Answer:
1.83% probability there are no car accidents on that stretch on Monday
Step-by-step explanation:
In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
In which
x is the number of sucesses
e = 2.71828 is the Euler number
[tex]\mu[/tex] is the mean in the given time interval.
The number of accidents on a certain section of I-40 averages 4 accidents per weekday independent across weekdays.
This means that [tex]\mu = 4[/tex]
What is the probability there are no car accidents on that stretch on Monday?
This is P(X = 0).
[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]
[tex]P(X = 0) = \frac{e^{-4}*(4)^{0}}{(0)!} = 0.0183[/tex]
1.83% probability there are no car accidents on that stretch on Monday
The probability of there being no car accidents on Monday on a certain section of I-40 can be calculated using the Poisson distribution. In this case, with an average of 4 accidents per weekday, the probability is approximately 1.83%.
Explanation:To calculate the probability of there being no car accidents on Monday, we can use the Poisson distribution. In this case, the average number of accidents per weekday is given as 4. The Poisson distribution can be used to calculate the probability of a specific number of events occurring in a given time period.
The formula for calculating the probability of x events occurring in a Poisson distribution is:
P(x) = (e^-λ * λ^x) / x!
Where λ is the average number of events, and x is the number of events we want to calculate the probability for.
In this case, the average number of accidents per weekday is 4, so λ = 4. And we want to calculate the probability of there being no accidents, so x = 0.
Using the formula, we can calculate:
P(0) = (e^-4 * 4^0) / 0! = (e^-4 * 1) / 1 = e^-4, approximately 0.0183Therefore, the probability of there being no car accidents on Monday is approximately 0.0183, or 1.83%.
Determine if the described set is a subspace. The subset of Rn (n even) consisting of vectors of the form v = v1 vn , such that v1 − v2 + v3 − v4 + v5 − − vn = 0. The set is a subspace. The set is not a subspace.
Answer:
The set is a subspace
Step-by-step explanation:
We need to check 3 things: whether the 0 vector is in the set, whether the sum of 2 elements of the set is an element of the set and whether the product of an element of the set for a real scalar is an element of the set.
0 is in the setYes: the 0 vector (0, 0, ..., 0) satysfies the set property: 0-0+0-0........-0 = 0.
Given 2 elements v = (v1, ..., vn), w = (w1, ..., wn), is the sum v+2 = (v1+w1, v2+w2, ..., vn+wn) an element of the set?Yes: Note that (v1+w1)-(v2+w2)+(v3+w3)- ..... - (vn+wn) = v1-v1+v3 - ... - vn + w1 - w2 + w3 - ... - wn = 0+0 = 0.
Given an element of the set v = (v1, ... ,vn), and a real number r, is rv = (rv1, ..., rvn) an element of the set?Yes: By taking r as common factor, we have rv1 - rv2 + rv3 - ... - rvn = r * (v1-v2+v3 - ... - vn) = r*0 = 0.
Thus, the described set is effectively a subspace.
The described set is a subspace of Rn (n even). It satisfies all three conditions of a subspace: containing the zero vector, closed under addition, and closed under scalar multiplication.
Explanation:The set described is a subspace of ℝn (where n is even).
To determine if the set is a subspace, we need to check if it satisfies three conditions:
It contains the zero vector: The zero vector satisfies v1 - v2 + v3 - v4 + v5 - ... - vn = 0, so it is in the set.
Since the set satisfies all three conditions, it is a subspace of ℝn (where n is even).
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In the final round of a TV game show, contestants have a chance to increase their current winnings of $1 million to $2 million. If they are wrong, their prize is decreased to $500,000. A contestant thinks his guess will be right 50 percent of the time. Should he play
Answer:
The contestant should not play.Step-by-step explanation:
As per the given question, their current winning is $1 million.
The probability of the guessing to be true is 50% = [tex]\frac{50}{100} = \frac{1}{2}[/tex].
There is also a possibility of 50% to be wrong, which can reduced the winning amount to $500,000 that is the half of the current amount.
Hence, the contestant should not play.
Let p1 be the proportion of successes in the first population and let p2 be the proportion of successes in the second population. Suppose that you are testing the hypotheses: H0:p1−p2=0Ha:p1−p2≠0 Futhermore suppose that z∗=1.73, find and input the p-value for this test. Round your answer to 4 decimal places.
Answer:
0.0836
Step-by-step explanation:
Given that p1 be the proportion of successes in the first population and let p2 be the proportion of successes in the second population
[tex]H_0:p_1-p_2=0\\H_a:p_1-p_2\neq 0[/tex]
is the hypotheses created
This is two tailed
Test statistic Z = 1.73
corresponding p value
= 0.08363
=0.0836 (after rounding off to 4 decimals)
The p-value for this test is 0.0802.
The p-value for this two-tailed test can be calculated by doubling the probability of observing a test statistic at least as extreme as the observed z-score of 1.73, under the assumption that the null hypothesis is true.
Given that the z-score is 1.73, we need to find the area under the standard normal distribution curve that lies beyond this z-score. This area represents the probability of observing a value at least as extreme as 1.73 standard deviations from the mean under the null hypothesis.
Using a standard normal distribution table or a calculator, we can find the probability corresponding to a z-score of 1.73. Let's denote this probability as P(Z > 1.73).
P(Z > 1.73) = 1 - P(Z < 1.73)
Using a standard normal distribution table or calculator, we find that P(Z < 1.73) is approximately 0.9599. Therefore:
P(Z > 1.73) = 1 - 0.9599 = 0.0401
Since this is a two-tailed test, we need to consider the probability of observing a z-score at least as extreme in either direction (both tails of the distribution). Thus, we multiply the single-tail probability by 2:
p-value = 2 * P(Z > 1.73) = 2 * 0.0401 = 0.0802
Rounded to four decimal places, the p-value is 0.0802.
Therefore, the p-value for this test is 0.0802.
The answer is: 0.0802.
Seventy percent of light aircraft that disappear while in flight in a certain country are subsequently discovered. Of the aircraft that are discovered, 60% have an emergency locator, whereas 90% of the aircraft not discovered do not have an emergency locator. Suppose that a light aircraft has disappeared.a) If it has an emergency locator, what is the probability that it will not be discovered?b) If it does not have an emergency locator, what is the probability that it will be discovered?c) If we consider 10 light aircraft that disappeared in flight with an emergency recorder, what is the probability that 7 of them are discovered?
Answer:
Figure out the various probabilities first, that will make the rest of the questions easier:
P(discovered) = .7
P(not discovered) = 1 - .7 = .3
P(locator|discovered) = .6
P(no locator|discovered) = 1 - .6 = .4
P(locator|not discovered) = 1 - .9 = .1
P(no locator|not discovered) = .9
P(discovered and locator) = .7 * .6 = .42
P(discovered and no locator) = .7 * .4 = .28
P(not discovered and locator) = .3 * .1 = .03
P(not discovered and no locator) = .3 * .9 = .27
a) The total probability that an aircraft has a locator is .42 + .03 = .45. So the probability it will not be discovered, given it has a locator, is .03/.45 = .067
b) The total probability that an aircraft does not have a locator is .28 + .27 = .55. So the probability it will be discovered, given it does not have a locator, is .28/.55 = .509
c) Probability that 7 are discovered = C(10,7) * P(discovered|locator)^7 * P(not discovered|locator)^3
We already figured out P(not discovered|locator) = .067, so P(discovered|locator) = 1-.067 = .933. C(10,7) = 10*9*8, so we can compute total probability: 10*9*8 * .933^7 * .067^3 = .133
Step-by-step explanation:
A 15-inch candle is lit and burns at a constant rate of 1.1 inches per hour. Let t represent the number of hours since the candle was lit, and suppose f is a function such that f ( t ) represents the remaining length of the candle (in inches) t hours after it was lit. Write a function formula for f . f ( t )
Answer: f(t) = 15 - 1.1t
Step-by-step explanation:
Let t represent the number of hours since the candle was lit.
A 15-inch candle is lit and burns at a constant rate of 1.1 inches per hour. This means that in t hours, the candle that would have burnt is 1.1t
The length of the candle that would be left after t hours is expressed as
15 - 1.1t
suppose f is a function such that f(t) represents the remaining length of the candle (in inches) t hours after it was lit, then
f(t) = 15 - 1.1t
4.47 Consider an experiment, the events A and B, and probabilities P(A) 5 0.55, P(B) 5 0.45, and P(A d B) 5 0.15. Find the probability of: a. A or B occurring. b. A and B occurring. c. Just A occurring. d. Just A or just B occurring.
Answer:
a) 0.85
b) 0.15
c) 0.40
d) 0.70
Step-by-step explanation:
P(A) = 0.55
P(B) = 0.45
P(A n B) = 0.15
a) Probability of A or B occurring = P(A u B) = P(A) + P(B) - P(A n B) = 0.55 + 0.45 - 0.15 = 0.85
b) Probability of A and B occurring = P(A n B) = 0.15
c) Probability of just A occurring = P(A n B') = P(A) - P(A n B) = 0.55 - 0.15 = 0.40
d) Probability of just A or just B occurring = P(A n B') + P(A' n B) = 0.4 + (0.45 - 0.15) = 0.4 + 0.3 = 0.70
Write the composite function in the form f(g(x)).[Identify the inner function u = g(x)and the outer function
y = f(u).]
$ y = e^{{\color{red}7}\sqrt{x}} $
(g(x), f(u)) = ??
and find the derivative
For what values of x does the graph of f have a horizontal tangent? (Use n as your integer variable. Enter your answers as a comma-separated list.)
f(x) = x ? 2 sin x
x=??
Answer:
a) (g(x), f(u)) = ( 7*√x , e^u )
b) y ' = 3.5 * e^(7*√x) / √x
Step-by-step explanation:
Given:
- The given function:
y = e^(7*√x)
Find:
- Express the given function as a composite of f(g(x)). Where, u = g(x) and y = f(u).
- Express the derivative of y, y'?
Solution:
- We will assume the exponent of the natural log to be the u. So u is:
u = g(x) = 7*√x
- Then y is a function of u as follows:
y = f(u) = e^u
- The composite function is as follows:
(g(x), f(u)) = ( 7*√x , e^u )
- The derivative of y is such that:
y = f(g(x))
y' = f' (g(x) ) * g'(x)
y' = f'(u) * g'(x)
y' = e^u* 3.5 / √x
- Hence,
y ' = 3.5 * e^(7*√x) / √x
A company has learned that the relationship between its advertising and sales shows diminishing marginal returns. That is, as it saturates consumers with ads, the benefits of increased advertising diminish. The company should expect to find linear association between its advertising and sales.
Answer:
A company has learned that the relationship between its advertising and sales shows diminishing marginal returns. That is, as it saturates consumers with ads, the benefits of increased advertising diminish. The company should expect to find a linear association between its advertising and sales - This statement is false
Step-by-step explanation:
According to the scenario given for the company, it was said that the marginal return diminished after a saturation point, therefore, the company should rather expect a non-linear pattern and not a linear pattern.
Therefore, the statement expressed in the question is false.
The dimensions of a closed rectangular box are measured x, y and z as 100 cm, 70 cm, and 30 cm, respectively, with a possible error of 0.2 cm in each dimension. The surface area and the volume of the box is given by the equations S(x, y, z) = 2xy + 2xz + 2yz, V(x, y, z) = xyz Find the linear approximation of S at the point (96, 69, 29). b. Suppose the box has been measured with a ruler that has one centimeter gradation, find the actual maximum error in measuring the surface of the box. c. Find L(101,71,31) -L(100,70,30) d. Use differentials to estimate the error in the measurement of the surface area of the box. e. Compare the answers of parts c to d and the d to b. What do you conclude? f. A coat of paint of thickness 0.0002 cm is applied to the exterior surface of the box. Use differentials to estimate the amount of the paint needed.
Answer:see the pictures attached
Step-by-step explanation:
Given that (-4,9) is on the graph of f(x), find the corresponding point for the function. f(4x).
Answer:
(-1,9)
Step-by-step explanation:
Suppose that 20% of the adult women in the United States dye or highlight their hair. We would like to know the probability that a SRS of size 200 would come within plus or minus 3 percentage points of this true value. In other words, find probability that pˆ takes a value between 0.17 and 0.23.
Answer:
71.08% probability that pˆ takes a value between 0.17 and 0.23.
Step-by-step explanation:
We use the binomial approxiation to the normal to solve this question.
Binomial probability distribution
Probability of exactly x sucesses on n repeated trials, with p probability.
Can be approximated to a normal distribution, using the expected value and the standard deviation.
The expected value of the binomial distribution is:
[tex]E(X) = np[/tex]
The standard deviation of the binomial distribution is:
[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]
Normal probability distribution
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
When we are approximating a binomial distribution to a normal one, we have that [tex]\mu = E(X)[/tex], [tex]\sigma = \sqrt{V(X)}[/tex].
In this problem, we have that:
[tex]p = 0.2, n = 200[/tex]. So
[tex]\mu = E(X) = np = 200*0.2 = 40[/tex]
[tex]\sigma = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{200*0.2*0.8} = 5.66[/tex]
In other words, find probability that pˆ takes a value between 0.17 and 0.23.
This probability is the pvalue of Z when X = 200*0.23 = 46 subtracted by the pvalue of Z when X = 200*0.17 = 34. So
X = 46
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{46 - 40}{5.66}[/tex]
[tex]Z = 1.06[/tex]
[tex]Z = 1.06[/tex] has a pvalue of 0.8554
X = 34
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{34 - 40}{5.66}[/tex]
[tex]Z = -1.06[/tex]
[tex]Z = -1.06[/tex] has a pvalue of 0.1446
0.8554 - 0.1446 = 0.7108
71.08% probability that pˆ takes a value between 0.17 and 0.23.
The probability that [tex]\( \hat{p} \)[/tex] takes a value between 0.17 and 0.23 is approximately 0.7108.
1. Given that 20% of adult women dye or highlight their hair, the true population proportion [tex]\( p \)[/tex] is 0.20.
2. We want to find the probability that a sample proportion [tex]\( \hat{p} \)[/tex] from a simple random sample (SRS) of size 200 falls within plus or minus 3 percentage points of this true value. In other words, we want to find [tex]\( P(0.17 < \hat{p} < 0.23) \)[/tex].
3. The standard error of [tex]\( \hat{p} \)[/tex] is given by:
[tex]\[ SE = \sqrt{\frac{p(1-p)}{n}} \][/tex]
where [tex]\( p = 0.20 \)[/tex] (the true population proportion) and [tex]\( n = 200 \)[/tex] (the sample size).
4. Calculate the standard error:
[tex]\[ SE = \sqrt{\frac{0.20 \times (1-0.20)}{200}} = \sqrt{\frac{0.20 \times 0.80}{200}} \][/tex]
[tex]\[ SE = \sqrt{\frac{0.16}{200}} = \sqrt{0.0008} \approx 0.0283 \][/tex]
5. Next, we find the z-scores corresponding to the values 0.17 and 0.23 using the standard normal distribution table:
[tex]\[ z_{0.17} = \frac{0.17 - 0.20}{0.0283} \approx -1.0601 \][/tex]
[tex]\[ z_{0.23} = \frac{0.23 - 0.20}{0.0283} \approx 1.0601 \][/tex]
6. Using the z-scores, we find the corresponding probabilities from the standard normal distribution table:
[tex]\[ P(\hat{p} < 0.17) \approx P(Z < -1.0601) \approx 0.1446 \][/tex]
[tex]\[ P(\hat{p} < 0.23) \approx P(Z < 1.0601) \approx 0.8554 \][/tex]
7. Therefore, the probability that [tex]\( \hat{p} \)[/tex] takes a value between 0.17 and 0.23 is approximately:
[tex]\[ P(0.17 < \hat{p} < 0.23) = P(\hat{p} < 0.23) - P(\hat{p} < 0.17) \][/tex]
[tex]\[ \approx 0.8554 - 0.1446 = 0.7108 \][/tex]
8. Alternatively, we can find this probability directly using the cumulative distribution function (CDF) of the standard normal distribution:
[tex]\[ P(0.17 < \hat{p} < 0.23) = P(-1.0601 < Z < 1.0601) \][/tex]
[tex]\[ \approx \Phi(1.0601) - \Phi(-1.0601) \][/tex]
[tex]\[ \approx 0.8554 - 0.1446 = 0.7108 \][/tex]
9. Therefore, the probability that[tex]\( \hat{p} \)[/tex] takes a value between 0.17 and 0.23 is approximately 0.7108.
A humanities professor assigns letter grades on a test according to the following scheme. A: Top 7% of scores B: Scores below the top 7% and above the bottom 56% C: Scores below the top 44% and above the bottom 19% D: Scores below the top 81% and above the bottom 6% F: Bottom 6% of scores Scores on the test are normally distributed with a mean of 72.1 and a standard deviation of 9.5. Find the minimum score required for an A grade. Round your answer to the nearest whole number, if necessary.
Answer:
The minimum score required for an A grade is 86.
Step-by-step explanation:
Problems of normally distributed samples can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
In this problem, we have that:
[tex]\mu = 72.1, \sigma = 9.5[/tex]
Find the minimum score required for an A grade.
Top 7%, which is the value of X when Z has a pvalue of 1-0.07 = 0.93. So it is X when Z = 1.475. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]1.475 = \frac{X - 72.1}{9.5}[/tex]
[tex]X - 72.1 = 1.475*9.5[/tex]
[tex]X = 86[/tex]
The minimum score required for an A grade is 86.
The equation r(t)= (3t+9)i+(sqrt(2)t)j+(t^2)k is the position of a particle in space at time t. Find the angle between the velocity and acceleration vectors at time t=0
What is the angle? ______radians
Answer:
[tex]\theta= \frac{\pi}{2} +\pi \cdot i[/tex], for all [tex]i = \mathbb{Z} \cup\{0\}[/tex]
Step-by-step explanation:
The velocity vector is found by deriving the position vector depending on the time:
[tex]\dot r(t)= v (t) = 3 \cdot i +\sqrt{2} \cdot j + 2\cdot t \cdot k[/tex]
In turn, acceleration vector is found by deriving the velocity vector depending on time:
[tex]\ddot r(t) = \dot v(t) = a(t) = 2 \cdot k[/tex]
Velocity and acceleration vectors at [tex]t = 0[/tex] are:
[tex]v(0) = 3\cdot i + \sqrt{2} \cdot j\\a(0) = 2 \cdot k\\[/tex]
Norms of both vectors are, respectively:
[tex]||v(0)||\approx 3.317\\||a(0)|| \approx 2[/tex]
The angle between both vectors is determined by using the following characteristic of a Dot Product:
[tex]\theta = \cos^{-1}(\frac{v(0) \bullet a(0)}{||v(0)||\cdot ||a(0)||})[/tex]
Given that cosine has a periodicity of [tex]\pi[/tex]. There is a family of solutions with the form:
[tex]\theta= \frac{\pi}{2} +\pi \cdot i[/tex], for all [tex]i = \mathbb{Z} \cup\{0\}[/tex]
Final answer:
π/2 radians, indicating the vectors are perpendicular at that instant.
Explanation:
The angle between the velocity and acceleration vectors at a given time can be found by first determining the velocity (νt) and acceleration (ν2t) vectors as the first and second derivatives of the position vector r(t). At time t=0, these derivatives can be calculated and then used to find the angle through the dot product and magnitude of these vectors.
For the given position vector
r(t) = (3t+9)i + (√2t)j + (t2)k,
the velocity vector v(t) is obtained by differentiating each component of r(t) with respect to time t, which gives
v(t) = (3)i + (√2)j + (2t)k.
Similarly, acceleration a(t) is the derivative of velocity v(t), which results in a(t) = (0)i + (0)j + (2)k.
At t=0, v(0) = (3)i + (√2)j and a(0) = (2)k. The angle θ between v(0) and a(0) is given by the cosine of the angle between the two vectors, which is calculated using the dot product formula:
θ = cos-1((v ⋅ a) / (|v||a|)).
Here, (v ⋅ a) is the dot product of v(0) and a(0), and |v| and |a| are the magnitudes of v(0) and a(0), respectively.
Since v(0) and a(0) are perpendicular at t=0, their dot product is 0, and the magnitudes of v(0) and a(0) do not affect the angle. Therefore, the angle θ is simply cos-1(0), which is π/2 radians, indicating the vectors are perpendicular.
Jack inherited a perpetuity-due, with annual payments of 15,000. He immediately exchanged the perpetuity for a 25-year annuity-due having the same present value. The annuity-due has annual payments of X. All the present values are based on an annual effective interest rate of 10% for the first 10 years and 8% thereafter. Calculate X.
Jack inherited a perpetuity-due, which he exchanged for a 25-year annuity-due. The present value of both are equal and based on different interest rates over the 25 years. The annual payment X of the annuity can be calculated using the formula for the present value of an annuity and the present value of the inherited perpetuity-due.
Explanation:This problem is about financial mathematics, specifically involving perpetuities and annuities. In a perpetuity-due, the payments are made at the beginning of each period indefinitely. An annuity-due is similar, but the payments only last a specified number of years.
The present value (PV) of a perpetuity-due with annual payments P and annual interest rate r is calculated by PV = P / r. Given that P is $15,000 and r is 10%, the present value of the perpetuity-due that Jack inherits is $150,000.
When Jack swaps this for a 25-year annuity-due, with the first 10 years at 10% interest and the next 15 years at 8% interest, we calculate the annual payment X using the formula for the present value of an annuity. This formula involves dividing the total present value by the sum of the present value factors for each year at the respective interest rates.
Consequently, X can be calculated as follows: X = (PV of perpetuity-due) / ∑ (discount factors for each year). A detailed calculation will give the exact value of X.
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You estimate that you can save $3,800 by selling your home yourself rather than using a real estate agent. What would be the future value of that amount if invested for five years at 5 percent? Use Exhibit_1-A. (Round time value factor to 3 decimal places and final answer to 2 decimal places.)
Answer:
$4848.8
Step-by-step explanation:
(1 + 0.05)⁵ = 1.276
FV = 3800 × 1.276
= 4848.8
Final answer:
The future value of $3,800 invested for five years at a 5% interest rate, using the formula [tex]FV = PV[/tex] x [tex](1 + r)^t[/tex], is approximately $4,849.07.
Explanation:
You are asking how to calculate the future value of an amount of money when invested at a certain interest rate over a set period of time. Specifically, you want to know the future value of $3,800 invested for five years at a 5% interest rate.
To calculate the future value (FV) of money we use the formula:
[tex]FV = PV X (1 + r)^t[/tex]
Where:
PV is the present value or initial amount ($3,800)
r is the annual interest rate (5%, or as a decimal, 0.05)
t is the time in years the money is invested (5 years)
Using the formula, we get:
FV = $3,800 x (1 + 0.05)5
FV = $3,800 x (1.276281)
FV = $3,800 x 1.276 (rounded to three decimal places)
FV = $4,849.07 (rounded to two decimal places)
Hence, the future value of $3,800 invested for five years at a 5% interest rate would be approximately $4,849.07.
Speeding on the I-5, Part I. The distribution of passenger vehicle speeds traveling on the Interstate 5 Freeway (I-5) in California is nearly normal with a mean of 72.6 miles/hour and a standard deviation of 4.78 miles/hour.47 (a) What percent of passenger vehicles travel slower than 80 miles/hour?(b) What percent of passenger vehicles travel between 60 and 80 miles/hour?(c) How fast to do the fastest 5% of passenger vehicles travel?(d) The speed limit on this stretch of the I-5 is 70 miles/hour. Approximate what percentage of the passenger vehicles travel above the speed limit on this stretch of the I-5.
Answer:
a) [tex]P(X<80)=P(\frac{X-\mu}{\sigma}<\frac{80-\mu}{\sigma})=P(Z<\frac{80-72.6}{4.78})=P(z<1.548)[/tex]
And we can find this probability using the normal standard distirbution or excel and we got:
[tex]P(z<1.548)=0.939[/tex]
And that correspond to 93.9 %
b) [tex]P(60<X<80)=P(\frac{60-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{80-\mu}{\sigma})=P(\frac{60-72.6}{4.78}<Z<\frac{80-72.6}{4.78})=P(-2.636<z<1.548)[/tex]
And we can find this probability with this difference:
[tex]P(-2.636<z<1.548)=P(z<1.548)-P(z<-2.636)[/tex]
And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.
[tex]P(-2.636<z<1.548)=P(z<1.548)-P(z<-2.636)=0.939-0.0042=0.935 [/tex]
So we have approximately 93.5%
c) [tex]z=1.64<\frac{a-72.6}{4.78}[/tex]
And if we solve for a we got
[tex]a=72.6 +1.64*4.78=80.439[/tex]
So the value of velocity that separates the bottom 95% of data from the top 5% is 80.439.
d) [tex]P(X>70)=P(\frac{X-\mu}{\sigma}>\frac{70-\mu}{\sigma})=P(Z>\frac{70-72.6}{4.78})=P(z>-0.544)[/tex]
And we can find this probability using the complement rule, normal standard distirbution or excel and we got:
[tex]P(z>-0.544)=1-P(z<-0.544) = 1-0.293=0.707 [/tex]
And that correspond to 70.7 %
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".
Part a
Let X the random variable that represent the vehicles speeds of a population, and for this case we know the distribution for X is given by:
[tex]X \sim N(72.6,4.78)[/tex]
Where [tex]\mu=72.6[/tex] and [tex]\sigma=4.78[/tex]
We are interested on this probability
[tex]P(X<80)[/tex]
And 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<80)=P(\frac{X-\mu}{\sigma}<\frac{80-\mu}{\sigma})=P(Z<\frac{80-72.6}{4.78})=P(z<1.548)[/tex]
And we can find this probability using the normal standard distirbution or excel and we got:
[tex]P(z<1.548)=0.939[/tex]
And that correspond to 93.9 %
Part b
We want this probability
[tex]P(60<X<80)=P(\frac{60-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{80-\mu}{\sigma})=P(\frac{60-72.6}{4.78}<Z<\frac{80-72.6}{4.78})=P(-2.636<z<1.548)[/tex]
And we can find this probability with this difference:
[tex]P(-2.636<z<1.548)=P(z<1.548)-P(z<-2.636)[/tex]
And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.
[tex]P(-2.636<z<1.548)=P(z<1.548)-P(z<-2.636)=0.939-0.0042=0.935 [/tex]
So we have approximately 93.5%
Part c
For this part we want to find a value a, such that we satisfy this condition:
[tex]P(X>a)=0.05[/tex] (a)
[tex]P(X<a)=0.95[/tex] (b)
Both conditions are equivalent on this case. We can use the z score again in order to find the value a.
As we can see on the figure attached the z value that satisfy the condition with 0.95 of the area on the left and 0.05 of the area on the right it's z=1.64. On this case P(Z<1.64)=0.95 and P(z>1.64)=0.05
If we use condition (b) from previous we have this:
[tex]P(X<a)=P(\frac{X-\mu}{\sigma}<\frac{a-\mu}{\sigma})=0.95[/tex]
[tex]P(z<\frac{a-\mu}{\sigma})=0.95[/tex]
But we know which value of z satisfy the previous equation so then we can do this:
[tex]z=1.64<\frac{a-72.6}{4.78}[/tex]
And if we solve for a we got
[tex]a=72.6 +1.64*4.78=80.439[/tex]
So the value of velocity that separates the bottom 95% of data from the top 5% is 80.439.
Part d
[tex]P(X>70)[/tex]
And 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>70)=P(\frac{X-\mu}{\sigma}>\frac{70-\mu}{\sigma})=P(Z>\frac{70-72.6}{4.78})=P(z>-0.544)[/tex]
And we can find this probability using the complement rule, normal standard distirbution or excel and we got:
[tex]P(z>-0.544)=1-P(z<-0.544) = 1-0.293=0.707 [/tex]
And that correspond to 70.7 %
a) What percentage of the area under the normal curve lies to the left of μ? % (b) What percentage of the area under the normal curve lies between μ − σ and μ + σ? % (c) What percentage of the area under the normal curve lies between μ − 3σ and μ + 3σ? %
Answer:
a) 50%
b) 68%
c) 99%
Step-by-step explanation:
for a standard normal curve ,
a) since the standard normal curve is symmetric and centred around μ , 50% of the curve lies at the left of μ and 50% lies to the right
b) according to the 68-95-99 rule, 68% of the standard normal curve lies from μ − σ and μ + σ
c) from the same rule , 99% of the standard normal curve lies from μ − 3σ and μ + 3σ
Answer:
a) 50%
b) 68%
c) 99.7%
Step-by-step explanation:
The Empirical Rule states that, for a normally distributed random variable:
68% of the measures are within 1 standard deviation of the mean.
95% of the measures are within 2 standard deviation of the mean.
99.7% of the measures are within 3 standard deviations of the mean.
The normal distribution is also symmetric, which means that 50% of the measures are below the mean and 50% are above.
In this problem, we have that:
Mean μ
Standard deviation σ
Area under the normal curve = percentage
a) What percentage of the area under the normal curve lies to the left of μ?
Normal distribution is symmetric, so the answer is 50%.
(b) What percentage of the area under the normal curve lies between μ − σ and μ + σ?
Within 1 standard deviation of the mean, so 68%.
(c) What percentage of the area under the normal curve lies between μ − 3σ and μ + 3σ?
Within 3 standard deviation of the mean, so 99.7%.
Birth weights of full-term babies in a certain area are normally distributed with mean 7.13 pounds and standard deviation 1.29 pounds. A newborn weighing 5.5 pounds or less is a low-weight baby. What is the probability that a randomly selected newborn is low-weight? Use the appropriate applet. Enter a number in decimal form, e.g. 0.68, not 68 or 68%.
Answer: probability that a randomly selected newborn is low-weight is 0.1038
Step-by-step explanation:
Since Birth weights of full-term babies in a certain area are normally distributed m, we would apply the formula for normal distribution which is expressed as
z = (x - µ)/σ
Where
x = birth weights of full-term babies.
µ = mean weight
σ = standard deviation
From the information given,
µ = 7.13 pounds
σ = 1.29 pounds
The probability that a randomly selected newborn is low-weight is expressed as
P(x ≤ 5.5)
For x = 5.5
z = (5.5 - 7.13)/1.29 = - 1.26
Looking at the normal distribution table, the probability corresponding to the z score is 0.1038
P(x ≤ 5.5) = 0.1038
Multiply.
(x - 6)(x - 4)
Answer:
[tex]x^{2} -10x+24[/tex]
Step-by-step explanation:
If a procedure meets all of the conditions of a binomial distribution except the number of trials is not fixed, then the geometric distribution can be used. The probability of getting the first success on the xth trial is given by P(x)equalsp (1 minus p )Superscript x minus 1, where p is the probability of success on any one trial. Subjects are randomly selected for a health survey. The probability that someone is a universal donor (with group O and type Rh negative blood) is 0.07. Find the probability that the first subject to be a universal blood donor is the seventh person selected. The probability is nothing. (Round to four decimal places as needed.)
Answer:
Given p = 0.07 as the probability that someone is a universal donor
In case of Geometric Distribution, Probability of getting the first success on nth trial is given by
[tex]P (X=n) = p (1-p) ^ {n-1}[/tex]
where p is the probability of success on any one trial and (1-p) shows the probability of failure.
So the probability of the first subject to be a universal blood donor will be the seventh person is
[tex]P (X=7) = 0.07 (1-0.07) ^ {7-1} = 0.07 (0.93) ^ 6 = 0.07*0.647 = 0.0453[/tex]
So the final probability is 0.0453
Find the area of a regular octagon with side length 8 cm. Round your answer to the nearest square centimeter.
A. 53 cm
B. 106 cm
C. 155 cm
D. 309 cm
Answer:
A≈309.02cm² Step by step
A student takes an exam containing 1414 multiple choice questions. The probability of choosing a correct answer by knowledgeable guessing is 0.30.3. At least 99 correct answers are required to pass. If the student makes knowledgeable guesses, what is the probability that he will pass? Round your answer to four decimal places.
Answer:
0.0082 = 0.82% probability that he will pass
Step-by-step explanation:
For each question, there are only two possible outcomes. Either the students guesses the correct answer, or he guesses the wrong answer. The probability of guessing the correct answer for a question is independent of other questions. So we use the binomial probability distribution to solve this question.
Binomial probability distribution
The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
In which [tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by the following formula.
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
And p is the probability of X happening.
In this problem we have that:
[tex]n = 14, p = 0.3[/tex].
If the student makes knowledgeable guesses, what is the probability that he will pass?
He needs to guess at least 9 answers correctly. So
[tex]P(X \geq 9) = P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14)[/tex]
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 9) = C_{14,9}.(0.3)^{9}.(0.7)^{5} = 0.0066[/tex]
[tex]P(X = 10) = C_{14,10}.(0.3)^{10}.(0.7)^{4} = 0.0014[/tex]
[tex]P(X = 11) = C_{14,11}.(0.3)^{11}.(0.7)^{3} = 0.0002[/tex]
[tex]P(X = 12) = C_{14,12}.(0.3)^{12}.(0.7)^{2} = 0.000024[/tex]
[tex]P(X = 13) = C_{14,13}.(0.3)^{13}.(0.7)^{1} = 0.000002[/tex]
[tex]P(X = 14) = C_{14,14}.(0.3)^{14}.(0.7)^{0} \cong 0 [/tex]
[tex]P(X \geq 9) = P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) = 0.0066 + 0.0014 + 0.0002 + 0.000024 + 0.000002 = 0.0082[/tex]
0.0082 = 0.82% probability that he will pass
Which polynomials are in standard form?
Choose all answers that apply:
5-2x
x^4-8x^2 -16
5x^3 + 4x^4 — 3x + 1
None of the above
Answer: The second option, x^4 - 8x^2 - 16.
Step-by-step explanation:
Polynomials in standard form start with the highest degree, from greatest to least exponent. After all terms with exponents are in order, alphabetical variables are next. In this case there's only x. Last are constant terms, which are by itself, with no variable next to it/an exponent to the right of it.
x^4 - 8x^2 - 16 is in standard form because it follows the criteria above. 4 is the highest degree since it's the highest exponent in the polynomial expression, which is why it starts off with x^4. Other terms with lesser exponents are next. In this case, it's 8x^2 with the less exponent of 2. Finally, it ends with your constant term, -16.
The standard form of the polynomial is,
⇒ x⁴ - 8x² - 16
Given that,
All the polynomials are,
⇒ 5 - 2x
⇒ x⁴ - 8x² - 16
⇒ 5x³ + 4x⁴ - 3x + 1
Since we know that,
In standard form, a polynomial is arranged in descending order of the exponents of its terms.
This means that the term with the highest degree is listed first, followed by the terms with lower degrees.
Hence, Based on this definition, the polynomial in standard form among the options is,
⇒ x⁴ - 8x² - 16
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There are 98 balls in a box. 25 of them are red, 19 of them are green, 30 of them are purple,and 24 of them are blue. Suppose Hao draws 22 balls from the box with replacement (hedraws the ball, records its color, and then puts it back into the box). Find the probabilitythat he draws 2 red balls, 5 green balls, 10 purple balls, and 5 blue balls.
Answer:
Since the balls are drawn with replacement, it means the individual probability remain constant,
I have solved this problem on paper (Figures Attached).
Thanks.
Answer:
Since this problem related to the replacement problem thus
Pr(2 red)=(25/98)(25/98)=0.0651
Pr(5 green)=(19/98)(19/98)(19/98)(19/98)(19/98)
Pr(5 green)=2.73*10^-4
Pr(10 purple)=((30/98))^10=7.22*10^-6
Pr(5 blue)=((24/98))^5=8.8*10^-4
If one wanted to find the probability of 10 customer arrivals in an hour at a service station, one would generally use the _____. a. hypergeometric probability distribution b. Poisson probability distribution c. exponential probability distribution d. binomial probability distribution
Answer:
b. Poisson probability distribution
Step-by-step explanation:
The Poisson distribution is the discrete probability distribution of the number of events occurring in a given time period. The poisson distribution is also use when few large demand is expected.
In this question; poisson distribution is use to find the probability of 10 customers arrivals in an hour at a service station.
To find the probability of 10 customer arrivals in an hour at a service station, one would use the Poisson probability distribution, which calculates the probability of a certain number of events happening in a set period of time.
Explanation:The correct answer is b. Poisson probability distribution. This type of distribution is often used to calculate the probability of a certain number of events happening in a set period of time. In this case, the 'events' are the arrivals of customers at a service center within an hour.
Here's a very simplified version of the steps to calculate a Poisson probability:
Step 1: Identify the average rate (λ) - this is the average number of times the event is happening per unit of measure (in your case, customer arrivals per hour).
Step 2: Use the formula for Poisson probability, which is P(x; λ) = e^-λ * λ^x / x! Where 'x' is the actual number of successes that result from the experiment, 'e' is approx 2.71828 and '!' denotes a factorial.
So, if we knew the average rate of customer arrivals, we could easily apply it to this formula to get the probability of 10 customer arrivals in an hour.
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The table in the shape of a circle has a diameter of 6 feet. How much fabric is needed to make a table cloth if it hangs 1 foot off the table all the way around?jdjdkndjcjjfnnfm
Answer: area of fabric needed is 50.24 ft²
Step-by-step explanation:
The table in the shape of a circle has a diameter of 6 feet. This means that the diameter of the fabric that would just fit the table is 6 feet. Therefore, the diameter of the fabric needed to make a table cloth if it hangs 1 foot off the table would be 6 + 1 + 1 = 8 feet
The formula for determining the area of a circle is expressed as
Area = πr²
Where
r represents radius of the circle.
π is a constant whose value is 3.14
Radius = diameter/2. Therefore
r = 8/2 = 4
Area of fabric = 3.14 × 4²
= 50.24 ft²
Final answer:
To make a tablecloth for a circular table with a diameter of 6 feet, and an overhang of 1 foot, the student would need approximately 50.27 square feet of fabric.
Explanation:
The student is asking about finding the amount of fabric needed to create a tablecloth for a circular table with specific dimensions. Given that the table has a diameter of 6 feet, and the tablecloth needs to hang 1 foot off the table all the way around, we need to calculate the diameter of the fabric required.
To solve this, we need to add the overhang to the diameter of the table, considering that the overhang occurs on both sides:
Diameter of the table: 6 feetOverhang on one side: 1 footOverhang on the other side: 1 footTotal diameter needed: 6 feet + 1 foot + 1 foot = 8 feetNow, to find the area of fabric needed, we apply the formula for the area of a circle which is π × radius². First, we find the radius by halving the diameter:
Radius of the fabric: 8 feet / 2 = 4 feetThen, we calculate the area:
Area of fabric: π × (4 feet)² = π × 16 feet²Finally, we can approximate π as 3.1416 to get an approximate area of:
Area of fabric ≈ 3.1416 × 16 feet² ≈ 50.2656 square feetTherefore, the student would need approximately 50.27 square feet of fabric to make the tablecloth.
Find the length of arc. DB. Leave your answer in terms of pi
Answer: Length of arc DB is 14π feet.
Step-by-step explanation:
The sum of the angles on a straight line is 180 degrees. This means that
m∠DAB + m∠CAB = 180
m∠DAB + 40 = 180
m∠DAB = 180 - 40
m∠DAB = 140°
The formula for determining the length of an arc is expressed as
Length of arc = θ/360 × 2πr
Where
θ represents the central angle.
r represents the radius of the circle.
π is a constant whose value is 3.14
From the information given,
r = 18 feet
θ = 140°
Therefore,
Length of arc DB = 140/360 × 2 × π × 18
Length of arc DB = 14π feet
A retail store stocks windbreaker jackets in small, medium, large, and extra large and all are available in blue or red. What are the combined choices and how many combined choices are there?
Answer:
Choices={SB,SR,MB,MR,LS,LM,XLB,XLR}
8 combined choices
Step-by-step explanation:
Combinations
We'll define two sets of options for the windbreaker jackets, one for the sizes and another for the colors. Being S=small, M=medium, L=large, and XL=extra large, then
Z={S,M,L,XL}
is the set of possible sizes for the windbreaker jackets. Now, being B=blue and R=red, the set of colors is
C={B,R}
The combined choices are found by the cartesian product of ZxC:
Choices={SB,SR,MB,MR,LS,LM,XLB,XLR}
Where MB, for example, is Medium-Blue
That is the sample space for all the possible combinations, there are 8 in total
According to the Counting Principle in Mathematics, there are 8 different combinations of sizes and colors for the windbreaker jackets available at the retail store.
Explanation:The retail store offers windbreaker jackets in four different sizes: small, medium, large, and extra large. Each of these sizes is available in two colors: blue and red. Therefore, using a concept in mathematics known as the Counting Principle, we can ascertain the number of combinations. The Counting Principle states that if one event can occur in m ways and another can occur in n ways, then the number of ways that both events can occur is m*n.
So in this scenario, we have 4 sizes (small, medium, large, extra-large) and 2 colors (blue, red). Applying the Counting Principle, there are a total of 4 * 2 = 8 different combinations of jackets that can be purchased from the store.
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based on the graph below, what is yhe total number of solutions to the equation f(x)= g(x)?
1
2
3
4
Answer:
Based on the graph below, what is the total number of solutions to the equation f(x)= g(x) will be 3.
Step-by-step explanation:
The intersection points of both graphs would be the total number of solutions to the equation f(x)= g(x).
From the given diagram, it is clear that both the graphs intersect at three locations points or intersection points. The approximations locations of The intersection points of both graphs are
(1.5, 4.125)(-1, 0), and (-2.5, -3.5)Therefore, based on the graph below, what is the total number of solutions to the equation f(x)= g(x) will be 3.
In a certain very large city, the Department of Transportation (D.O.T.) has organized a complex system of bus transportation. In an advertising campaign, citizens are encouraged to use the new "GO-D.O.T!" system and head for the nearest bus stop to be transported to and from the central city. Suppose that at one of the bus stops the amount of time (in minutes) that a commuter must wait for a bus is a uniformly distributed random variable, T.
The possible values of T run from 0 minutes to 20 minutes.
(a) What is the probability that a randomly selected commuter will spend more than 7 minutes waiting for GO-D.O.T?
(b) What is the standard deviation?
(c) What is the probability that a randomly selected commuter will spend longer than 10 minutes but no more than 17 minutes waiting for the GO-D.O.T?
(d) What is the average waiting time?
Answer:
a) Probability that a randomly selected commuter will spend more than 7 minutes waiting for GO-D.O.T = P(7 < x ≤ 20) = 0.65
b) Standard deviation of the uniform distribution = 5.77 minutes
c) Probability that a randomly selected commuter will spend longer than 10 minutes but no more than 17 minutes waiting for the GO-D.O.T = P(10 < x < 17) = 0.35
d) average waiting time for the uniform distribution = 10 minutes.
Step-by-step explanation:
This is a uniform distribution problem with lower limit of 0 minute and upper limit of 20 minutes.
a = 0, b = 20
Probability = f(x) = [1/(b-a)] ∫ dx (with the definite integral evaluated between the two intervals whose probability is required.
a) Probability that a randomly selected commuter will spend more than 7 minutes waiting for GO-D.O.T
P(7 < x ≤ 20) = f(x) = [1/(b-a)] ∫²⁰₇ dx
P(7 < x ≤ 20) = (20-7)/(20-0) = (13/20) = 0.65
b) Standard deviation of the uniform distribution
Standard deviation of a uniform distribution is given as
σ = √[(b-a)²/12]
σ = √[(20-0)²/12]
σ = √[20²/12]
σ = 5.77 minutes
c) Probability that a randomly selected commuter will spend longer than 10 minutes but no more than 17 minutes waiting for the GO-D.O.T = P(10 < x < 17)
P(10 < x < 17) = (17-10)/(20-0)
P(10 < x < 17) = (7/20) = 0.35
d) The average waiting time.
The average of a uniform distribution = (b+a)/2
Average waiting time = (20+0)/2
Average waiting time = 10 minutes
Hope this Helps!!!