 9t. It becomes
80 + 9×7
d) in all, she will earn
80 + 9×7 = $143
The graph of the function f(x) = –(x + 3)(x – 1) is shown below. On a coordinate plane, a parabola opens down. It goes through (negative 3, 0), has a vertex at (negative 1, 4), and goes through (1, 0).
What is true about the domain and range of the function?
A. The domain is all real numbers less than or equal to 4, and the range is all real numbers such that –3 ≤ x ≤ 1.
B. The domain is all real numbers such that –3 ≤ x ≤ 1, and the range is all real numbers less than or equal to 4.
C. The domain is all real numbers, and the range is all real numbers less than or equal to 4.
D. The domain is all real numbers less than or equal to 4, and the range is all real numbers.
Answer:
The Domain is "All Real numbers", and the Range is "All real number less than or equal to 4" which coincides with option C in your problem.
Step-by-step explanation:
Recall that the Domain of a function is the set of all x-values for which there is a y-value obtained by the rule defined by the function.
In this case, whatever real number you enter for "x" in your functional expression, you will find another real value "y". That means that the actual Domain of your function is the full Real number line (all Real numbers).
Recall as well that the Range of a function is the set of y-values that are being "called" (or generated) as you use all the values for x in the Domain. In our case, it is great that they give the graph of the function (see attached image), so you can visualize that the function's graph is that of a "parabola" with branches opening downwards. You can see as well that the parabola presents a maximum value when x = -1, that means that any other value of x you use cannot give you as result a y-value that goes above that maximum.
If you evaluate that maximum value of the vertical coordinate by replacing "x" with "-1" in the actual function, you get:
[tex]f(x)=-(x+3)*(x-1)\\f(-1)=-(-1+3)*(-1-1)\\f(-1)=-(2)*(-2)\\f(-1)=4[/tex]
That means that the maximum y-value one can get from this function is "4". That is, the actual Range of the function can be any number that is smaller or equal to 4.
Bottom line: The Domain is "All Real numbers", and the Range is "All real number less than or equal to 4".
Answer:
its C
Step-by-step explanation:
edge 2020
A farmer needs to enclose three sides of a garden with a fence (the fourth side is a cliff wall). The farmer has 43 feet of fence and wants the garden to have an area of 228 sq-feet. What should the dimensions of the garden be? (For the purpose of this problem, the width will be the smaller dimension(needing two sides); the length will be the longer dimension (needing one side). Additionally? The length should be as long as possible.)
Answer:
9.5 ft wide by 24 ft long
Step-by-step explanation:
A graphing calculator shows there are two solutions to the system of equations ...
2x + y = 43 . . . . . . fence length when x= width
xy = 228 . . . . . . . area
The solutions are ...
(width, length) = (9.5, 24) or (12, 19)
Since we want the length as long as possible, the choice of dimensions is ...
length = 24 feet; width = 9.5 feet.
I need help with 3 please!
Answer:
Step-by-step explanation:
PQ=24
PS=19
PR=42
TQ=10
QR=19
SR=24
PT=21
SQ=20
m∠QRS=180-m∠PQR=180-106=74°
m∠PQS=m∠QSR=49°
m∠RPS=m∠PRQ=m∠QRS-m∠PRS=74-35=39°
m∠PSQ=m∠RQS=m∠PQR-m∠PQS=106-49=57°
m∠PQR=106°
m∠ QSR=49°
M∠PRS=35°
Find the vertices and foci of the hyperbola with equation quantity x plus 4 squared divided by 9 minus the quantity of y plus 3 squared divided by 16 = 1.
Answer:
vertices: (-7, -3), (-1, -3)foci: (-9, -3), (1, -3)Step-by-step explanation:
For a hyperbola of the form ...
[tex]\dfrac{(x-h)^2}{a^2}-\dfrac{(y-k)^2}{b^2}=1[/tex]
The vertices are located at (h±a, k), and the foci are located at (h±c, k), where ...
[tex]c=\sqrt{a^2+b^2}[/tex]
Here, we have (h, k) = (-4, -3), a=3, b=4, and c=√(9+16) = 5.
So, the points of interest are ...
vertices: (-4±3, -3) . . . . shown red on the graphfoci: (-4±5, -3) . . . . . . . . shown green on the graphAnswer:
previous was correct
Step-by-step explanation:
Identify the asymptotes of each function. 1/(x-3) - 6
[tex]\bf \cfrac{1}{x-3}-6\implies \stackrel{\textit{using the LCD of x-3}}{\cfrac{1-(x-3)6}{x-3}}\implies \cfrac{1-6x+18}{x-3}\implies \cfrac{-6x+19}{x-3}[/tex]
for the vertical asymptote, we simply zero out the denominator and solve for "x"
x - 3 = 0
x = 3 <---- that's the only vertical asymptote
for the horizontal asymptote, well, let's notice the degrees of the numerator and denominator, in this case, the degree of the numerator is 1, and the degree of the denominator is 1, thus when that occurs, the horizontal asymptote occurs at the fraction from the leading terms' coefficients.
[tex]\bf \cfrac{-6x+19}{1x-3}\implies \stackrel{\textit{horizontal asymptote}}{\cfrac{-6}{1}\implies -6 = y}[/tex]
Computer output from a regression analysis is provided. Coefficients: Estimate Std. Error t value p-value (Intercept) 7.2960 14.5444 0.502 0.62200 X 1.6370 0.5453 3.002 0.00765 We want to do the hypothesis test to see if the slope in the population is different from zero? That is, do the hypothesis test to see if we have a statistically significant linear relationship. What is your decision on the hypothesis test and why? Use a level of significance of .05.
Answer:
Step-by-step explanation:
Hello!
You need to test the hypothesis that the slope of the regression is cero.
I've run in the statistic software the given data for Y and X and estimated the regression line:
Yi= 7.82 -1.60Xi
Where
a= 7.82
b=-1.60
Sb= 3.38
The hypothesis is:
H₀: β = 0
H₁: β ≠ 0
α: 0.05
This is a two-tailed test, the null hypothesis states that the slope of the regression is cero, this means that if the null hypothesis is true, there is no linear regression between Y and X.
The statistic for this test is a Student-t
t= b - β ~t[tex]_{n-2}[/tex]
Sb
The critical values are:
Left: [tex]t_{n-2; \alpha /2} = t_{2; 0.025} = -4.303[/tex]
Right: [tex]t_{n-2; \alpha /2} = t_{2; 0.975} = 4.303[/tex]
t= -1.60 - 0 = -0.47
3.38
the p-value is also two-tailed, you can calculate it by hand:
P(t ≤ -0.47) + (1 - P(t ≤ 0.47) = 0.3423 + (1 - 0.6603) =0.6820
With the level of significance of 5%, the decision is to not reject the null hypothesis. This means that the slope of the regression is equal to cero, i.e. there is no linear regression between the two variables.
I hope this helps!
The hypothesis test, utilizing a t-statistic of 3.002 and a p-value of 0.00765, rejects the null hypothesis, demonstrating a statistically significant linear relationship. The slope coefficient (β₁ = 1.637) suggests a moderately strong relationship.
Hypothesis Test for a Linear Relationship
In this hypothesis test, we are trying to determine whether there is a statistically significant linear relationship between the independent and dependent variables. The null hypothesis (H₀) states that there is no linear relationship (β₁ = 0), while the alternative hypothesis (H₁) states that there is a linear relationship (β₁ ≠ 0).
The test statistic used in this case is the t-statistic, which is calculated as the ratio of the estimated slope (β₁) to its standard error. The p-value is the probability of obtaining a test statistic as extreme or more extreme than the observed test statistic, assuming that the null hypothesis is true.
In this particular example, the t-statistic is 3.002 and the p-value is 0.00765. The level of significance, which is the threshold for rejecting the null hypothesis, is typically set at 0.05.
Since the p-value (0.00765) is less than the level of significance (0.05), we reject the null hypothesis. This means that we have statistically significant evidence to conclude that there is a linear relationship between the independent and dependent variables. In other words, the slope of the regression line is not equal to zero, indicating that there is a change in the dependent variable as the independent variable changes.
Interpretation of Results
The rejection of the null hypothesis in this case indicates that there is a statistically significant linear relationship between the independent and dependent variables. This means that we can be confident that the observed relationship is not due to chance. The magnitude of the t-statistic (3.002) suggests that the relationship is moderately strong.
The interpretation of the slope coefficient (β₁) is that for every one-unit increase in the independent variable, the dependent variable is expected to increase by 1.637 units, on average. The standard error (0.5453) indicates the variability of the slope estimate.
In conclusion, the hypothesis test provides strong evidence to support the existence of a statistically significant linear relationship between the independent and dependent variables. The magnitude of the slope coefficient indicates that the relationship is moderately strong.
Learn more about hypothesis test, here:
https://brainly.com/question/34171008
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Juliet runs out of gas in Barnhaven, South Carolina. She walks 7 mi west and then 4 mi south looking for a gas station. How far is she from her starting point?
Answer: she is 8.06 miles from her starting point
Step-by-step explanation:
The diagram in the attached photo describes Juliet's movement from her starting point to her current position.
A triangle ABC is formed
AB = distance that she walked towards west
BC = distance that she walked towards south
AC= x = distance that she is from her starting point
The diagram is a right angle triangle. So we can find the distance that she is from her starting point using Pythagoras theorem.
Hypotenuse^2 = opposite^2 + adjacent^2
Hypotenuse = x
Opposite = 7
Adjacent = 4
x^2 = 7^2 + 4^2
x^2 = 49 + 16 = 65
x = √65 = 8.06 miles
If 1/3 of the school is 6th graders and 1/2 are girls what fraction are 6th grade girls
Answer: 1/6
Step-by-step explanation:
Find the value of x.
A. 4
B. 5
C. 6
D. 9
Answer:
C. 6
Step-by-step explanation:
You can try the answers to see which satisfies the Pythagorean theorem:
A. 4² +7² = 16+49 ≠ 117
B. 5² +8² = 25 +64 ≠ 117
C. 6² +9² = 36 +81 = 117 . . . . this (C) is the correct choice
and, for completeness, ...
D. 9² +12² = 81 +144 ≠ 117
_____
Working out
You can also actually work the problem. The Pythagorean theorem tells you ...
x² + (x+3)² = (√117)²
2x² +6x +9 = 117
2x² +6x = 108 . . . . subtract 9
x² +3x = 54 . . . . . . divide by 2
x(x +3) = 54 . . . . . . factor
Now, you can compare to your memorized times tables, where you find 6×9 = 54, so you know that x=6.
__
In case times tables are a challenge, you can continue to complete the square:
x² +3x +1.5² = 54 +1.5²
(x +1.5)² = 56.25 = 7.5² . . . . write as squares
x = 7.5 -1.5 = 6 . . . . . . . . . . . square root, subtract 1.5
The value of x is 6.
_____
The other solution to the quadratic equation is x=-9, but negative segment lengths make no sense. We acknowledge, then ignore, that extraneous solution.
The accounting department analyzes the variance of the weekly unit costs reported by two production departments. A sample of 16 cost reports for each of the two departments shows cost variances of 2.3 and 5.6, respectively. Is this sample sufficient to conclude that the two production departments differ in terms of unit cost variance? Use a = .10.
Calculate the value of the test statistic (to 2 decimals).
Answer:
Test Statistic is 2.34
Step-by-step explanation:
File Attached. kindly view it.
A newsletter publisher believes that over 64%64% of their readers own a Rolls Royce. For marketing purposes, a potential advertiser wants to confirm this claim. After performing a test at the 0.010.01 level of significance, the advertiser decides to reject the null hypothesis. What is the conclusion regarding the publisher's claim?
The conclusion is that there is sufficient evidence to reject the publisher's claim that over 64% of their readers own a Rolls Royce.
When conducting a hypothesis test, there are two competing hypotheses: the null hypothesis (H0) and the alternative hypothesis (H1). In this scenario:
Null hypothesis (H0): The proportion of readers who own a Rolls Royce is equal to or less than 64%.
Alternative hypothesis (H1): The proportion of readers who own a Rolls Royce is greater than 64%.
The significance level, denoted by α, represents the probability of rejecting the null hypothesis when it is actually true. In this case, α = 0.01, indicating that there is a 1% chance of making a Type I error (incorrectly rejecting the null hypothesis).
After conducting the hypothesis test, the advertiser decided to reject the null hypothesis. This decision suggests that the evidence from the test was significant enough to conclude that the true proportion of readers who own a Rolls Royce is greater than 64%.
In summary, based on the test results at the 0.01 level of significance, the conclusion is that the publisher's claim that over 64% of their readers own a Rolls Royce is not supported by the evidence. The data suggest that the proportion may be higher than 64%.
please help
1 through 5
Answer:
Step-by-step explanation:
A boat tour company finds that if the price p , charged for an one-hour harbor tour, is $20, the average number of passengers per week, x , is 300. When the price is reduced to $18, the average number of passengers per week increases to 360.The demand and supply curves of a certain brand of running shoes are given by
p=0(x)=1 1 2-0.04x, and p-S(x)-0.06x+42, where p is the price in dollars and x is the quantity sold.
(a) Assuming that the demand curve p D(x) is linear, find its formula: P-D(x)
(b) Assuming that the equilibrium price is $22 per tour, the equilibrium demand is E
(c) The consumers' surplus at that demand is $ . (Use a fraction for the coefficient.)
Answer
The answer and procedures of the exercise are attached in a microsoft word document.
Explanation
Please consider the data provided by the exercise. If you have any question please write me back. All the exercises are solved in a single sheet with the formulas indications.
Answer:
a) p(x) = (-1/30)*x + 30
b) xe = 240
c) The consumers' surplus at that demand is $ 960.00
Step-by-step explanation:
Given info
p₁ = 20
x₁ = 300
p₂ = 18
x₂ = 360
a) We can apply the following equation
p - p₁ = m*(x - x₁)
where
m is the slope, which can be obtained as follows
m = (p₂-p₁) / (x₂-x₁)
⇒ m = (18-20) / (360-300) = -2/60 = -1/30
then
p-20 = (-1/30)*(x-300)
⇒ 30*p - 600 = -x + 300
⇒ p(x) = (-1/30)*x + 30
b) If p(x) = 22
⇒ 22 = (-1/30)*x + 30
⇒ xe = 240 passengers
c) If x = 0 ⇒ p(0) = (-1/30)*(0) + 30 = 30
then
we can get h as follows
h = p(0) - p(240) = 30 - 22 = 8
if b = x(equlibrium) = 240
we apply
A = b*h/2 ⇒ A = 240*8/2 = 960
The consumers' surplus at that demand is $ 960.00
A small town has 2100 inhabitants. At 8 AM, 80 people have heard a rumor. By noon half the town has heard it. At what time will 90% of the population have heard the rumor? (Do not round k in your calculation. Round the final answer to one decimal place.) hours after the beginning
Answer:
2.7 PM
Step-by-step explanation:
Since, the rate of spread is proportional to the product of fraction y of people who have heard the rumour and the fraction who have not heard,
[tex]\frac{dy}{dt}=ky(1-y)[/tex]
[tex]\frac{dy}{y(1-y)}=kdt[/tex]
[tex](\frac{1}{y}+\frac{1}{1-y})dy = kdt[/tex]
Integrating both sides,
[tex]\int (\frac{1}{y}+\frac{1}{1-y})dy = \int kdt[/tex]
[tex]\ln y - \ln (1-y) = kt + C[/tex]
[tex]\ln (\frac{y}{1-y}) = kt + C[/tex]
[tex]\frac{y}{1-y}=e^{kt + C}[/tex]
[tex]y = e^{kt+C} - y e^{kt+C}[/tex]
[tex]y(1+e^{kt+C}) = e^{kt+C}[/tex]
[tex]y = \frac{e^{kt+C}}{1+e^{kt+C}}[/tex]
If t = 0, ( at 8 AM ), y = [tex]\frac{80}{2100}[/tex]
[tex]\frac{80}{2100}= \frac{e^{0+C}}{1+e^{0+C}}[/tex]
[tex]\frac{4}{105}=\frac{e^C}{1+e^C}[/tex]
[tex]4 + 4e^C = 105e^C[/tex]
[tex]4 = 101e^C[/tex]
[tex]\implies e^C=\frac{4}{101}[/tex]
Now, at noon, i.e t = 4, y = [tex]\frac{1}{2}[/tex]
[tex]\frac{1}{2}=\frac{e^{4k}.\frac{4}{101}}{1+e^{4k}.\frac{4}{101}}[/tex]
[tex]\frac{1}{2}=\frac{4e^{4k}}{101+4e^{4k}}[/tex]
[tex]101 + 4e^{4k}=8 e^{4k}[/tex]
[tex]101 = 4e^{4k}[/tex]
[tex]\frac{101}{4}=e^{4k}[/tex]
[tex](\frac{101}{4})^\frac{1}{4} = e^k[/tex]
If [tex]y = \frac{90}{100}=\frac{9}{10}[/tex]
[tex]\frac{9}{10}= \frac{(\frac{101}{4})^\frac{t}{4}\times \frac{4}{101}}{1+(\frac{101}{4})^\frac{t}{4}\times \frac{4}{101}}[/tex]
Using graphing calculator,
t ≈ 6.722,
Hence, after 6.722 hours since 8 AM, i.e. on 2.7 PM ( approx ) the 90% of the population have heard the rumour.
If WZ is the perpendicular bisector of VY , what conclusion can you make?
VWZ=YWZ
VWZ=WVZ
VYW=YWV
WYZ=ZWV
Answer:
[tex]\triangle VWZ = \triangle YWZ[/tex]. Justification is given below.
Step-by-step explanation:
Given:
[tex]\angle WVZ = \angle WYZ[/tex]
WZ is the perpendicular bisector of VY.
∴[tex]\angle WZV = \angle WZY = 90\°\\and\\VZ = YZ[/tex]
In [tex]\triangle VWZ and \triangle YWZ[/tex]
[tex]\angle WVZ = \angle WYZ[/tex] Given
[tex]VZ = YZ[/tex]
[tex]\angle WZV = \angle WZY = 90\°[/tex]
∴ [tex]\triangle VWZ \cong \triangle YWZ[/tex] by ASA test
[tex]\triangle VWZ = \triangle YWZ[/tex]
Here the correspondence of the vertices of a triangle should be match hence the option is first one that is.
[tex]\triangle VWZ = \triangle YWZ[/tex]
Answer:
Step-by-step explanation:
VWZ=YWZ
A random sample of 49 text books purchased at a local bookstore showed an average price of $122 with a population standard deviation of $15. Let u (new) be the true mean cost of a text book sold by this store. Construct a confidence interval with a 90% degree of confidence. Clearly label the following:
a. Point estimate
b. Critical value,
c. Margin of error
d. Confidence interval
e. Interpretation (confidence statement).
Answer:
point estimate is $122critical value for the 90% confidence level (1.645)margin of error is $3.52590% confidence interval is $122±3.525there is 90% probability that true population average price of text books is in the range $122±$3.525Step-by-step explanation:
Confidence Interval can be calculated using P±ME where
P is the point estimate for the mean cost of a text book ( $122 )ME is the margin of error from the meanAnd margin of error (ME) can be calculated using the formula
ME=[tex]\frac{z*s}{\sqrt{N} }[/tex] where
z is the critical value for the 90% confidence level (1.645)s is the population standard deviation ($15) N is the sample size (49)Margin of error, ME=[tex]\frac{1.645*15}{\sqrt{49} }[/tex] = 3.525
Then 90% confidence interval is $122±3.525
To interpret this, there is 90% probability that true population average price of text books is in the range $122±$3.525
The mean life span of a brand name tire is 50,000 miles. Assume that the life spans of the tires are normally distributed, and the population standard deviation is 800 miles.
a. If you select one tire, what is the probability that its life span is less than 48,500 miles?
b. If you select 100 tires, what is the probability that their mean life span is more than 50,200 miles?
Answer:
a) [tex]P(X <48500)=0.0304[/tex]
b) [tex]P(\bar X>50200)=1-0.994=0.0062[/tex]
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Let X the random variable that represent the mean life span of a brand name tire, and for this case we know the distribution for X is given by:
[tex]X \sim N(\mu=50000,\sigma=800)[/tex]
Part a
We want this probability:
[tex]P(X<48500)[/tex]
The best way to solve this problem is using the normal standard distribution and the z score given by:
[tex]z=\frac{x-\mu}{\sigma}[/tex]
If we apply this formula to our probability we got this:
[tex]P(X <48500)=P(Z<\frac{48500-50000}{800})=P(Z<-1.875)=0.0304[/tex]
Part b
Let [tex]\bar X[/tex] represent the sample mean, the distribution for the sample mean is given by:
[tex]\bar X \sim N(\mu,\frac{\sigma}{\sqrt{n}})[/tex]
On this case [tex]\bar X \sim N(50000,\frac{800}{\sqrt{100}})[/tex]
We want this probability:
[tex]P(\bar X>50200)=1-P(\bar X<50200)[/tex]
The best way to solve this problem is using the normal standard distribution and the z score given by:
[tex]z=\frac{x-\mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
If we apply this formula to our probability we got this:
[tex]P(\bar X >50200)=1-P(Z<\frac{50200-50000}{\frac{800}{\sqrt{100}}})=1-P(Z<2.5)[/tex]
[tex]P(\bar X>50200)=1-0.994=0.0062[/tex]
Final answer:
To determine the probability that a tire's lifespan is less than a certain value, you calculate its Z-score and look up the probability in a standard normal distribution table. For sample means, use the standard error to calculate Z-scores. In hypothesis testing, compare the p-value to α to decide on the claim.
Explanation:
Calculating Probabilities for Normally Distributed Tires' Lifespan
To calculate the probability that a single tire has a lifespan of less than 48,500 miles, we can use the Z-score formula. The formula for the Z-score is (X - μ) / σ, where X is the value we are looking at (48,500 miles), μ is the mean (50,000 miles), and σ is the population standard deviation (800 miles). Calculating the Z-score gives us (48,500 - 50,000) / 800 = -1.875. We can then look up this Z-score on a standard normal distribution table or use a calculator to find the corresponding probability.
For part b, looking at the probability for the mean of 100 tires being more than 50,200 miles, we would use the standard error of the mean, σ/√ n, where n is the sample size (100 in this case). We calculate the Z-score with this new standard deviation and find the probability corresponding to that Z-score using the same method.
It is essential to remember that for larger samples, the standard deviation of the mean decreases, and the calculations have to reflect this change.
Hypothesis Testing for Tires' Lifespan
Using a hypothesis test, we can determine if there is enough evidence to support or reject a claim about the population mean. In a hypothesis test, we compare the p-value to the level of significance (α = 0.05). If the p-value is lower than α, we reject the null hypothesis. In the example given with an alpha of 0.05 and a p-value of 0.0103, we would reject the null hypothesis, concluding that the average lifespan of the tires is likely less than the claimed 50,000 miles.
The Aluminum Association reports that the average American uses 56.8 pounds of aluminum in a year. A random sample of 50 households is monitored for one year to determine aluminum usage. If the population standard deviation of annual usage is 12.1 pounds, what is the probability that the sample mean will be each of the following? Appendix A Statistical Tablesa. More than 58 poundsb. More than 57 poundsc. Between 55 and 57 poundsd. Less than 53 poundse. Less than 48 pounds
Answer:
[tex]\mu = 56.8[/tex]
[tex]\sigma = 12.1[/tex]
A)what is the probability that the sample mean will be More than 58 pounds
P(x>58)
Formula : [tex]Z=\frac{x-\mu}{\sigma}[/tex]
Substitute the values :
[tex]Z=\frac{58-56.8}{12.1}[/tex]
[tex]Z=0.09917[/tex]
refer the z table
P(x<58)=0.5359
P(X>58)=1-P(x<58)=1-0.5359=0.4641
Hence the probability that the sample mean will be More than 58 pounds is 0.4641
B)what is the probability that the sample mean will be More than 57 pounds
P(x>57)
Formula : [tex]Z=\frac{x-\mu}{\sigma}[/tex]
Substitute the values :
[tex]Z=\frac{57-56.8}{12.1}[/tex]
[tex]Z=0.0165[/tex]
refer the z table
P(x<57)=0.5040
P(X>57)=1-P(x<57)=1-0.5040=0.496
Hence the probability that the sample mean will be More than 57 pounds is 0.496
C)what is the probability that the sample mean will be Between 55 and 57 pound
Formula : [tex]Z=\frac{x-\mu}{\sigma}[/tex]
Substitute the values :
[tex]Z=\frac{57-56.8}{12.1}[/tex]
[tex]Z=0.0165[/tex]
refer the z table
P(x<57)=0.5040
Formula : [tex]Z=\frac{x-\mu}{\sigma}[/tex]
Substitute the values :
[tex]Z=\frac{55-56.8}{12.1}[/tex]
[tex]Z=-0.1487[/tex]
refer the z table
P(x<55)=0.4443
P(55<x<57)=P9x<57)-P(x<55) =0.5040-0.4443=0.0597
Hence the probability that the sample mean will be Between 55 and 57 pounds is 0.0597
D)what is the probability that the sample mean will be Less than 53 pounds
Formula : [tex]Z=\frac{x-\mu}{\sigma}[/tex]
Substitute the values :
[tex]Z=\frac{53-56.8}{12.1}[/tex]
[tex]Z=−0.314[/tex]
refer the z table
P(x<53)=0.3783
The probability that the sample mean will be Less than 53 pounds is 0.3783
E)what is the probability that the sample mean will be Less than 48 pounds
Formula : [tex]Z=\frac{x-\mu}{\sigma}[/tex]
Substitute the values :
[tex]Z=\frac{48-56.8}{12.1}[/tex]
[tex]Z=−0.727[/tex]
refer the z table
P(x<48)=0.2358
The probability that the sample mean will be Less than 48 pounds is 0.2358
Let R+ denote the set of positive real numbers. Let f : R × R+ → R be given by f(x, y) = x/y. (a) Is f an injective function? Prove your answer. (b) Is f a surjective function? Again, prove your answer. (c) Is f a bijection? Prove your answer.
Let [tex]f:\Bbb R\times\Bbb R^+\to\Bbb R[/tex].
a. [tex]f[/tex] is injective if [tex]f(x_1,y_1)=f(x_2,y_2)[/tex] for any two points [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex], then the two points must be identical with [tex]x_1=x_2[/tex] and [tex]y_1=y_2[/tex].
[tex]f[/tex] is not injective because we can pick two points for which [tex]f[/tex] gives the same value:
[tex]f(2,1)=\dfrac21=2[/tex]
[tex]f(4,2)=\dfrac42=2[/tex]
b. [tex]f[/tex] is surjective if there exists [tex](x,y)[/tex] for which every point in the codomain [tex]\Bbb R[/tex] is obtained by [tex]f(x,y)[/tex].
This should be somewhat obvious if you consider 3 different cases:
[tex]f(x,y)<0[/tex] and can take on any negative real number for any choice of [tex]x<0[/tex][tex]f(x,y)=0[/tex] if and only if [tex]x=0[/tex][tex]f(x,y)>0[/tex] for any choice of [tex]x>0[/tex]So [tex]f[/tex] is surjective.
c. [tex]f[/tex] is bijective if it is both injective and surjective. The conclusions above show [tex]f[/tex] is not bijective.
Construct a confidence interval of the population proportion at the given level of confidence. x = 125, n = 250, 90 % confidence.
The 90% confidence interval is ____________?
(Use ascending order. Round to three decimal places as needed.)
Answer: 90% confidence interval would be (0.448, 0.552).
Step-by-step explanation:
Since we have given that
x = 125
n = 250
So, [tex]\hat{p}=\dfrac{x}{n}=\dfrac{125}{250}=0.5[/tex]
We need to find the 90% confidence interval.
so, z = 1.64
So, interval would be
[tex]\hat{p}\pm z\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}\\\\=0.5\pm 1.64\sqrt{\dfrac{0.5\times 0.5}{250}}\\\\=0.5\pm 0.0519\\\\=(0.5-0.0519,0.5+0.0519)\\\\=(0.448,0.552)[/tex]
Hence, 90% confidence interval would be (0.448, 0.552)
Final answer:
The 90% confidence interval is (0.462, 0.538). To calculate a confidence interval, use the formula p' ± z * √(p'q'/n). In this case, with x = 125, n = 250, and 90% confidence, the interval is (0.462, 0.538).
Explanation:
The 90% confidence interval is (0.462, 0.538).
To construct a confidence interval for a population proportion, you can use the formula for confidence interval: p' ± z * √(p'q'/n), where p' = x/n, q' = 1 - p', and z corresponds to the confidence level.
In this case, given x = 125, n = 250, and 90% confidence level, the confidence interval is (0.462, 0.538).
The speed limit posted on a local highway is 75 mph. Is the average speed on that stretch of highway significantly more than 75mph? Forty vehicle’s speeds were recorded by speed detection devices. What would be the correct alternative hypothesis?
Answer: [tex]H_a: \mu >75[/tex]
Step-by-step explanation:
Alternative hypothesis [tex](H_a)[/tex]: It is a statement which always indicates that there is a significant difference between the groups being tested.
It always contains by < , > or ≠ sign .
Given claim : Is the average speed on that stretch of highway significantly more than 75mph?
Here objective is whether the average speed on that stretch of highway significantly more than 75mph.
Let [tex]\mu[/tex] be the population mean speed on that stretch of highway significantly more than 75mph.
Then, [tex]H_a: \mu >75[/tex]
Hence , the correct alternative hypothesis would be : [tex]H_a: \mu >75[/tex]
The correct alternative hypothesis for this question is that the average speed on the highway is significantly greater than 75mph.
Explanation:The correct alternative hypothesis for this question would be that the average speed on that stretch of highway is significantly greater than 75mph.
To determine if the average speed is significantly more than 75mph, a statistical test can be conducted. This test would compare the speeds of the 40 recorded vehicles to the 75mph speed limit and calculate the average speed. If the average speed is significantly higher than 75mph, it would support the alternative hypothesis.
For example, if the average speed of the 40 recorded vehicles is found to be 80mph with a small p-value, it would indicate that the average speed on that stretch of highway is significantly greater than 75mph.
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According to the United Nations, in the year 2002, the population of the world was 6.1 billion people and was growing at an annual rate of about 1.5%. If this pattern were to continue, then every year, the population would be 1.015 times the population of the previous year. Thus, if P(t) is the world population (in billions) t years after the base year 2002.
(a) What was the population in 2004?
(b) What will the population be in 2010?
Answer:
(a) 6.2843 billion
(b) 6.8716 billion
Step-by-step explanation:
Since there is a constant growth rate of 1.5% per year, and the population in 2002 was 6.1 billion people. The general equation for the total world population, in billions, after 2002 is:
[tex]P(t) = 6.1*(1+0.015)^t[/tex]
With t being the time, in years, after 2002.
a) What was the population in 2004?
[tex]t=2004-2002=2\\P(2) = 6.1*(1.015)^2\\P(2) = 6.2843 \ billion[/tex]
b) What was the population in 2010?
[tex]t=2010-2002=8\\P(8) = 6.1*(1.015)^8\\P(8) = 6.8716 \ billion[/tex]
The total resistance R of two resistors connected in parallel circuit is given by 1/R = 1/R_1 + 1/R_2. Approximate the change in R as R_1 is decreased from 12 ohms to 11 ohms and R_2 is increased from 10 ohms to 11 ohms. Compute the actual change.
Answer:
a) Approximate the change in R is 0.5 ohm.
b) The actual change in R is 0.04 ohm.
Step-by-step explanation:
Given : The total resistance R of two resistors connected in parallel circuit is given by [tex]\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}[/tex]
To find :
a) Approximate the change in R ?
b) Compute the actual change.
Solution :
a) Approximate the change in R
[tex]R_1=12\ ohm[/tex] and [tex]R_2=10\ ohm[/tex]
[tex]R_1[/tex] is decreased from 12 ohms to 11 ohms.
i.e. [tex]\triangle R_1=21-11=1\ ohm[/tex]
[tex]R_2[/tex] is increased from 10 ohms to 11 ohms.
i.e. [tex]\triangle R_2=11-10=1\ ohm[/tex]
The change in R is given by,
[tex]\frac{1}{\triangle R}=\frac{1}{\triangle R_1}+\frac{1}{\triangle R_2}[/tex]
[tex]\frac{1}{\triangle R}=\frac{\triangle R_2+\triangle R_1}{(\triangle R_1)(\triangle R_2)}[/tex]
[tex]\triangle R=\frac{(\triangle R_1)(\triangle R_2)}{\triangle R_2+\triangle R_1}[/tex]
[tex]\triangle R=\frac{(1)(1)}{1+1}[/tex]
[tex]\triangle R=\frac{1}{2}[/tex]
[tex]\triangle R=0.5\ ohm[/tex]
b) The actual change in Resistance
[tex]\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}[/tex]
[tex]\frac{1}{R}=\frac{1}{12}+\frac{1}{10}[/tex]
[tex]R=\frac{10\times 12}{10+12}[/tex]
[tex]R=\frac{120}{22}[/tex]
[tex]R=5.46\ ohm[/tex]
When resistances are charged, [tex]R_1=R_2=11[/tex]
[tex]\frac{1}{R'}=\frac{1}{R_1}+\frac{1}{R_2}[/tex]
[tex]\frac{1}{R'}=\frac{1}{11}+\frac{1}{11}[/tex]
[tex]R'=\frac{11}{2}[/tex]
[tex]R'=5.5\ ohm[/tex]
Change in resistance is given by,
[tex]C=R'-R[/tex]
[tex]C=5.5-5.46[/tex]
[tex]C=0.04\ ohm[/tex]
Final answer:
The actual change in total resistance R of a parallel circuit as R_1 is decreased from 12 ohms to 11 ohms and R_2 is increased from 10 ohms to 11 ohms is calculated using the formula 1/R = 1/R_1 + 1/R_2.
Explanation:
The question is asking for the change in total resistance R of a parallel circuit when the resistances R_1 and R_2 are changed from 12 ohms to 11 ohms and 10 ohms to 11 ohms, respectively. To calculate the actual change, we can use the formula:
1/R = 1/R_1 + 1/R_2
Before the change, the total resistance is:
1/R_initial = 1/12 + 1/10
After the change, it becomes:
1/R_final = 1/11 + 1/11
By calculating both 1/R_initial and 1/R_final, we determine R_initial and R_final separately and find the difference between them to get the actual change in resistance.
In a competition between players X and Y, the first player to win three games in a row or a total of four games wins. How many ways can the competition be played if X wins the first game and Y wins the second and third games? (Hint: Draw a tree.)
Answer:
The competition can be played in 7 different ways
Step-by-step explanation:
First game : X
Second game : Y
Third game : Y
After that either X or Y can win
¹Case - 1: Y wins the fourth game:The competition ends with Y winning as he/she won 3 games in a row
Case - 2: X wins the fourth game:more games are required to decide the winner
Sub-case - a: X wins the fifth game:more games are required to decide the winner
²Sub-sub-case - i: X wins the sixth game:The competition ends with X winning as he/she won 3 games in a row
³⁻⁴Sub-sub-case - ii: Y wins the sixth game:The competition ends after the seventh game. Whoever wins seventh game wins the competition as he/she would've won 4 games in total by then.
Sub-case - b: Y wins the fifth game:more games are required to decide the winner
⁵Sub-sub-case - i: Y wins the sixth game:The competition ends with Y winning as he/she won 4 games in total
⁶⁻⁷Sub-sub-case - ii: X wins the sixth game:The competition ends after the seventh game. Whoever wins seventh game wins the competition as he/she would've won 4 games in total by then.
∴The competition can be played in 7 different ways
The problem requires combinatorial analysis to find all potential sequences of wins leading to player X or Y's victory. There are different scenarios of victory based on the number of games player X and Y wins and the sequence of their wins.
Explanation:This problem showcases combinatorial analysis where we consider the different sequences of wins that leads either team to victory. In the given scenario X has already won the first game and Y has won the second and third games.
From this point, some possible winning combinations could be:
X winning next three games in a rowY winning next game, X winning three in a rowY winning next two games, X winning next three gamesY winning next game, X winning the next one, then Y wins, followed by another X winX wins next game, Y wins the two after, and X wins the next twoThese are representative of the different sequences of wins that could occur. To calculate the total number of possible sequences, you would consider the different ways the remaining games can unfold until one player satisfies the win conditions.
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In 1992 there was an earthquake at Little Skull Mountain, Nevada, measuring 5.5 on the Richter scale. In 1994 there was an earthquake near Double Spring Flat, Nevada, measuring 6.0 on the Richter scale. How did the power of the Double Spring Flat quake compare with that of the Little Skull Mountain quake? (Round your answer to two decimal places.)
Suppose that a magnet high school includes grades 11 and 12, with half of the students in each grade. 60% of the senior class and 10% of the junior class are taking calculus. Suppose a calculus student is randomly selected to accompany the math teachers to a conference. What is the probability that the student is a junior? (Enter your answer as a fraction.)
Answer: Our required probability is [tex]\dfrac{1}{7}[/tex]
Step-by-step explanation:
Since we have given that
P(Junior ) = [tex]\dfrac{1}{2}[/tex]
P(Senior) = [tex]\dfrac{1}{2}[/tex]
Let the given event be 'C' taking calculus.
P(C|J) = 10% = 0.10
P(C|S) = 60% = 0.60
We need to find the probability that the student is a junior.
So, our required probability is given by
[tex]P(J|C)=\dfrac{P(J).P(C|J)}{P(S).P(C|S)+P(J).P(C|J)}\\\\P(J|C)=\dfrac{0.5\times 0.1}{0.5\times 0.1+0.5\times 0.6}\\\\P(J|C)=\dfrac{0.05}{0.05+0.3}\\\\P(J|C)=\dfrac{0.05}{0.35}\\\\P(J|C)=\dfrac{5}{35}\\\\P(J|C)=\dfrac{1}{7}[/tex]
Hence, our required probability is [tex]\dfrac{1}{7}[/tex]
A random sample of 300 CitiBank VISA cardholder accounts indicated a sample mean debt of 1,220 with a sample standard deviation of 840. Construct a 95 percent confidence interval estimate of the average debt of all cardholders.
Answer: (1124.5619, 1315.4381)
Step-by-step explanation:
The confidence interval for population mean[tex](\mu)[/tex] when populatin standard deviation is unknown :-
[tex]\overline{x}\pm t^*\dfrac{s}{\sqrt{n}}[/tex]
, where [tex]\overline{x}[/tex] = Sample mean
[tex]s[/tex] =Sample standard deviation
t* = Critical t-value.
Given : n= 300
Degree of freedom : df = n-1 = 299
[tex]\overline{x}=1220[/tex]
[tex]s=840[/tex]
Confidence interval = 95%
Significance level : [tex]\alpha=1-0.95=0.05[/tex]
Using t-distribution table ,
The critical value for 95% Confidence interval for significance level 0.05 and df = 299 : [tex]t^*=t_{\alpha/2,\ df}=t_{0.025,\ 299}=1.9679[/tex]
Then, a 95% confidence interval estimate of the average debt of all cardholders will be :-
[tex]1220\pm (1.9679)\dfrac{840}{\sqrt{300}}[/tex]
[tex]=1220\pm (1.9679)\dfrac{840}{17.3205080757}[/tex]
[tex]=1220\pm (1.9679)(48.4974226119)[/tex]
[tex]\approx1220\pm 95.4381=(1220-95.438,\ 1220+95.438)\\\\=(1124.5619,\ 1315.4381)[/tex]
Hence, a 95% confidence interval estimate of the average debt of all cardholders is (1124.5619, 1315.4381) .
A television camera is positioned 4000 ft from the base of a rocket launching pad. The angle of elevation of the camera has to change at the correct rate in order to keep the rocket in sight. Also, the mechanism for focusing the camera has to take into account the increasing distance from the camera to the rising rocket. Let’s assume the rocket rises vertically and its speed is 600 ft/s when it has risen 3000 ft.
(a) How fast is the distance from the television camera to the rocket changing at that moment?
(b) If the television camera is always kept aimed at the rocket, how fast is the camera’s angle of elevation changing at that same moment?
Answer:
a) the time speed for distance from the television camera to the rocket is changing at rate of 360 ft/sec
b) camera’s angle of elevation changing at the rate of 0.096 radians/ sec
Step-by-step explanation:
details and step by step explanation is in the images below
image I is for Solution of a)
image 2 is for Solution of b)
In a sample of 83 walking canes, the average length was found to be 34.9in. with a standard deviation of 1.5. Give a point estimate for the population standard deviation of the length of the walking canes. Round your answer to two decimal places, if necessary.
Answer:
The point estimate for the population standard deviation of the length of the walking canes is 0.16.
Step-by-step explanation:
The Central Limit Theorem estabilishes that, for a random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], a large sample size can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\frac{\sigma}{\sqrt{n}}[/tex].
In this problem
The point estimate of the standard deviation is the standard deviation of the sample.
In a sample of 83 walking canes, the average length was found to be 34.9in. with a standard deviation of 1.5. By the Central Limit Theorem, we have that:
[tex]s = \frac{1.5}{\sqrt{83}} = 0.16[/tex]
The point estimate for the population standard deviation of the length of the walking canes is 0.16.