Answer:
Its magnitude will be larger than 0.004.
Step-by-step explanation:
When a divisor is less than 1, the quotient will be greater than the dividend.
When the divisor is "almost zero", the quotient will be much greater than the dividend. Here, the dividend may be considered to be "almost zero", so we cannot say anything about the actual quotient except to say its magnitude will be greater than the dividend.
_____
The dividend is positive, so the quotient will have the same sign as the divisor. (Negative divisors can be "almost zero," too.)
yo already knowww :D
Answer:
There are NO parallel sides.
Step-by-step explanation:
Look at them.
Answer: Bottom left, lamins have no parrellel sides
Step-by-step explanation: All of them have no parrellel sides. Hope this helps!
Which of the following relationships represents a function?
A. (-3,4), (6,2), (-7,1), (2,2)
B. (-3,0), (6,3), (-7,1), (6,5)
C. (-3,4), (6,6), (-3,3), (2,2)
D. (-3,4), (6,6), (6,3), (2,2)
Answer:
A. (-3,4), (6,2), (-7,1), (2,2)
Step-by-step explanation:
Functions cannot have the x repeating. All the other answers have one of their x-values repeating, so they are not functions.
In ABCD, the measure of ZD=90°, BD = 20, CB = 29, and DC = 21. What is the value
of the cosine of C to the nearest hundredth?
Answer:
0.72
Step-by-step explanation:
The mnemonic SOH CAH TOA reminds you that the relation of interest is ...
Cos = Adjacent/Hypotenuse
Then the cosine of angle C is ...
cos(C) = CD/CB = 21/29
cos(C) ≈ 0.72
Answer:
Step-by-step explanation:
Sequence B: The bacteria on a sponge multiply rapidly. This sequence describes the
growth in bacteria over time.
3, 6, 12, 24, 48, 96
the bottor in each sequence and show your work. 1
Step-by-step explanation:
The bacteria is always doubled because in the sequence we multiply the number by 2
3 x 2 = 6 x 2 = 12 x 2 = 24 x 2 = 48 x 2 = 96
A national park keeps track of how many people per car enter the park.Today, 57 cars had 4 people, 61 cars had 2 people, 9 cars had 1 person, and 5 cars had 5 people. What is the average number of people per car
The average number of people per car visiting the national park is approximately 2.91.
Explanation:In order to find the average number of people per car, you would first multiply the number of cars by the number of people in each.
Therefore, 57 cars had 4 people (57*4=228), 61 cars had 2 people (61*2=122), 9 cars had 1 person (9*1=9), and 5 cars had 5 people (5*5=25). After this, add all the results (228+122+9+25 = 384).
The total number of cars is the sum of all cars, which is (57+61+9+5 = 132).
Now, to find the average number of people per car, you would divide the total number of people, which is 384 by the total number of cars, which is 132. Therefore, the average number of people per car is 384 / 132 = about 2.91 people per car.
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Historically, voter turnout for political elections in Texas have been reported to be 54%. You have been assigned by a polling company to test the hypothesis that voter turnout during the most recent election was higher than 54%. You have collected a random sample of 90 registered voters from this elections and found that 54 actually voted. Assume that the true proportion of voter turnout is 0.67. Calculate the probability of a Type II error using α= 0.02.
Answer:
The probability of a Type II error is 0.3446.
Step-by-step explanation:
A type II error is a statistical word used within the circumstance of hypothesis testing that defines the error that take place when one is unsuccessful to discard a null hypothesis that is truly false. It is symbolized by β i.e.
β = Probability of accepting H₀ when H₀ is false
= P (Accept H₀ | H₀ is false)
In this case we need to test the hypothesis whether the voter turnout during the most recent elections in Texas was higher than 54%.
The hypothesis can be defined as:
H₀: The voter turnout during the most recent elections in Texas was 54%, i.e. p = 0.54.
Hₐ: The voter turnout during the most recent elections in Texas was higher than 54%, i.e. p > 0.54.
A type II error would be committed if we conclude that the voter turnout during the most recent elections in Texas was 54%, when in fact it was higher.
The information provided is:
X = 54
n = 90
α = 0.02
true proportion = p = 0.67
Compute the mean and standard deviation as follows:
[tex]\mu=p=0.54[/tex]
[tex]\sigma=\sqrt{\frac{0.54(1-0.54)}{90}}=0.053[/tex]
Acceptance region = P (Z < z₀.₀₂)
The value z₀.₀₂ is 0.6478.
Compute the sample proportion as follows:
[tex]z=\frac{\hat p-\mu}{\sigma}\\2.054=\frac{\hat p-0.54}{0.053}\\\hat p=0.6489[/tex]
Compute the value of β as follows:
β = P (p < 0.54 | p = 0.67)
[tex]=P(\frac{\hat p-\mu}{\sigma}<\frac{0.6489-0.67}{0.053})\\=P(Z<-0.40)\\=0.34458\\\approx 0.3446[/tex]
Thus, the probability of a Type II error is 0.3446.
Need help It is working with functions and I need assistance
Answer: its 4. Look at the graph, x is horizontal and y is vertical (first go sideways then go up.) if coordinates are x,y then first go sideways when looking for x. they give you x. 7. then we go up. the graph intersects 7 at 4, therefore the coords are 7, 4 and your y coordinate is 4.
Step-by-step explanation:
Mark has 153 hot dogs and 171 hamburgers. He wants to put the same number of hot dogs and hamburgers on each tray. What is the greatest number of trays Mark can use to accomplish this?
Answer:
Mark will use 153 no of trays to accomplish his task.
Final answer:
Mark can use 9 trays to distribute 153 hot dogs and 171 hamburgers evenly by finding the Greatest Common Divisor (GCD) of the two numbers, which is 9.
Explanation:
Mark has 153 hot dogs and 171 hamburgers and wants to distribute them evenly across the greatest number of trays. To do this, Mark needs to find the Greatest Common Divisor (GCD) of the two numbers, which is the largest number that can evenly divide both 153 hot dogs and 171 hamburgers. The GCD of 153 and 171 is 9.
Step 1: List the factors of 153 (1, 3, 9, 17, 51, 153) and 171 (1, 3, 9, 19, 57, 171).Step 2: Identify the largest factor that appears in both lists, which is 9.Step 3: Divide the number of hot dogs and hamburgers by the GCD to find the number of items per tray. For hot dogs: 153 / 9 = 17 hot dogs per tray. For hamburgers: 171 / 9 = 19 hamburgers per tray.Therefore, Mark can use 9 trays, with 17 hot dogs and 19 hamburgers on each tray, to distribute them evenly.
Nathan wants to buy a sweatshirt and is trying to determine the better buy. He has a 20% coupon for the in-store purchase. The store charges $44 for the sweatshirt he wants. However, he found the same sweatshirt online for $38 and gets 15% off as a first-time buyer. There are no shipping charges. Which is the better buy? By how much? Use the Polya problem solving method to solve this problem.
Answer:
$38 + 15% off
Step-by-step explanation:
to do this, multiply each original price by the coupon percentage, (38 being .15, and 44 being .4,) and subtract the answers. when this is done and you compare, you will see that the $38 sweatshirt is less money by $2.90.
The cost of the sweatshirt both in-store and online after applying the respective discounts shows that it is better to buy the sweatshirt online, saving Nathan $2.90.
Explanation:This problem relates to the mathematics field of percentage or discount calculations. To solve this problem, we need to calculate the cost of the sweatshirt after applying the discounts both in-store and online.
First, let's calculate the amount of the 20% discount for the in-store purchase. To find the discount, we multiply the cost of the sweatshirt by the discount rate: $44 * 20% = $8.80. So the total cost for the sweatshirt in-store after the discount would be $44 - $8.80 = $35.20.
Secondly, let's calculate the online price with a 15% discount. The discount on the online price would be calculated as: $38 * 15% = $5.70. Therefore, the cost of the sweatshirt online after the discount is $38 - $5.70 = $32.30.
Based on these calculations, buying online is the better deal by $35.20 - $32.30 = $2.90.
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Subtract breaking apart 143-79
To subtract 143-79 by breaking apart, round 79 to 80, subtract it from 143 to get 63, and add 1 back for rounding to get the final answer of 64.
Explanation:The question involves breaking apart numbers to subtract 143-79. This is a technique used in math to simplify subtraction by splitting numbers into parts that are easier to work with.
Step-by-Step Explanation:First, let's round 79 to the nearest tens, which is 80.Now, subtract 80 from 143: 143 - 80 = 63.Since we rounded up 79 to 80, we subtracted one extra unit from 143. So, we need to add that 1 back to the result: 63 + 1 = 64.The final answer is 64.
Which investment has the highest liquidity and can be converted into cash easy
Answer: Stocks will have the highest liquidity and convertability to cash
I need on dis plz because we’re in quarantine and I’m also dumb
Matty jogs 9 km/hr. Compute Matty's speed in m/s.
Step-by-step explanation:
1km = 9000 m
1 min = 60 sec
So 60 min = 3600 sec
In this way Matty jogs 9000m /3600sec
Matty's speed of 9 km/hr is equivalent to 2.5 m/s when converted using the relationship where 1 km equals 1000 meters and 1 hour equals 3600 seconds.
Explanation:To compute Matty's speed in meters per second (m/s), we need to convert from kilometers per hour (km/h) to m/s. To do this, we can use the conversion rate where 1 km equals 1000 meters, and 1 hour equals 3600 seconds.
First, we convert Matty's speed to meters:
9 km/h = 9 × 1000 m/h = 9000 m/h.
Then, we convert hours to seconds:
9000 m/h × (1 h / 3600 s) = 9000 m / 3600 s = 2.5 m/s.
Therefore, Matty's speed in meters per second is 2.5 m/s.
What’s the area of the circle in terms of pi
Answer:
A = πr²
Step-by-step explanation:
The equations of two lines are y=4x+2 and 6x-y=4 . What is the value of x in the solution for this system of equations?
Answer: x = 3
Step-by-step explanation
This is a simultaneous equation. So to find x , we solve to ascertained the coordinate or point of intersection of the too lines.
y = 4x + 2, and 6x - y = 4
To solve , we need to rearrange first
y = 4x + 2 will be
4x - y = -2 -------------------------1
Equation 2 is in order
6x - y = 4 -------------------------- 2
Now solve the two equation for x by using any methods you are familiar with.
4x - y = -2
6x - y = 4, now subtract equation 2 from 1 in order to eliminate y,
we now have
-2x = -6, now divide through by 2
x = 3.
Hence y could be find by substitution for x in any of the equation above and now check.
4x - y = -2
4(3) - y = -2
12 - y = = -2
y = 12 + 2
y = 14.
Now let check
6x - y = 4
6(3) - 14
18 - 14
= 4.
Therefore, the value of x is 3
Jay owns some DVDs. Lamar lives next to Jay and three houses from Freda. Lamar has 2 fewer DVDs than Jay. Davina lives on the other side of Jay and has three times as many DVD's as Lamar. Bart lives in house 2 and has 4 more DVD's than Jay. If Tara had 13 more DVD's, she would have four times as many as Jay. Jay acquired all of his DVD from freda, who lives in house 6. Freda had 17 DVD before she gave "p" of them to Jay. the person in house 5 owns the most DVD's wat is the value of p?
Answer:
6
Step-by-step explanation:
Freda lives in house 6. Lamar lives 3 houses from Freda, so lives in house 3. Jay lives next to Lamar, but not in house 2, which is Bart's. So, Jay lives in house 4 and Davina lives in house 5, on the other side of Jay from Lamar.
Davina lives in house 5, so has the most DVDs.
__
Since Freda gave p DVDs to Jay, she has 17-p. Lamar has p-2, Bart has p+4, Tara has 4p-13, and Davina has the most at 3(p-2).
We want Davina to have the most DVDs, so there are some inequalities that must be true:
3(p-2) > 4p-13 ⇒ p < 7
3(p-2) > p +4 ⇒ p > 5
The only value of p satisfying both of these requirements is p = 6.
_____
Tara lives in house 1 and has 11 DVDs.
Bart lives in house 2 and has 10 DVDs.
Lamar lives in house 3 and has 4 DVDs.
Jay lives in house 4 and has 6 DVDs.
Davina lives in house 5 and has 12 DVDs. (the most)
Freda lives in house 6 and has 11 DVDs.
A manufacturer wishes to estimate the proportion of dishwashers leaving the factory
that is defective. How large a sample should be tested in order to be 99% confident
that the true proportion of defective dishwashers is estimated to within a margin of
error of 3%?
1843.27
1800
1843
1844
The sample size needed with 99% confidence and 3% margin of error is 1843.
To estimate the sample size needed:
Determine the critical value for 99% confidence, which is 2.576.
Calculate the minimum sample size using the formula:
n = (Z^2 * p * q) / E^2
Substitute Z = 2.576, p = 0.5, q = 0.5, and E = 0.03.
Calculate to get the sample size, which is approximately 1843.
The times between the arrivals of customers at a taxi stand are independent and have a distribution F with mean F. Assume an unlimited supply of cabs, such as might occur at an airport. Suppose that each customer pays a random fare with distribution G and mean G. Let W.t/ be the total fares paid up to time t. Find limt!1EW.t/=t.
Answer:
Check the explanation
Step-by-step explanation:
Let
\(W(t) = W_1 + W_2 + ... + W_n\)
where W_i denotes the individual fare of the customer.
All W_i are independent of each other.
By formula for random sums,
E(W(t)) = E(Wi) * E(n)
\(E(Wi) = \mu_G\)
Mean inter arrival time = \(\mu_F\)
Therefore, mean number of customers per unit time = \(1 / \mu_F\)
=> mean number of customers in t time = \(t / \mu_F\)
=> \(E(n) = t / \mu_F\)
What is the radius and diameter of the following circle 7cm
Answer:
7
3.5
Step-by-step explanation:
Answer:
The radius of the circle is [tex]\(7 \, \text{cm}\)[/tex], and the diameter is [tex]\(14 \, \text{cm}\)[/tex].
Explanation:
The radius [tex](\(r\))[/tex] of a circle is half of its diameter ([tex]\(d\)[/tex]), and the diameter ([tex]\(d\)[/tex]) is twice the radius ([tex]\(r\)[/tex]). So, we can find the radius and diameter of the circle with a given radius of 7 cm.
Given:
Radius ([tex]\(r\)[/tex]) = 7 cm
To find the diameter ([tex]\(d\)[/tex]), we use the relationship:
[tex]\[d = 2r\][/tex]
Substituting the given value of the radius:
[tex]\[d = 2 \times 7\][/tex]
[tex]\[d = 14\][/tex]
So, the diameter of the circle is [tex]\(14 \, \text{cm}\)[/tex].
Therefore, the radius of the circle is [tex]\(7 \, \text{cm}\)[/tex], and the diameter is [tex]\(14 \, \text{cm}\).[/tex]
Question:
What is the diameter of a circle whose radius is 7 cm?
Suppose there are two full bowls of cookies. Bowl #1 has 12 chocolate chip and 24 plain cookies, while bowl #2 has 22 of each. Our friend Fred picks a bowl at random, and then picks a cookie at random. The cookie turns out to be a plain one. What is the probability that Fred picked Bowl #1? 56.8966
Answer:
0.5
Step-by-step explanation:
Simplify the Square root of 72a6b3c2
For the single roots -1 and 2,the graph ——— the x-axis at the intercepts
/crosses/
/does not intersect/
/touches/
—————————
For the double root 3 the graph ——— at the intercepts
/crosses/
/does not intersect/
/touches/
Answer:
crosses and touches is %100 RIGHT
Step-by-step explanation:
For the single roots -1 and 2,the graph crosses the x-axis at the intercepts
For the double root 3 the graph touches at the intercepts
What are intercepts?"These are the point at which the graph of the function intersects the x-axis or Y-axis"
What are roots of function?"These are the values for which the function equals zero."
For given question,
We have been given a function f(x) = (x + 1)(x - 2)(x - 3)²
To find the roots of function f(x)
⇒ f(x) = 0
⇒ (x + 1)(x - 2)(x - 3)² = 0
⇒ x + 1 = 0 or x - 2 = 0 or (x - 3)²=0
⇒ x = -1 or x = 2 or x = 3
This means x = -1, 2, 3 are the roots of the function f(x)
From the graph of the function we can observe that the for the single roots -1 and 2,the graph of the function f(x) crosses the x-axis at the intercepts x = -1 and x = 2 respectively.
And for the double root 3 the graph of the function f(x) touches at the intercept x = 3.
Therefore, For the single roots -1 and 2,the graph crosses the x-axis at the intercepts.
For the double root 3 the graph touches at the intercepts.
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What is the opposite of the coordinate for point D? A) −33 B) −34 C) 33 D) 34
will mark the brainiest
The opposite of a coordinate in mathematics is obtained by changing the sign of the given number. Without the specific coordinate for point D, the opposite cannot be determined from the provided options.
The question seems to refer to the notion of opposite coordinates in a coordinate system. In mathematics, particularly in the context of a coordinate plane, the opposite of a coordinate is simply the same number with its sign changed. If the original coordinate for point D is not given in the question, it's impossible to determine the correct opposite coordinate from the options provided.
4.7 kg = __
What equals 4.7 kg =____g
Answer:
If your asking what 4.7kg is to grams then it is 4700g
Answer:
4700 g
Step-by-step explanation:
1 kg = 1000g
Using this rule, 4.7 kg would be 4700g
4.7 x 1000 = 4700
Mrs. Potts is putting a fence around her rectangular garden the length of the garden is 8 yards the width of the garden is 4 yards how many feet of fencing does Mrs. Pott need
Answer:
Mrs. Potts needs 72 feet of fencing.
Step-by-step explanation:
You find the perimeter of the rectangular garden, which is 24 yards. You then convert the yards to feet. There are 3 feet in a yard. You multiply 24 by 3 to get your final answer of 72 feet.
A butterfly population is decreasing at a rate of 0.82% per year. There are currently about 100,000 butterflies in the population. How many butterflies will there be in the population in 250 years?
Answer:
12,765
Step-by-step explanation:
The exponential formula for the population can be written as ...
population = (initial population)(1 -decay rate)^t
where t is in years, and the decay rate is the loss per year.
The given numbers make this ...
population = 100,000(0.9918^t)
In 250 years, the population will be about ...
population = 100,000(0.9918^250) ≈ 12,765.15
There will be about 12,765 butterflies in 250 years.
In 250 years, there will be approximately 12,765 butterflies in the population.
This figure is obtained by applying the exponential decay formula with a 0.82% decrease rate per year from an initial population of 100,000 butterflies.
To determine the butterfly population in 250 years given a yearly decrease rate of 0.82%, we utilize the exponential decay formula:
[tex]\[ P(t) = P_0 \times (1 - r)^t \][/tex]
where:
- (P(t)) represents the population after (t) years,
- (P_0) is the initial population, which is 100,000,
- (r) is the annual decrease rate in decimal form, which is 0.82% or 0.0082,
- (t) is the time in years, here 250 years.
By substituting the given values into the formula:
[tex]\[ P(250) = 100,000 \times (1 - 0.0082)^{250} \][/tex]
Upon calculation, the population after 250 years is found to be approximately 12,765, rounding to the nearest whole number for practical purposes. This result is based on the principle of exponential decay, reflecting how a consistent percentage decrease affects the population over a prolonged period.
NEED HELP WITH THESE THREE QUESTIONS!!!
1.) Use the binomial probability formula to find P(x) given that n = 8, p = 0.31, and x = 4.
2.) Suppose that 5 fair coins are tossed all at once. What is the probability that all 5 of them land heads up?
3.)An unprepared student is given a 14 question multiple choice quiz on Reptiles from the Star Wars Saga. Each question has 5 possible answers of which only one is correct. What is the probability that this student answers correctly on less than 4 of these questions?
Answer:
(a) The value of P (X = 4) is 0.1465.
(b) The probability that all 5 of them land heads up is 0.0313.
(c) The probability that the student answers correctly on less than 4 of these questions is 0.6980.
Step-by-step explanation:
A Binomial distribution is the probability distribution of the number of successes, X in n independent trials with each trial having an equal probability of success, p.
The probability mass function of a Binomial distribution is:
[tex]P(X=x)={n\choose x}p^{x}(1-p)^{n-x};\ x=0,1,2,3...,\ 0<p<1[/tex]
(1)
The information provided is:
X = 4
n = 8
p = 0.31
Compute the value of P (X = 4) as follows:
[tex]P(X=4)={8\choose 4}0.31^{4}(1-0.31)^{8-4}\\=70\times 0.00923521\times 0.22667121\\=0.146535\\\approx 0.1465[/tex]
Thus, the value of P (X = 4) is 0.1465.
(2)
The probability of heads, on tossing a single fair coin is, p = 0.50.
It is provided that n = 5 fair coins are tossed together.
Compute the probability that all 5 of them land heads up as follows:
[tex]P(X=5)={5\choose 5}0.50^{5}(1-0.50)^{5-5}\\=1\times 0.03125\times 1\\=0.03125\\\approx 0.0313[/tex]
Thus, the probability that all 5 of them land heads up is 0.0313.
(3)
There are 5 possible answers for every multiple choice question. Only one of the five options is correct.
The probability of selecting the correct answer is, p = 0.20.
Number of multiple choice questions, n = 14.
Compute the probability that the student answers correctly on less than 4 of these questions as follows:
P (X < 4) = P (X = 0) + P (X = 1) + P (X = 2) + P (X = 3)
[tex]=\sum\limits^{3}_{x=0}{{14\choose x}0.20^{x}(1-0.20)^{14-x}}\\=0.0439+0.1539+0.2501+0.2501\\=0.6980[/tex]
Thus, the probability that the student answers correctly on less than 4 of these questions is 0.6980.
The final probabilities are P(4 successes) ≈ 0.146, P(5 heads) ≈ 0.03125, and P(less than 4 correct answers) ≈ 0.5972.
Solutions to Probability Questions
Let's tackle each of your probability questions step-by-step:
Question 1 To find P(x) using the binomial probability formula, given n = 8, p = 0.31, and x = 4, we use the formula:
P(x) = (ⁿCₓ) * pˣ * (1-p)ⁿ⁻ˣWhere:
(ⁿCₓ) = n! / (x! * (n-x)!)pˣ = p raised to the power of x(1-p)ⁿ⁻ˣ = (1-p) raised to the power of (n-x)Calculating:
(⁸C₄) = 8! / (4! * 4!) = 70p⁴ = 0.31⁴ ≈ 0.009235(1-p)⁸⁻⁴ = (0.69)⁴ ≈ 0.226981So, P(4) ≈ 70 * 0.009235 * 0.226981 ≈ 0.146.
Question 2 If 5 fair coins are tossed, the probability that all 5 of them land heads up is calculated as:
P(5 Heads) = (1/2)⁵ = 1/32 ≈ 0.03125.Question 3 For a student answering a multiple-choice quiz with 14 questions where each question has 5 possible answers, and we are to find the probability of correctly answering less than 4 questions, we use the binomial distribution with n = 14, p = 0.2 (since only one answer out of five is correct), and x < 4:
P(X < 4) = P(0) + P(1) + P(2) + P(3)P(x) = (¹⁴Cₓ) * (0.2)ˣ * (0.8)¹⁴⁻ˣCalculating for x = 0, 1, 2, 3 and summing the probabilities:P(0) ≈ 0.0282, P(1) ≈ 0.099, P(2) ≈ 0.209, P(3) ≈ 0.261.P(X < 4) = 0.0282 + 0.099 + 0.209 + 0.261 ≈ 0.5972.
The number of people who enter an elevator on the ground floor is a Poisson random variable with mean 10. If there are N floors above the ground floor and if each person is equally likely to get off at any one of these N floors, independently of where the others get off, compute the expected number of stops that the elevator will make before discharging all of its passengers.
Answer:
Expected no of stops=N(1-exp(-10/N)
Step-by-step explanation:
Using equation
X=∑[tex]I_{m}[/tex]
where m lies from 1 to N
solve equation below
E(X)=N(1-exp(-10/N)
Given Information:
Distribution = Poisson
Mean = 10
Required Information:
Expected number of stops = ?
Answer:
[tex]E(N) = N(1 - e^{-0.1N} )[/tex]
Explanation:
The number of people entering on the elevator is a Poisson random variable.
There are N floors and we want to find out the expected number of stops that the elevator will make before discharging all of its passengers.
Mean = μ = 10
The expected number of stops is given by
[tex]E(N) = N(1 - e^{-mN} )[/tex]
Where m is the decay rate and is given by
Decay rate = m = 1 /μ = 1/10 = 0.10
Therefore, the expected number of stops is
[tex]E(N) = N(1 - e^{-0.1N} )[/tex]
If we know the number of floor (N) then we can the corresponding expected value.
Compute the mean and standard deviation of the sampling distribution of the sample mean when you plan to take an SRS of size 49 from a population with mean 420 and standard deviation 21. Now repeat the calculations for a sample size of 576. Explain the effect of the increase on the sample mean and standard deviation.
Answer:
The mean of the sampling distribution(SRS 49) of the sample mean is 420 and the standard deviation is 3.
The mean of the sampling distribution(SRS 576) of the sample mean is 420 and the standard deviation is 0.875.
By the Central Limit Theorem, the sample size does not influence the sample mean, but it does decrease the standard deviation of the sample
Step-by-step explanation:
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
Population
mean = 420, standard deviation = 21.
Sample of 49
Mean = 420, standard deviation [tex]s = \frac{21}{\sqrt{49}} = 3[/tex]
The mean of the sampling distribution of the sample mean is 420 and the standard deviation is 3.
Sample of 49
Mean = 420, standard deviation [tex]s = \frac{21}{\sqrt{576}} = 0.875[/tex]
The mean of the sampling distribution of the sample mean is 420 and the standard deviation is 0.875.
Increasing the sample size from 49 to 576 in a simple random sample from a population with a mean of 420 and a standard deviation of 21 keeps the mean of the sampling distribution the same but decreases the standard deviation.
Explanation:When taking a simple random sample (SRS) from a population where the mean (μ) is 420 and the standard deviation (σ) is 21, and using a sample size (n) of 49, the sampling distribution of the sample mean will have:
A mean (μ_x) equal to the population mean (μ), which is 420.A standard deviation (σ_x), also known as the standard error (SE), calculated using the formula σ/√n, which in this case is 21/√49 = 21/7 = 3.When the sample size increases to 576, the sampling distribution of the sample mean will still have a mean of 420, but the standard deviation will decrease because it is inversely proportional to the square root of the sample size. The new standard deviation will be 21/√576 = 21/24 = 0.875.
The effect of increasing the sample size is that while the mean of the sampling distribution remains the same, the standard deviation decreases, leading to a more narrow distribution. This indicates that there is less variability in the sample means, and they will be closer to the population mean, which is in accordance with the Central Limit Theorem.
A quiz consists of 10 true or false questions. To pass the quiz a student must answer at least eight questions correctly.
If the student guesses on each question, what is the probability that the student will pass the quiz?
Answer:
The probability of the student will pass the quiz = .0546
Step-by-step explanation:
Given -
Total no of question = 10
If the student guesses on each question there are two outcomes true of false
the probability of guesses question correctly = [tex]\frac{1}{2}[/tex]
the probability of success is (p) = [tex]\frac{1}{2}[/tex]
the probability of guesses question incorrectly = [tex]\frac{1}{2}[/tex]
the probability of failure is (q) = 1- p = [tex]\frac{1}{2}[/tex]
If the student guesses on each question he must answered at least 8 question correctly
the probability of the student will pass the quiz = [tex]P(X\geq8 )[/tex]
= P(X = 8 ) + P(X = 9) + P(X = 10 )
= [tex]\binom{10}{8}(p)^{8}(q)^{10 - 8} + \binom{10}{9}(p)^{9}(q)^{10 - 9} + \binom{10}{10}(p)^{10}(q)^{10 - 10}[/tex]
= [tex]\frac{10!}{(2!)(8!)}(\frac{1}{2})^{8}(\frac{1}{2})^{10 - 8} +\frac{10!}{(1!)(9!)} (\frac{1}{2})^{9}(\frac{1}{2})^{10 - 9} + \frac{10!}{(0!)(10!)}(\frac{1}{2})^{10}(\frac{1}{2})^{10 - 10}[/tex]
= [tex]45\times\frac{1}{2^{10}} + 10\times\frac{1}{2^{10}} + 1\times\frac{1}{2^{10}}[/tex]
= [tex]\frac{56}{2^{10}}[/tex]
= .0546
Final answer:
The probability of a student passing the true or false quiz by guessing and getting at least 8 out of 10 questions correct is 7/128. This is calculated by finding the binomial probabilities for 8, 9, and 10 correct guesses and summing them.
Explanation:
To determine the probability that the student passes the quiz by guessing, we need to calculate the chances of them getting at least 8 out of 10 true or false questions correct. Since each question can only be true or false, there's a 1/2 chance of guessing each question correctly, and therefore, a 1/2 chance of guessing incorrectly.
The scenarios in which a student can pass are by getting 8, 9, or 10 questions correct. We will use the binomial probability formula, which is P(X=k) = (n choose k) * (p)^k * (1-p)^(n-k), where 'n' is the number of trials (questions), 'k' is the number of successes (correct answers), and 'p' is the probability of success on an individual trial (1/2 for true/false questions).
The probability of getting exactly 8 questions right is (10 choose 8) * (1/2)^8 * (1/2)^(10-8).
The probability of getting exactly 9 questions right is (10 choose 9) * (1/2)^9 * (1/2)^(10-9).
The probability of getting all 10 questions right is (10 choose 10) * (1/2)^10 * (1/2)^(10-10).
We add these individual probabilities together to find the total probability of passing the quiz.
Using a calculator or the binomial coefficients, we find:
P(getting 8 right) = 45 * (1/2)^10,
P(getting 9 right) = 10 * (1/2)^10,
P(getting 10 right) = 1 * (1/2)^10.
Adding these together gives us the total probability:
P(8 or more correct) = [45 + 10 + 1] * (1/2)^10 = 56 * (1/2)^10
After simplifying, we find that the probability of passing the quiz with at least 8 correct answers is thus 56/1024, which can be reduced to 7/128.