The correct hypotheses to be tested are:
H0: p = 0.5 (The proportion of songs loaded by Rina is 0.5, meaning Ed and Rina loaded an equal proportion of songs.)
Ha: p = 0.5 (The proportion of songs loaded by Rina is not equal to 0.5, meaning Ed and Rina loaded different proportions of songs.)
To determine how strong the evidence is that Ed and Rina have each loaded a different proportion of songs into the player, we can perform a hypothesis test. The test statistic for a proportion in a binomial setting is given by:
[tex]\[ z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \][/tex]
where:
- [tex]\(\hat{p}\)[/tex] is the sample proportion of successes (in this case, the proportion of songs by Rina),
- [tex]\(p_0\)[/tex] is the hypothesized proportion under the null hypothesis (0.5),
- [tex]\(n\)[/tex] is the sample size (53 songs).
Given that 35 out of 53 songs were by Rina, we calculate [tex]\(\hat{p}\)[/tex] as:
[tex]\[ \hat{p} = \frac{35}{53} \approx 0.660 \][/tex]
Now we can calculate the test statistic [tex]\(z\)[/tex] :
[tex]\[ z = \frac{0.660 - 0.5}{\sqrt{\frac{0.5(1-0.5)}{53}}} = \frac{0.160}{\sqrt{\frac{0.25}{53}}} \approx \frac{0.160}{\sqrt{0.004717}} \approx \frac{0.160}{0.0686} \approx 2.33 \][/tex]
The P-value for a two-tailed test is the probability of observing a test statistic as extreme as or more extreme than the one observed, under the assumption that the null hypothesis is true. We can find the P-value by looking up the z-score in a standard normal distribution table or using a calculator:
[tex]\[ P\text{-value} = 2 \times P(Z > |2.33|) \approx 2 \times 0.0099 \approx 0.0198 \][/tex]
Since the P-value (0.0198) is less than the significance level of 0.05, we reject the null hypothesis in favour of the alternative hypothesis. This means there is strong evidence to suggest that Ed and Rina have each loaded a different proportion of songs into the player.
Before we conclude, we must check the conditions for the use of this test:
1. The samples are independent. Each song selection is made independently of the others.
2. The number of successes and failures are each at least 10. In this case, Rina's songs (35) and Ed's songs (18) are both greater than 10, so this condition is satisfied.
3. The sample size is less than 10% of the population size. The population size is 3476 songs, and the sample size is 53 songs, which is much less than 10% of the population, so this condition is also satisfied.
Since all conditions are met, we can confidently conclude that there is strong evidence that Ed and Rina have each loaded a different proportion of songs into the player.
The number of bald eagles in a state during the winters from 1996 to 2002 can be modeled by the quartic function
f(x)= -3.357x4 + 35.171x3 - 99.868x2 + 41.098x + 177.252
where x is the number of years since 1996. Find the number of bald eagles in the state in the winter of 2001,
Answer:
f(2x+1)sin(9[tex]F(2x+1)sin(90.9292)\pi[/tex]- 4.3784)
Step-by-step explanation:
The number of bald eagles is 26.
3=9-y solve for variable
Answer:
y=6
Step-by-step explanation:
3=9-y
y+3=9
y=6
Answer:
y=6
Step-by-step explanation:
3=9-y
3-9 = -y
-6 = -y
y=6
Hotel Rooms and Beyond: Error-and-erasure correction leveraging the CRT (58 pts) Note: for this entire problem, you can use properties of the Chinese Remainder Theorem that we discussed in lecture, notes, homework, and discussion without proof and without having to specify all the details. Alice is staying at a hotel and she wants to share her room number with Bob by leaving a sequence of notes in a list of pre-arranged locations.
i) There are only 100 possible hotel rooms, labeled 0 to 99.
ii) Alice takes her room number p and computes the remainders y; = p mod pi. The specific p; that she uses are pı = 3, p2 = 5,23 = 7,24 = 11, and ps = 13.
iii) She writes y, on the i-th note and places the note in location i. (Both Alice and Bob know which location corresponds to which numbers i and p.) This sequence of notes can be viewed as a codeword y(p). For example, if Alice is in room 51, she sends the codeword y(51)=(0,1,2,7,12) since 51 mod 3=0, 51 mod 5= 1, 51 mod 7=2, 51 mod 11 = 7, 51 mod 13= 12.
(a) Unfortunately, there is a chance that some of Alice's notes get blown away by the wind. Those missing notes will be treated as erasures and denoted by X It turns out that Alice is in room 51 as above, so she sends the codeword (0,1,2,7,12). Bob gets the received symbols (0,1,2,X,X). Explain how Bob can leverage the CRT to identify Alice's room number. (Alt + A)
(b) Generalizing the previous part, prove that in the case of up to two erasures anywhere, your scheme will always correctly identify the room number. For this part, feel free to just use the properties of the CRT along with the following facts. • Pi
Answer:
Step-by-step explanation:
From the information given we know that
[tex]p \equiv 0 \,\,\,\, \text{mod(3)}\\p \equiv 1 \,\,\,\, \text{mod(5)}\\p \equiv 2 \,\,\,\, \text{mod(7)}\\[/tex]
And we know as well that
[tex]p \equiv x \,\,\,\, \text{mod(11)}\\p \equiv x \,\,\,\, \text{mod(13)}[/tex]
Remember what that the Chinese reminder theorem states.
Theorem:
Let p,q be coprimes, then the system of equations
[tex]x \equiv a \,\,\,\, mod(p)\\x \equiv b \,\,\,\, mod(q)[/tex]
has a unique solution [tex]mod(pq)[/tex].
Now, if you read the proof of the theorem you will notice that if
[tex]q_1 = q^{-1} \,\, mod(p) , p_1 = p^{-1} \,\,mod(q)[/tex]
the the solution looks like this.
[tex]x = aqq_1 + bpp_1[/tex]
Now. you can easily generalize what I just stated for multiple equations and you will see that if you apply the theorem for this case it is straightforward that
[tex]p \equiv 0*35*[35^{-1}]_3+1*21*[21^{-1}]_5+2*15[15^{-1}]_7 \,\,\,\,\,\,\,\, mod(3*5*7)\\p \equiv 1*21*1+2*15*1 \,\,\,\,\,\,\,\,mod(105) \\p \equiv 1*21*1+2*15*1 \,\,\,\,\,\,\,\, \\p \equiv 51[/tex]
Therefore, Alice is in room 51.
(b)
Using the Chinese reminder theorem you need less than 2 erasures. The process is very similar.
Answer:
Step-by-step explanation:
From the information given we know that
And we know as well that
Remember what that the Chinese reminder theorem states.
Theorem:
Let p,q be coprimes, then the system of equations
has a unique solution .
Now, if you read the proof of the theorem you will notice that if
the the solution looks like this.
Now. you can easily generalize what I just stated for multiple equations and you will see that if you apply the theorem for this case it is straightforward that
Therefore, Alice is in room 51.
(b)
Using the Chinese reminder theorem you need less than 2 erasures. The process is very similar.
Step-by-step explanation:
Dakota earned $4.50 in interest in Account A and $15 in interest in Account B after 18 months. If the simple interest rate is 3% for Account A and 4% for Account B, which account has the greater principal? Explain.
this is probably easy for most people but i need help ASAP
Answer:
Account B
Step-by-step explanation:
The product of two is 450.the first number is half the second number
Answer:
I don't know you're options but the answer could be 30.
What is the percent of 18% of 87
The answer is 15.66
Answer:
15.66
Step-by-step explanation:
[tex]18\% \: of \: 87 \\ \\ = \frac{18}{100} \times 87 \\ \\ = 0.18 \times 87 \\ \\ = 15.66 \\ [/tex]
is 4.1 or 4.009 greater
Answer:
Among 4.1 and 4.009 The greater one is 4.1
Hope it will help.
Pls Help if u Know thx
Answer:
x = 42
Step-by-step explanation:
The two angles are complementary so the add to 90 degrees.
x+48 = 90
Subtract 48 from each side
x+48-48=90-48
x = 42
Answer:
The angle x°=42.
Step-by-step explanation:
∠PQS equals 90° because it's a right angle (denoted with the square on the bottom).
∠PQS = ∠PQR + ∠RQS
So, ∠RQS = ∠PQS - ∠PQR where ∠PQR = 48°
Plug in the Values:
∠RQS = 90° - 48° = 42°
The area of a rectangle is 100cm2. After a dilation, the area is 25cm2. What is the scale factor?
Answer:
The scale factor is 1/2
Step-by-step explanation:
we know that
A dilation is a non rigid transformation that produces similar figures
If two figures are similar, then the ratio of its areas is equal to the scale factor squared
Let
z ---> the scale factor
x ---> the area after dilation
y ---> the original area
[tex]z^2=\frac{x}{y}[/tex]
we have
[tex]x=25\ cm^2\\y=100\ cm^2[/tex]
substitute
[tex]z^2=\frac{25}{100}[/tex]
[tex]z^2=\frac{1}{4}[/tex]
[tex]z=\frac{1}{2}[/tex]
Leo's family keeps recyclables in a cylindrical trash can. Today Leo bought a larger can with a radius and height that are twice the radius and
height of the old can. Leo claims the new and old cans are geometrically similar figures.
Which statement is true?
A. The two cans are similar figures, and the volume of the new can is 2 times the volume of the old can.
B. The two cans are similar figures, and the volume of the new can is 8 times the volume of the old can.
C. The two cans are not similar figures, and the volume of the new can is 2 times the volume of the old can.
D. The two cans are not similar figures, and the volume of the new can is 8 times the volume of the old can.
E. The two cans are similar figures, and the volume of the new can is 4 times the volume of the old can.
Answer:
The answer is B
Step-by-step explanation:
the question states that they are similar, so B is automatically an option. It's 8 times because the radius and height are being doubled. Logically there are more factors to be A and E.
B. The two cans are similar figures, and the volume of the new can is 8 times the volume of the old can.
Write down all of the prime numbers between 15 and 25.
Answer:
17,19,23
that the prime number
Answer:
A prime number is a number whose only factors are one and itself
Prime numbers between 15 and 25:
17, 19, 23
What is the slope that passes through points -4 and 1?
On the planet of Mercury, 4-year-olds average 3.2 hours a day unsupervised. Most of the unsupervised children live in rural areas, considered safe. Suppose that the standard deviation is 1.7 hours and the amount of time spent alone is normally distributed. We randomly survey one Mercurian 4-year-old living in a rural area. We are interested in the amount of time X the child spends alone per day.
The question is incomplete! Complete question along with answer and step by step explanation is provided below.
Question:
On the planet of Mercury, 4-year-olds average 3.2 hours a day unsupervised. Most of the unsupervised children live in rural areas, considered safe. Suppose that the standard deviation is 1.7 hours and the amount of time spent alone is normally distributed. We randomly survey one Mercurian 4-year-old living in a rural area. We are interested in the amount of time X the child spends alone per day.
a. In words, define the random variable X
b. What is X ~N(,)
c. Find the probability that the child spends less than 2 hours per day unsupervised.
d. What percent of the children spend over 12 hours per day unsupervised?
Given Information:
Mean = μ = 3.2 hours
Standard deviation = σ = 1.7 hours
Required Information:
a. In words, define the random variable X
b. X ~N(,) = ?
c. P(X < 2) = ?
d. P(X > 12) = ?
Answer:
a) X is the number of hours in a day that a 4-year-old child spends being unsupervised.
b) X ~N(μ,σ) = X ~N(3.2, 1.7)
c) P(X < 2) = 23.88%
d) P(X > 12) = 0%
Explanation:
a)
Let X is the number of hours in a day that a 4-year-old child spends being unsupervised.
b)
X ~N(μ,σ) = X ~N(3.2, 1.7)
Where 3.2 is the average number of hours that 4-year-old child spends being unsupervised and 1.7 is the standard deviation.
c)
We want to find out the probability that a child spends less than 2 hours per day unsupervised.
P(X < 2) = P(Z < (x - μ)/σ)
P(X < 2) = P(Z < (2 - 3.2)/1.7)
P(X < 2) = P(Z < (- 1.2)/1.7)
P(X < 2) = P(Z < -0.71)
The z-score corresponding to -0.71 is 0.2388
P(X < 2) = 0.2388
P(X < 2) = 23.88%
Therefore, the probability that a child spends less than 2 hours per day unsupervised is 23.88%
d)
We want to find out the probability that a child spends over 12 hours per day unsupervised.
P(X > 12) = 1 - P(X < 12 )
P(X > 12) = 1 - P(X < (x - μ)/σ)
P(X > 12) = 1 - P(X < (12 - 3.2)/1.7)
P(X > 12) = 1 - P(X < 8.8/1.7)
P(X > 12) = 1 - P(X < 5.18)
The z-score corresponding to 5.18 is 1
P(X > 12) = 1 - 1
P(X > 12) = 0
Therefore, the probability that a child spends over 12 hours per day unsupervised is 0%
The question discusses unsupervised hours of Mercurian children and it's a statistics problem involving normal distribution where the average unsupervised time is 3.2 hours with a standard deviation of 1.7 hours.
Explanation:This question is related to statistics, specifically about normal distribution. The mean unsupervised time for 4-year-olds on Mercury is 3.2 hours, with a standard deviation of 1.7 hours. If we randomly survey a 4-year-old Mercurian child living in a rural area, the time they likely spend alone, denoted as X, will range according to this distribution.
Normal distributions can be defined by two parameters: the mean (μ) and the standard deviation (σ). In this case, μ = 3.2 hours and σ = 1.7 hours. Therefore, we can say that the time X spent by this sampled child alone is normally distributed with these parameters.
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Triangle PQR is reflected over the line y = x.
What is the coordinate of the image point R'?
R (2,4)
R (2, -4)
R(-2,4)
Answer:
-2,4
Step-by-step explanation:
i just did it
Answer:
-2,4
Step-by-step explanation:
i just did it on ed 2020
The face of a clock is divided into 12 equal parts. The radius of the clock face is 6 inches. Assume the hands of the clock will form a central angle. Which statements about the clock are accurate? Check all that apply.
Step-by-step explanation:
Given, the face of a clock is divided into 12 equal parts.
Angle of each part = [tex]\frac{360}{12}[/tex] = 30°
(i) When one hand points at 2 and the other points at 4, this is can be divided into two parts, 2 to 3 and 3 to 4.
The angle formed = 2 (30) = 60°
Option (i) is correct
(ii) The circumference of the clock is ,
Circumference of circle = 2πr,
where r is the radius = 6 and π = 3.14.
Substituting the values in the formula, we get
Circumference of circle = 37.68.
Option (ii) is wrong.
(iii) With one hand at 5 and the other at 10, this is 5 parts
The angle formed= 30(5) = 150°.
The arc length =[tex]\frac{150}{360}[/tex](37.68) = 15.7
Option (iii) is correct
(iv) When one hand points at 1 and the other points at 9, this is 4 parts,
30(4) = 120°. T
Option (iv) is wrong
(v) The length of the minor arc from 11 to 2, this is 3 parts
3(30) = 90°
minor arc from 7 to 10 is 3(30) = 90°
Option (v) is correct
Answer: options 1,3,5
What is the value of the interquartile range of the data below?
+
10
20
30
40
50
A(6)
B(12)
C(14)
D(24)
16 < b - 8, if b = 22
Answer:
False
Step-by-step explanation:
Substitute 22 in for b
16 < b - 8
16 < 22 - 8
16 < 14
False, 14 is not greater than 16
The solution:
16 < b - 8
Add 8 to both sides
24 < b
An item originally priced at $55 is marked 25% off. What is the sale price?
Answer:
41.25$
Step-by-step explanation:
Answer:
41.25
Step-by-step explanation:
55/4=13.75
So 13.75 is 25% of 55
So then you would do 55-13.75
Because that is 25% off
The final answer would be $41.25 for the sale price.
The maintenance department at the main campus of a large state university receives daily requests to replace fluorecent lightbulbs. The distribution of the number of daily requests is bell-shaped and has a mean of 63 and a standard deviation of 10. Using the 68-95-99.7 rule, what is the approximate percentage of lightbulb replacement requests numbering between 63 and 83
Answer:
47.5% of lightbulb replacement requests numbering between 63 and 83
Step-by-step explanation:
The Empirical Rule(68-95-99.7 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.
In this problem, we have that:
Mean = 63
Standard deviation = 10
What is the approximate percentage of lightbulb replacement requests numbering between 63 and 83
63 is the mean
83 = 63 + 2*20
So 83 is two standard deviations above the mean.
The normal distribution is symmetric, so 50% of the measures are above the mean and 50% below the mean.
Of those above the mean, 95% are within 2 standard deviations of the mean.
So
0.5*95% = 47.5%
47.5% of lightbulb replacement requests numbering between 63 and 83
answer like gauss 1+3+5+7+...=999
1+3+5+7+...+999 =
= 1+2+3+4+...+500
+1+2+3+...+499
= 2·(1+2+3+...+499) + 500
= 2·(499·500)/2 + 500
= 499·500 + 500
= 500·(499 + 1)
= 500·500
= 250.000
To find the sum of the series 1 + 3 + 5 + 7 + ... + 999, we recognize this is the sum of the first 500 odd numbers. Using the property that the sum of the first n odd numbers is n², we find the sum to be 500², which is 250000.
Summation of Odd Numbers:
To find the sum of the series 1 + 3 + 5 + 7 + ... + 999 using Gauss's method, we first recognize that this sequence is a series of the first 500 odd numbers.
An interesting property to note is that the sum of the first n odd numbers is n².
For example:
→ 1 = 1 (which is 1¹)
→ 1 + 3 = 4 (which is 2²)
→ 1 + 3 + 5 = 9 (which is 3³)
And so on...
In general, the [tex]n_{th[/tex] odd number can be expressed as 2n - 1.
For 999, to find its position in the series, note that 999 is the 500th odd number (since 2*500 - 1 = 999).
Thus, the sum of the first 500 odd numbers is:
→ Sum = 500²
= 250000
→ So, the sum of the series 1 + 3 + 5 + 7 + ... + 999 is 250000.
50 points! ASAP!
Tyler build a dollhouse for his sister shown in the diagram below. Find the volume of the dollhouse. Explain your method for finding the volume of the dollhouse.
(See attached image)
Answer:
V = 18.75
Method: First find the area of the pyramid on top, then find the area of the rectangular prism below.
V of the pyramid = 3.75 ft³
V of the rectangular prism = 15 ft³
3.75 + 15 = 18.75
Hope this helped!! :)
First, find the volume of the pyramid on top:
(2.5 * 3 * 1.5) / 3 = 3.25
Then, find the volume of the rectangular prism:
2.5 * 3 * 2 = 15
Add the two volumes together:
3.25 + 15 = 18.25
To explain: you found the volume of each separate part and added it together.
Jennifer ran 356 m more than Tyler. Tyler ran a distance of 2 km 30 m how many total meters did Tyler and Jennifer Run
Answer:
Tyler ran 2030 meters. Jennifer ran 2386. Together they ran 4416 meters.
Step-by-step explanation:
Turn the km to m then add them together to get Tyler's distance. From Tyler's distance add what Jennifer ran more of to Tyler's to get Jennifer's distance. Then add together what they ran to get the total amount of meters they ran.
Answer:586
Step-by-step explanation:
A certain college classifies students according to credits earned. A studentwith less than 7 credits is a Freshman. At least 7 credits are required to bea Sophomore, 16 to be a Junior and 26 to be classified as a Senior. Write aprogram that calculates class standing from the number of credits earned.
Answer and explanation:
Here is code:
credit = int(input("Enter credits : "))
if credit < 7:
print("You are a Freshman")
elif credit >= 7 and credit < 16:
print("You are a Sophomore")
elif credit >= 16 and credit < 26:
print("You are a Junior")
elif credit >= 26:
print("You are a Senior")
Output: check image
To calculate a student's class standing, use conditional statements to determine the appropriate class based on the number of credits earned.
To write a program that calculates a student's class standing based on the number of credits earned, you can use conditional statements. Here's a step-by-step explanation:
Take the input of the number of credits earned from the user.
Use conditional statements (if, else if) to check the number of credits and assign the appropriate class standing.
If the number of credits is less than 7, then the student is a Freshman. If it is at least 7 and less than 16, then the student is a Sophomore.
If it is at least 16 and less than 26, then the student is a Junior. Otherwise, the student is a Senior.
Display the class standing to the user.
Here's an example pseudocode:
credits = input('Enter the number of credits earned: ')
if credits < 7:
print('Class Standing: Freshman')
elif credits < 16:
print('Class Standing: Sophomore')
elif credits < 26:
print('Class Standing: Junior')
else:
print('Class Standing: Senior')
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You are the engineer in charge of evaluating a new product for potential use in a hydraulic system. The manufacturer has sent 100 randomly chosen samples for you to test. You plan to go through with the deal as long as you can prove that less than 8% of the products you subsequently purchase will be defective. The engineering and business teams at your company believe a 90% confidence level is appropriate for the test.
(a) Specify the null and alternative hypotheses for your test.
(b) Describe (in words) a Type I error for this deal.
(c) Out of the 100 samples provided by the manufacturer, at most how many can be defective for you to agree to use the new product?
(d) For better or worse, your boss convinces you to go through with the deal. Turns out the minimum order is 5000 pieces. Assuming you purchase that many pieces of the new product, and that you found 6 defective pieces out of the 100, generate a 90% two-sided confidence interval for the number of pieces that will be flawed.
Answer:
a) The null hypothesis is represented as
H₀: p ≥ 0.08
The alternative hypothesis is represented as
Hₐ: p < 0.08
b) A type I error for this question would be that
we conclude that the proportion of defective products is less than 8% when in reality, the proportion of defective products, is more than or equal to 8%.
c) At most, the number of defective products in the sample for you to agree to use the new product = 7
d) If minimum of 5000 pieces are purchased, 90% confidence interval for minimum number of flawed pieces will be (103, 497)
Step-by-step explanation:
For hypothesis testing, the first thing to define is the null and alternative hypothesis.
The null hypothesis plays the devil's advocate and is usually stating the opposite of the theory is being tested. It usually maintains that random chance is responsible for the outcome or results of any experimental study/hypothesis testing. It usually contains the signs =, ≤ and ≥ depending on the directions of the test.
While, the alternative hypothesis takes the other side of the hypothesis; that there is indeed a significant difference between two proportions being compared. It usually confirms the the theory being tested by the experimental setup. It usually maintains that other than random chance, there are significant factors affecting the outcome or results of the experimental study/hypothesis testing. It usually contains the signs ≠, < and > depending on the directions of the test
For this question, we want to prove that less than 8% of the products we subsequently purchase will be defective.
So, the null hypothesis will be that there is not enough evidence in the sample to say that less than 8% of the products we subsequently purchase will be defective. That is, the proportion of the sample that are defective is more than or equal to 8%.
And the alternative hypothesis is that there is enough evidence in the sample to say that less than 8% of the products we subsequently purchase will be defective.
Mathematically,
The null hypothesis is represented as
H₀: p ≥ 0.08
The alternative hypothesis is represented as
Hₐ: p < 0.08
b) A type I error involves rejecting the null hypothesis and accepting the alternative hypothesis when in reality, the null hypothesis is true. It involves saying that there is enough evidence in the sample to say that less than 8% of the products we subsequently purchase will be defective when in reality, there isn't enough evidence to arrive at this conclusion.
That is, the proportion of defective products in reality, is more than or equal to 8% and we have concluded that the proportion is less than 8%.
c) Out of the 100 samples provided by the manufacturer, at most how many can be defective for you to agree to use the new product?
The engineer agrees to use the new product when less than 8% of the products we subsequently purchase will be defective.
8% of the product = 0.08 × 100 = 8.
Meaning that the engineer agrees to subsequently purchase the product if less than 8 out of 100 are defective.
So, the maximum number of defective product in the sample that will still let the engineer purchase the products will be 7.
(d) For better or worse, your boss convinces you to go through with the deal. Turns out the minimum order is 5000 pieces. Assuming you purchase that many pieces of the new product, and that you found 6 defective pieces out of the 100, generate a 90% two-sided confidence interval for the number of pieces that will be flawed.
Confidence Interval for the population proportion is basically an interval of range of values where the true population proportion can be found with a certain level of confidence.
Mathematically,
Confidence Interval = (Sample proportion) ± (Margin of error)
Sample proportion = 0.495
Margin of Error is the width of the confidence interval about the mean.
It is given mathematically as,
Margin of Error = (Critical value) × (standard Error)
Critical value at 90% confidence interval for sample size of 100 using the t-tables since information on the population standard deviation.
Degree of freedom = n - 1 = 100 - 1 = 99
Significance level = (100-90)/2 = 5% = 0.05
Critical value = t(0.05, 99) = 1.660
Standard error of the mean = σₓ = √[p(1-p)/n]
p = 0.06
n = sample size = 100
σₓ = (0.06/√100) = 0.006
σₓ = √[0.06(0.94)/100] = 0.0237486842 = 0.02375
90% Confidence Interval = (Sample proportion) ± [(Critical value) × (standard Error)]
CI = 0.06 ± (1.660 × 0.02375)
CI = 0.06 ± 0.039425
90% CI = (0.020575, 0.099425)
90% Confidence interval = (0.0206, 0.0994)
If minimum of 5000 pieces are purchased, 90% confidence interval for minimum number of flawed pieces will be
5000 × (0.0206, 0.0994) = (103, 497)
Hope this Helps!!!
(a) The null hypothesis [tex]\(H_0\)[/tex] is that the proportion of defective products is 8% or less (b) A Type I error occurs when the null hypothesis is true (the actual proportion of defective products is 8% or less), but we incorrectly reject it (c) at most 12 defective products can be found in the sample for the deal to proceed. (d) the 90% two-sided confidence interval for the number of defective pieces in the order of 5000 is from 105 to 496.
(a) The null hypothesis [tex]\(H_0\)[/tex] is that the proportion of defective products is 8% or less. The alternative hypothesis [tex]\(H_1\)[/tex] is that the proportion of defective products is greater than 8%. Mathematically, this can be expressed as:
[tex]\(H_0: p \leq 0.08\) \(H_1: p > 0.08\)[/tex]
(b) A Type I error occurs when the null hypothesis is true (the actual proportion of defective products is 8% or less), but we incorrectly reject it, concluding that the proportion of defective products is greater than 8%. This would mean unnecessarily turning down a good deal and potentially incurring additional costs to find another supplier.
(c) To ensure a 90% confidence level with a maximum defective rate of 8%, we can use the binomial distribution to find the maximum number of defective products allowed in the sample of 100. The formula for a binomial confidence interval is given by:
[tex]\(n \cdot p \pm Z_{\alpha/2} \sqrt{n \cdot p \cdot (1 - p)}\)[/tex]
where [tex]\(n\)[/tex] is the sample size, [tex]\(p\)[/tex] is the defect rate, and [tex]\(Z_{\alpha/2}\)[/tex] is the Z-score corresponding to the desired confidence level. For a 90% confidence level, [tex]\(Z_{\alpha/2} = 1.645\)[/tex]. Plugging in the values:
[tex]\(100 \cdot 0.08 \pm 1.645 \sqrt{100 \cdot 0.08 \cdot (1 - 0.08)}\)[/tex]
[tex]\(8 \pm 1.645 \sqrt{100 \cdot 0.08 \cdot 0.92}\)[/tex]
[tex]\(8 \pm 1.645 \sqrt{7.36}\)[/tex]
[tex]\(8 \pm 1.645 \cdot 2.713\)[/tex]
[tex]\(8 \pm 4.46\)[/tex]
The interval is from [tex]\(8 - 4.46\) to \(8 + 4.46\)[/tex], which gives us a range from approximately 3.54 to 12.46. Since we cannot have a fraction of a defective product, we round down to 3. Therefore, at most 12 defective products can be found in the sample for the deal to proceed.
(d) To generate a 90% two-sided confidence interval for the number of defective pieces out of 5000, given that 6 defective pieces were found out of 100, we first calculate the sample proportion of defective products:
[tex]\(\hat{p} = \frac{6}{100} = 0.06\)[/tex]
The formula for the confidence interval is:
[tex]\(\hat{p} \pm Z_{\alpha/2} \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}\)[/tex]
where [tex]\(n\)[/tex] is the sample size (100 in this case), and [tex]\(Z_{\alpha/2}\)[/tex] is the Z-score for a 90% confidence level (1.645). Plugging in the values:
[tex]\(0.06 \pm 1.645 \sqrt{\frac{0.06(1 - 0.06)}{100}}\)[/tex]
[tex]\(0.06 \pm 1.645 \sqrt{\frac{0.06 \cdot 0.94}{100}}\)[/tex]
[tex]\(0.06 \pm 1.645 \sqrt{\frac{0.0564}{100}}\)[/tex]
[tex]\(0.06 \pm 1.645 \cdot \sqrt{0.000564}\)[/tex]
[tex]\(0.06 \pm 1.645 \cdot 0.0237\)[/tex]
[tex]\(0.06 \pm 0.0391\)[/tex]
The interval is from [tex]\(0.06 - 0.0391\) to \(0.06 + 0.0391\)[/tex], which gives us a range from approximately 0.0209 to 0.0991. To find the number of defective pieces in the order of 5000, we multiply these proportions by 5000:
Lower bound: [tex]\(0.0209 \cdot 5000 = 104.5\)[/tex](round to 105)
Upper bound: [tex]\(0.0991 \cdot 5000 = 495.5\)[/tex] (round to 496)
Therefore, the 90% two-sided confidence interval for the number of defective pieces in the order of 5000 is from 105 to 496.
If m∠A = 87° and m∠B = 32°, find m∠1.
Answer:
61 degrees
Step-by-step explanation:
87+32=119
180 (total degrees for triangle)-119=67 degrees
g Which of the following is NOT true about the distribution function for a normal distribution? It ranges from 0 to 1. It increases as the quantity increases. It generally has a bell shape when graphed. It returns the probability that the outcome from the normal distribution is a certain quantity or lower.
Final answer:
The incorrect statement about the normal distribution is that its distribution function generally has a bell shape when graphed. This shape relates to the probability density function, not the cumulative distribution function, which actually has an S-shaped curve.
Explanation:
When we discuss the properties of a normal distribution, we are dealing with a continuous probability distribution that is widely used across many fields. The statement in question is identifying characteristics of the cumulative distribution function (CDF) of a normal distribution. The correct attributes of this function are that it ranges from 0 to 1, it increases as the quantity increases, and it returns the probability that the outcome from the normal distribution is a certain quantity or lower. However, the statement that the distribution function generally has a bell shape when graphed is incorrect regarding the CDF. The bell shape is a characteristic of the probability density function (PDF) and not the CDF, which increases from 0 to 1 in an S-shaped curve.
The standard normal distribution, which is a special case of the normal distribution with a mean (μ) of zero and a standard deviation (σ) of one, is used for a variety of applications in psychology, business, engineering, and other fields. Understanding the properties of the normal distribution — particularly the standard normal distribution — is crucial for interpreting data and using statistical methods.
It is also important to note that the integral of the normal distribution across its entirety (from ∞ to ∞) is equal to one. This area under the curve represents the total probability of all outcomes and justifies why the CDF ranges from 0 at the minimum to 1 at the maximum end of the distribution.
In the United States, 36 percent of the people have a blood type that is A positive. From a random sample of 150 people from Norway, 66 had a blood type that was A positive. Consider a hypothesis test to investigate whether the proportion of people in Norway with a blood type of A positive is different from that in the United States.
Determine the standard deviation used to calculate the test statistic for the one-sample z-test.
Final answer:
The standard deviation used to calculate the test statistic for the one-sample z-test, when investigating the proportion of people in Norway with A positive blood type against the U.S. proportion, is 0.0392.
Explanation:
To calculate the standard deviation used to calculate the test statistic for a one-sample z-test in this scenario, where we are testing whether the proportion of people in Norway with a blood type of A positive is different from that in the United States, we use the formula for the standard deviation of a proportion, which is [tex]\(\sqrt{\frac{p(1-p)}{n}}\)[/tex], where p is the proportion in the population (0.36 in this case, representing 36%), and n is the sample size (150 in this case).
Plugging in the values: [tex]\(\sqrt{\frac{0.36(1-0.36)}{150}}\) = \(\sqrt{\frac{0.36(0.64)}{150}}\) = \(\sqrt{\frac{0.2304}{150}}\) = \(\sqrt{0.001536}\) = 0.0392.[/tex]
So, the standard deviation used to calculate the test statistic for this hypothesis test is 0.0392.
The standard deviation used to calculate the test statistic for the one-sample z-test is approximately 0.0379.
To determine the standard deviation for the one-sample z-test, we use the formula for the standard deviation of a sample proportion, which is given by:
[tex]\[ \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \][/tex]
Given that the population proportion p of people with A positive blood type in the United States is 0.36, and the sample size n from Norway is 150, we can plug these values into the formula:
[tex]\[ \sigma_{\hat{p}} = \sqrt{\frac{0.36(1-0.36)}{150}} \][/tex]
[tex]\[ \sigma_{\hat{p}} = \sqrt{\frac{0.36 \times 0.64}{150}} \][/tex]
[tex]\[ \sigma_{\hat{p}} \approx 0.0379 \][/tex]
Therefore, the standard deviation used in the calculation of the test statistic for the one-sample z-test is approximately 0.0379.
Traders often buy foreign currency in hope of making money when the currency's value changes. For example, on a particular day, one U.S. dollar could purchase 0.8167 Euros, and one Euro could purchase 145.8038 yen. Let f (x )represent the number of Euros you can buy with x dollars, and let g (x )represent the number of yen you can buy with x Euros.
If you have $100, you can calculate the number of Euros by plugging x = 100 into the f(x) function: f(100) = 0.8167 * 100 = 81.67 Euros. Therefore, with $100, you can buy 81.67 Euros.
If you possess 50 Euros, you can determine the number of yen by substituting x = 50 into the g(x) function: g(50) = 145.8038 * 50 = 7290.19 yen. Hence, with 50 Euros, you can buy 7290.19 yen.
Traders engage in foreign currency trading to potentially profit from fluctuations in currency values. In this example, the exchange rates are as follows: 1 U.S. dollar buys 0.8167 Euros, and 1 Euro buys 145.8038 yen. We are asked to define two functions: f(x) represents the number of Euros you can buy with x dollars, and g(x) represents the number of yen you can buy with x Euros.
1. To determine the number of Euros you can buy with x dollars, multiply the exchange rate between dollars and Euros by x. In this case, the exchange rate is 0.8167 Euros per dollar. Therefore, the function f(x) is f(x) = 0.8167x.
2. To find the number of yen you can buy with x Euros, multiply the exchange rate between Euros and yen by x. In this case, the exchange rate is 145.8038 yen per Euro. The function g(x) is g(x) = 145.8038x.
For example:
- If you have $100, you can calculate the number of Euros by plugging x = 100 into the f(x) function: f(100) = 0.8167 * 100 = 81.67 Euros. Therefore, with $100, you can buy 81.67 Euros.
- If you possess 50 Euros, you can determine the number of yen by substituting x = 50 into the g(x) function: g(50) = 145.8038 * 50 = 7290.19 yen. Hence, with 50 Euros, you can buy 7290.19 yen.
These functions enable traders to evaluate the quantity of foreign currency they can acquire or exchange based on the prevailing exchange rates.
To Learn more about exchange rates here:
https://brainly.com/question/2202418
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HELPPP ASAPP PLZZ:) Question and answers are in the image.
Answer:
C. all real numbers except 3/4
Step-by-step explanation:
f(x) and g(x) are both defined for all real numbers. However, the ratio f/g will be undefined where g(x) = 0. That occurs when ...
4x -3 = 0
4x = 3 . . . . . add 3
x = 3/4 . . . . . divide by 4
The value of x = 3/4 makes f/g undefined, so must be excluded from the domain.
An exponential distribution is formed by the time it takes for a person to choose a birthday gift. The average time it takes for a person to choose a birthday gift is 41 minutes. Given that it has already taken 24 minutes for a person to choose a birthday gift,what is the probability that it will take more than an additional 34 minutes
Answer:
43.62% probability that it will take more than an additional 34 minutes
Step-by-step explanation:
To solve this question, we need to understand the exponential distribution and the conditional probability formula.
Exponential distribution:
The exponential probability distribution, with mean m, is described by the following equation:
[tex]f(x) = \mu e^{-\mu x}[/tex]
In which [tex]\mu = \frac{1}{m}[/tex] is the decay parameter.
The probability that x is lower or equal to a is given by:
[tex]P(X \leq x) = \int\limits^a_0 {f(x)} \, dx[/tex]
Which has the following solution:
[tex]P(X \leq x) = 1 - e^{-\mu x}[/tex]
The probability of finding a value higher than x is:
[tex]P(X > x) = 1 - P(X \leq x) = 1 - (1 - e^{-\mu x}) = e^{-\mu x}[/tex]
Conditional probability formula:
[tex]P(B|A) = \frac{P(A \cap B)}{P(A)}[/tex]
In which
P(B|A) is the probability of event B happening, given that A happened.
[tex]P(A \cap B)[/tex] is the probability of both A and B happening.
P(A) is the probability of A happening.
In this question:
Event A: Taking more than 24 minutes.
Event B: Taking ore than 24+34 = 58 minutes.
P(A)
More than 24, use the exponential distribution.
Mean of 41, so [tex]m = 41, \mu = \frac{1}{41} = 0.0244[/tex]
[tex]P(A) = P(X > 24) = e^{-0.0244*24} = 0.5568[/tex]
Intersection:
More than 24 and more than 58, the intersection is more than 58. So
[tex]P(A \cap B) = P(X > 58) = e^{-0.0244*58} = 0.2429[/tex]
Then:
[tex]P(B|A) = \frac{0.2429}{0.5568} = 0.4362[/tex]
43.62% probability that it will take more than an additional 34 minutes