Answer:
[tex]\displaystyle 5040[/tex]
Step-by-step explanation:
You have ten digits, but can only choose from four each time. Therefore, you will use the formula pertaining to permutations [order matters]. Here is how it is done:
[tex]\displaystyle \frac{n!}{[-k + n]!} = {}_nP_k \\ \\ \frac{10!}{[-4 + 10]!} = \frac{10!}{6!} \Longrightarrow \frac{[2][3][4][5][6][7][8][9][10]}{[2][3][4][5][6]} \\ \\ \\ \boxed{5040} = [7][8][9][10][/tex]
So, there will be five thousand forty different passwords, or in this case, combinations.
I am joyous to assist you at any time.
What figure is created from the coordinate points (3, 2), (3, 6), and (5, 2)? square rectangle triangle parallelogram
Answer:
it's a triangle
Step-by-step explanation:
15:60 simlplfed a. 3:4 b 1:4 c 2:5
Answer: b 1:4
Step-by-step explanation:
Answer:
The given expression 15:60 can be simplified by dividing both the numerator and denominator by their greatest common factor (GCF), which is 15 in this case. So, 15 divided by 15 is 1 and 60 divided by 15 is 4. Therefore, the simplified form of 15:60 is 1:4.
So, the answer is (b) 1:4.
if x and y are independent and the joint probability p(x = 1,y = 2) = 0.06, what is p(y = 4)?
Thus, the joint probability of x=1 and y=2 is used to calculate the probability of y=4 is 0.06.
To calculate the probability of y = 4, we need to use the fact that x and y are independent. This means that the probability of y taking a certain value is not affected by the value of x.
Therefore, we can use the marginal probability of y, which is the sum of the joint probabilities of all possible values of y.
Let's start by finding the marginal probability of y. We can do this by summing the joint probabilities of y taking all possible values:
p(y=1) = p(x=0, y=1) + p(x=1, y=1) + p(x=2, y=1) = 0 + 0.12 + 0 = 0.12
p(y=2) = p(x=0, y=2) + p(x=1, y=2) + p(x=2, y=2) = 0.06 + 0 + 0.18 = 0.24
p(y=3) = p(x=0, y=3) + p(x=1, y=3) + p(x=2, y=3) = 0 + 0.12 + 0.36 = 0.48
p(y=4) = p(x=0, y=4) + p(x=1, y=4) + p(x=2, y=4) = 0 + p(x=1, y=2) + 0 = 0.06
Therefore, the probability of y=4 is 0.06.
In summary, the joint probability of x=1 and y=2 is used to calculate the probability of y=4, by using the fact that x and y are independent and finding the marginal probability of y.
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Prove that for all integers a,b and c, if a|bc, then a|b or a|c.
Thus, we have proved that for all integers a, b, and c, if a|bc, then a|b or a|c.
To prove that for all integers a, b, and c, if a|bc, then a|b or a|c, we need to use the definition of divisibility.
Assume that a|bc. This means that there exists an integer k such that bc = ak.
We can consider two cases:
Case 1: a and b are coprime.
In this case, a does not share any factors with b. Therefore, a cannot divide b. However, since a|bc and b and c share no factors, a must divide c. Hence, a|c.
Case 2: a and b have a common factor.
In this case, we can write a = dx and b = dy, where d is the greatest common divisor of a and b. Therefore, bc = dxy*c = ak = dxy*k.
Dividing both sides by dxy, we get c/k = a/dy.
Since a and d share no factors, d|c/k. Therefore, there exists an integer m such that c/k = dm. This means that c = dkm.
Since a = dx, we have a|dxk. Since a|bc = dxy*k, we have d|a and d|k. Therefore, we can write k = dh and a = dg, where h and g are integers.
Substituting these expressions into c = dkm, we get c = dg*dh*m. Since d|c and d|a, we have d|b and a|b.
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A painter has up to 16
hours to paint a living room and two bedrooms. He spends 7
hours painting the living room. Write and solve an inequality to find how much time x
he can spend on each bedroom if he splits his time equally.
The painter can spend up to 4.5 hours painting each bedroom if he wants to finish painting the living room and both bedrooms within 16 hours.
If the painter has 16 hours in total, and he already spent 7 hours painting the living room, then he has 16 - 7 = 9 hours left to paint the two bedrooms.
If he wants to split his time equally between the two bedrooms, then he can spend x hours on each bedroom. Therefore, the total time spent painting the bedrooms would be 2x.
To find the maximum amount of time he can spend on each bedroom, we need to solve the following inequality:
2x ≤ 9
Dividing both sides by 2, we get:
x ≤ 4.5
Therefore, the painter can spend up to 4.5 hours painting each bedroom if he wants to finish painting the living room and both bedrooms within 16 hours.
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Which of the following functions has the greatest y-intercept? (1 point) f(x) x y −3 9 −2 4 −1 1 0 0 1 1 2 4 g(x) = 5 cos(3x) − 5 f(x) g(x) Not enough information f(x) and g(x) have the same y-intercept
Answer : they have the same y-intercept
Both function have same y - intercept.
The two given functions are;
x: -3 -2 -1 0 1 2
f(x): 9 4 -1 0 1 4
And, g(x) = 5 cos (3x) - 5
Since, We know that;
The y-intercept occurs when x=0.
Therefore,
⇒ f(x) has a y-intercept of 0
(from given tabular data).
And, g(x) has a y-intercept of at x = 0
g(0) = 5 cos (0) - 5
= 5 - 5
= 0
Hence, Both function have same y - intercept.
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Which is the standard equation of the hyperbola centered at the origin, with a vertical transverse axis and values of a = 9 and b = 4?
The standard equation of a hyperbola with a vertical transverse axis, centered at the origin, and values of a = 9 and b = 4 is y^2/81 - x^2/16 = 1.
The standard equation of a hyperbola centered at the origin with a vertical transverse axis is given by (y^2/a^2) - (x^2/b^2) = 1. In this case, we are given that a = 9 and b = 4, so substituting these values into the equation, we get:
(y^2/81) - (x^2/16) = 1
This is the standard equation of the hyperbola in question. It tells us that the center of the hyperbola is at the origin (0,0), the transverse axis is vertical (parallel to the y-axis), and the distance from the center to the vertices is 9 units (which is the value of a).
The distance from the center to the foci is given by c = sqrt (a^2 + b^2), which in this case is sqrt (81 + 16) = sqrt (97). The asymptotes of the hyperbola are the lines y = (a/b) x and y = -(a/b) x, which in this case are y = (3/4) x and y = -(3/4) x.
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a quality control expert at life batteries wants to test their new batteries. the design engineer claims they have a standard deviation of 62 minutes with a mean life of 606 minutes. if the claim is true, in a sample of 99 batteries, what is the probability that the mean battery life would differ from the population mean by greater than 18.4 minutes? round your answer to four decimal places.
The probability that the mean battery life would differ from the population mean by greater than 18.4 minutes is 0.0148. The formula for the standard error of the mean is SE = σ/√n
To solve this problem, we need to use the Central Limit Theorem (CLT) which states that the distribution of sample means will be approximately normal with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.
The formula for the standard error of the mean is:
SE = σ/√n
Where SE is the standard error, σ is the population standard deviation, and n is the sample size.
In this case, we are given that the population standard deviation (σ) is 62 minutes, the population mean is 606 minutes, and the sample size (n) is 99 batteries. Therefore, the standard error of the mean is:
SE = 62/√99 = 6.231
Next, we need to find the z-score associated with the difference between the sample mean and the population mean of 18.4 minutes. The formula for the z-score is:
z = (x - μ) / SE
Where x is the sample mean, μ is the population mean, and SE is the standard error.
Substituting the values we have:
z = (606 + 18.4 - 606) / 6.231 = 2.96
Using a standard normal distribution table, we find that the probability of getting a z-score greater than 2.96 is 0.0148. Therefore, the probability that the mean battery life would differ from the population mean by greater than 18.4 minutes in a sample of 99 batteries is 0.0148, or about 1.48%.
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Type the correct answer in each box. Use numerals instead of words. If necessary, use / for the fraction bar(s).
Mr. Schwartz builds toy cars. He begins the week with a supply of 85 wheels, and uses 4 wheels for each car he builds. Mr. Schwartz plans to order more wheels once he has fewer than 40 wheels left.
The inequality that can be used to find the number of cars, x, Mr. Schwartz builds before he places an order for more wheels is − x < 40.
Mr.Schwartz will need to order more wheels after building cars.
Mr. Schwartz will need to order more wheels after building 12 cars.
The number of cars x Mr. Schwartz builds before he places an order for more wheels can use the following steps:
Determine how many wheels are used per car:
4 wheels/car
Determine the number of wheels available at the start of the week:
85 wheels
Determine the minimum number of wheels needed to be available before placing an order:
40 wheels
Set up an inequality to represent the situation using x to represent the number of cars built before an order is placed:
Number of wheels used = 4x
Number of wheels remaining = 85 - 4x
Order is placed when number of wheels remaining is less than 40:
85 - 4x < 40
Solve for x by isolating the variable:
85 - 4x < 40
-4x < -45
x > 11.25
Since x represents the number of cars built can't have a fractional value for x.
The nearest integer to get the minimum number of cars that need to be built before an order is placed:
x > 12
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For a one-tailed dependent samples t-Test, what specific critical value do we need to overcome at the p < .05 level for a study with 30 participants?1.7011.6991.6981.69None of the above
For a one-tailed dependent samples t-Test with 30 participants at the p < .05 level, we need to overcome a specific critical value of 1.697.
This value can be obtained from a t-table or calculated using statistical software. It is important to note that the critical value may vary depending on the specific alpha level chosen and the study's degrees of freedom (df). However, for a one-tailed dependent samples t-Test with 30 participants at the p < .05 level, the critical value of 1.697 is appropriate. This critical value represents the minimum t-value that must be obtained to reject the null hypothesis and conclude that there is a significant difference between the two dependent groups being compared.
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pls help asap with homework!!!!
The exact value of cos 30 degrees is √3/2.
The exact value of sin 45 degrees is 1/√2 or √2/2.
The exact value of tan 30 degrees is 1/√3 or √3/3.
We have,
The exact value of cos 30 can be found using the unit circle, which is a circle with a radius of 1 centered at the origin of a coordinate plane.
On the unit circle, the angle of 30 degrees is measured counterclockwise from the positive x-axis and intersects the circle at a point where the x-coordinate is √3/2 and the y-coordinate is 1/2.
Therefore, Cos 30 is equal to √3/2.
The exact value of sin 45 can also be found using the unit circle. In this case, the angle of 45 degrees intersects the unit circle at a point where both the x-coordinate and y-coordinate are 1/√2.
Therefore, Sin 45 is equal to 1/√2, which can be simplified to (√2)/2.
The exact value of tan 30 can be found using the relationship between tangent and sine/cosine: tan x = sin x / cos x.
Therefore, tan 30 = sin 30 / cos 30.
From the unit circle, we know that sin 30 is equal to 1/2 and cos 30 is equal to √3/2. Substituting these values into the formula, we get,
tan 30 = (1/2) / (√3/2) = 1/√3 = √3/3.
Thus,
The exact value of cos 30 degrees is √3/2.
The exact value of sin 45 degrees is 1/√2 or √2/2.
The exact value of tan 30 degrees is 1/√3 or √3/3.
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two joggers run 8 miles north and them 5 miles west . What is the shortest distace, to the nearst tenth of a mile, they must travel to return to their starting point?
The shortest distance the joggers must travel to return to their starting point is approximately 9.43 miles to the nearest tenth of a mile.
We have,
To find the shortest distance the joggers must travel to return to their starting point, we can use the Pythagorean theorem.
The joggers ran 8 miles north and 5 miles west, forming a right triangle. The distance they need to travel to return to their starting point is the hypotenuse of this right triangle.
Using the Pythagorean theorem (a² + b² = c²), where a and b are the lengths of the legs and c is the length of the hypotenuse, we can calculate the distance:
a = 8 miles (north)
b = 5 miles (west)
c = √(a² + b²)
c = √(8² + 5²)
c = √(64 + 25)
c = √89
Calculating the square root of 89:
c ≈ 9.43 miles
Therefore,
The shortest distance the joggers must travel to return to their starting point is approximately 9.43 miles to the nearest tenth of a mile.
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Find the slope of a line that is perpendicular to the line containing the points (–2, –1) and (2, –3).
A. m = –2
B. m = 2
C. m = -1/2
D. m = –1
find the nth term of the geometric sequence whose initial term is a 1 = 5.5 a1=5.5 and common ratio is 3 3 .
The nth term of this geometric sequence is simply 5.5, regardless of the value of n.
The nth term of a geometric sequence with first term a1 and common ratio r is given by the formula:
an = a1 * r^(n-1)
In this case, the first term a1 is given as 5.5 and the common ratio r is 3/3 = 1. Therefore, the formula becomes:
an = 5.5 * 1^(n-1) = 5.5 * 1 = 5.5
So, the nth term of this geometric sequence is simply 5.5, regardless of the value of n.
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What is the percentage equivalent to 21 over 70?
30%
35%
49%
92%
Answer:
The percentage equivalent of 21/70 can be found by dividing 21 by 70 and then multiplying by 100 to convert to a percentage.
In order to find a percentage just divide two numbers which the numerator divided by the denominator x 100 = to answer%
(21/70) x 100 = 30%
Therefore, the answer is 30%.
Quadrilateral PQRS is a parallelogram. What is m∠PKS?
Answer:
C) 90°Step-by-step explanation:
The angle PKS is marked with a small square.
It is the indication of a right angle, hence the measure of this angle is:
m∠PKS = 90°The measure of an angle is 64.9°. What is the measure of its supplementary angle?
What is the value of x in the figure below?
The value of x in the right triangle is 10.2 units.
How to find the side of a right triangle?A right angle triangle is a triangle that has one of its angles as 90 degrees.
The sum of angles in a triangle is 180 degrees.
The triangles are similar. Similar triangles are the triangles that have corresponding sides in proportion to each other and corresponding angles equal to each other.
Let's use the similarity to find x as follows:
8 / x = x / 5 + 8
8 / x = x / 13
cross multiply
x² = 13 × 8
x² = 104
square root both sides
x = √104
x = 10.1980390272
x ≈ 10.2 units
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Each day for the next 4 days, there is a 70% chance of a thunderstorm. Use the simulation shown, where the digits 1 through 7 represent days with a thunderstorm, to estimate the probability of a thunderstorm on at least 3 of the next 4 days. Round your answer to the nearest tenth of a percent if necessary. Here are the numbers: 2074 4306 1636 6761 8419 4460 4164 9567 7351 9716 9967 6119 6021 7751 6344 4044 4988 7456 3472 1749 5094 1018 7693 2957 6424 3030 7632 8857 8607 9362 What is the probability in percent form? I will give you 42 points! Please hurry!!
The estimated probability of a thunderstorm on at least 3 of the next 4 days is 58.3%.
To estimate the probability of a thunderstorm on at least 3 of the next 4 days, we can analyze the given simulation results.
Out of the provided simulation results, the outcomes with thunderstorms on at least 3 of the next 4 days are:
2074, 4306, 1636, 6761, 8419, 4460, 4164, 9567, 7351, 9716, 9967, 6119, 6021, 7751, 6344, 4044, 4988, 7456, 3472, 1749, 5094, 1018, 7693, 2957, 6424, 3030, 7632, 8857, 8607, 9362 (a total of 29 outcomes)
Now, let's calculate the probability of having thunderstorms on at least 3 out of 4 days:
P(at least 3 out of 4 days) = P(3 days) + P(4 days)
= (0.70)³ + (0.70)⁴
= 0.343 + 0.2401
= 0.5831
To convert this probability to a percentage, we multiply by 100:
P(at least 3 out of 4 days) ≈ 0.5831 x 100 ≈ 58.3%
Therefore, the estimated probability of a thunderstorm on at least 3 of the next 4 days is 58.3%.
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Find the probability that a randomly
selected point within the circle falls in the
red-shaded square
The probability that a randomly selected point within the circle falls in the red-shaded square is: 0.50
What is the probability of selection?The formula for the area of a circle is:
A = πr²
where:
A is area
r is radius
Thus:
A = π * 4²
A = 50.265 unit²
Area of square = side * side
Area = 5 * 5
Area = 25 unit²
Thus:
P(selected area falls in the read square) = 25/50.265
P(selected area falls in the read square) = 0.497 ≈ 0.50
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what was the approximate number of mobile cellular subscriptions per 100 people in zimbabwe in 2003? the display gives the result of performing an exponential regression on a data set where the inputs are years since 2000 and the outputs are the corresponding mobile cellular subscriptions per 100 people in zimbabwe.
The number of mobile cellular subscriptions per 100 people in Zimbabwe in 2003 is 4.
Exponential model that shows the number of mobile cellular subscriptions per 100 people in Zimbabwe since 2000,
Exponential model is a mathematical function which express in the form of a = [tex]e^{x}[/tex] where, x is power and the function is increasing exponentially.
[tex]y = ab^{x}[/tex]
Where,
a=1.067905095,
b=1.501755837,
y = [tex]1.067905095(1.501755837 )^{x}[/tex]
Thus, for finding the number of subscriptions in 2003,
x = 3,
Hence, the number of mobile cellular subscriptions per 100 people in Zimbabwe in 2003 is
y = [tex]1.067905095(1.501755837 )^{3}[/tex] = 4
The number of mobile cellular subscriptions per 100 people in Zimbabwe in 2003 is 4.
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The given question is incomplete the complete question is :
Answer:4
Step-by-step explanation:
Find and plot the following parametric curves in the phase plane for −[infinity] < t < [infinity] by eliminating time from the equations. a) x = 3sin(2πt) , y = 4cos(2πt) d) x = e−2t , y = −2e−2t
The plot of the parametric curves in the phase plane is illustrated below.
To eliminate time, we need to express one of the variables in terms of the other and then plot the resulting equation in the xy-plane. Let's start with the first curve, x = 3sin(2πt) and y = 4cos(2πt).
To eliminate time, we can use the trigonometric identity sin²(θ) + cos²(θ) = 1 to express sin(2πt) in terms of cos(2πt):
sin²(2πt) + cos²(2πt) = 1
sin(2πt) = ±√(1 - cos²(2πt))
We can then substitute this expression for sin(2πt) in the equation for x:
x = 3sin(2πt) = 3*±√(1 - cos²(2πt))
Squaring both sides and rearranging, we get:
(x/3)² + (y/4)² = 1
This is the equation of an ellipse centered at the origin with semi-axes of length 3 and 4. We can graph this ellipse in the xy-plane to get the phase portrait of the first curve.
For the second curve, x = [tex]e^{-2t}[/tex] and y = -2 [tex]e^{-2t}[/tex] , we can eliminate time by solving for [tex]e^{-2t}[/tex] in terms of y and substituting into the equation for x:
[tex]e^{-2t}[/tex] = -y/2
x = [tex]e^{-2t}[/tex] = -y/2
This is the equation of a line passing through the origin with slope -1/2. We can graph this line in the xy-plane to get the phase portrait of the second curve.
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The distance that Karen ran each day of 5 days is shown I. The table above . What was the average distance that Karen ran per day ?
The average distance run by Karen per day comes out to be 4.8 miles.
Average refers to the ratio of the sum of the data to the number of data given. It is also called the mean of the data.
Mean = n₁ + n₂ + ...... nₐ / a
a is the number of data
Given:
Monday = 4 miles
Tuesday = 5 miles
Wednesday = 3 miles
Thursday = 6 miles
Friday = 6 miles
Sum = 4 + 5 + 3 + 6 + 6 = 24 miles
Number of data = 5
Average = 24 / 5
= 4.8 miles
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The complete question is:
Monday = 4 miles
Tuesday = 5 miles
Wednesday = 3 miles
Thursday = 6 miles
Friday = 6 miles
The distance that Karen ran each day for 5 days is shown in The table above. What was the average distance that Karen ran per day?
Which of the following is NOT TRUE about the transformation given below?
y=-2e^(x-3)+4
Group of answer choices
Up 4 units
Vertical shrink by 2
Reflection across x-axis
Right 3 units
Answer:
Vertical shrink by 2
Step-by-step explanation:
Multiplying by 2 stretches the function. It does not shrink it.
Answer: Vertical shrink by 2
what is game theory? how does game theory help us understand oligopoly? watch the following video clips about game theory. did the videos help you understand this concept? share your thoughts.
Game theory is a branch of mathematics that deals with the study of decision-making in situations where two or more individuals or groups are involved.
It aims to predict the outcomes of these decisions by analyzing the strategies and incentives of each player. In the context of oligopoly, game theory helps us understand how firms interact and compete with each other in a market. Oligopoly is a market structure where a small number of large firms dominate the market. In this situation, firms must take into account the actions and reactions of their competitors when making decisions about pricing, production, and advertising. Game theory provides a framework for analyzing the strategic behavior of firms in an oligopolistic market. It helps us understand how firms may collude or engage in strategic behavior to gain an advantage over their competitors. It also provides insights into how changes in market conditions or government regulations can affect the behavior of firms in an oligopoly. I can say that visual aids and real-world examples can be helpful in understanding complex concepts like game theory. Video clips can be a great way to illustrate key ideas and make them more accessible to a wider audience. However, it is important to remember that game theory is a highly technical field and may require a significant amount of study and practice to fully grasp its concepts and applications.
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Find the area of this square 5in. 6in.
Answer:
= 30 square inches.
Step-by-step explanation:
This pic is blurry.
This is a parallelogram. The formula for area is base x height. From your question, I'm guessing the dimensions are 5 inches and 6 inches.
area = 5 x 6 = 30 square inches
Another model for a growth function for a limited population is given by the Gompertz function, which is a solution to the differential equation dP/dt=c ln(K/P)P where c is a constant and K is the carrying capacity. Answer the following questions. 1. Solve the differential equation with a constant c=0.05, carrying capacity K=3000, and initial population P0=1000. 2. With c=0.05, c = 0.05 , K=3000, and P0=1000, find limt→[infinity]P(t).
The final answer is the population P(t) will continue to increase without bound as time goes on, reaching values larger than the carrying capacity K = 3000. To solve the differential equation [tex]\frac{dP}{dt}[/tex] = c ln(K/P)P.
We can use the separation of variables.
Solve the differential equation with c = 0.05, K = 3000, and P0 = 1000:
First, separate the variables:
[tex]dP/ln(K/P)P[/tex] = c dt
Integrate both sides:
∫(1/P) dP = ∫c dt
ln(P) = ct + C1
Taking the exponential of both sides:
[tex]e^(ln(P)) = e^(ct + C1)[/tex]
[tex]P = e^(ct + C1)\\[/tex]
To find C1, substitute the initial condition P0 = 1000 when t = 0:
1000 = [tex]e^(c * 0 + C1)[/tex]
1000 = [tex]e^(C1)[/tex]
Taking the natural logarithm of both sides:
ln(1000) = C1
So the equation becomes:
P = [tex]e^(ct + ln(1000))[/tex]
Now, substitute the values c = 0.05, K = 3000:
P = [tex]e^(0.05t + ln(1000))[/tex]
To find the limit as t approaches infinity (lim(t→∞) P(t)), we examine the behavior of the exponential term [tex]e^(0.05t)[/tex]as t grows without bound.
As t approaches infinity, [tex]e^(0.05t)[/tex]also approaches infinity. Therefore, the limit of P(t) as t approaches infinity is also infinity.
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how many license plates consisting of three letters followed by three digits contain no letter or digit twice?
There are 11,232,000 license plates consisting of three letters followed by three digits with no letter or digit repeated.
We can solve this problem using permutation, as we need to find the number of arrangements of three letters followed by three digits with no repetition.
There are 26 letters in the alphabet, so we can choose the first letter in 26 ways. For the second letter, we can choose from the remaining 25 letters, as we cannot use the same letter twice. Similarly, we can choose the third letter in 24 ways.
For the first digit, we have 10 options (0 to 9). For the second digit, we can choose from the remaining 9 digits (we cannot use the same digit twice). Similarly, we can choose the third digit in 8 ways.
Using the multiplication principle, we can find the total number of possible license plates:
26 × 25 × 24 × 10 × 9 × 8 = 11,232,000
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Find the x- and y- intercept in 3x+2y=24
The x and y intercept of the equation is (8,12)
What is linear equation?A linear equation is an algebraic equation of the form y=mx+b. It involves only a constant and a first-order term, where m is the slope and b is the y-intercept.
For 3x +2y = 24
we need to put it to the standard form
2y = 24 - 3x
divide both sides by 2
y = 12 - 3/2x
Here b is 12 and m is -3/2
therefore the y intercept is 12
when y = 0
0 = 12 -3/2 x
3/2 x = 12
3x = 24
divide both sides by 3
x = 24/3 = 8
therefore the x intercept is 8
The x and y intercept of the equation is ( 8,12)
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for the following scenarios, state the null hypothesis and the alternative hypothesis to be used when a hypothesis test is performed
i) Null hypothesis: Proportion of Americans who believe that nuclear capabilities of other countries seriously threaten world peace is equal to 2/3.
Alternative hypothesis: Proportion of Americans who believe that nuclear capabilities of other countries seriously threaten world peace is different from 2/3.
ii) Null hypothesis: Proportion of gaming headsets with manufacturing flaws that make game-play impossible is less than or equal to 9%. Alternative hypothesis: Proportion of gaming headsets with manufacturing flaws that make game-play impossible is greater than 9%.
iii) Null hypothesis: Mean water temperature downstream from JP's Power Plant's cooling tower discharge pipe is less than or equal to 106°F. Alternative hypothesis: Mean water temperature downstream from JP's Power Plant's cooling tower discharge pipe is greater than 106°F.
In hypothesis testing, the null hypothesis is the default assumption, while the alternative hypothesis is what the researcher wants to prove.
The next step is to conduct a hypothesis test, where a test statistic is calculated, and its probability of occurrence is determined assuming that the null hypothesis is true.
If this probability is very small (usually less than 5%), then the null hypothesis is rejected in favor of the alternative hypothesis.
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Complete Question
1. For the following scenarios, state the null hypothesis and the alternative hypothesis to be used when a hypothesis test is performed.
Scenario i) After the Cold War, it was claimed that two of three Americans say that the chances of world peace are seriously threatened by the nuclear capabilities of other countries. Is there evidence that this proportion is actually different? To investigate this, a random sample of 400 Americans was taken, and it was found that only 248 hold this view.
Scenario ii) JP wants to test whether at least 9% of gaming headsets have manufacturing flaws that make game-play impossible. A sample of 150 headsets revealed that 12 contained a defect.
Scenario iii) The mean water temperature downstream from JP's Power Plant's cooling tower discharge pipe should be no more than 106°F. Past experience has indicated that the standard deviation of temperature is 2°F. The water temperature is measured on nine randomly chosen days, and the average temperature is found to be 99°F