To construct a 95% confidence interval for the population proportion, calculate the sample proportion p' and its complement q', determine the Z-score for 95% confidence, calculate the margin of error using the formula E = Z*sqrt((p'q')/n), and add/subtract E from p' to get the lower and upper bounds.
Explanation:To construct a 95 percent confidence interval for the population proportion p using the given sample data, we must first calculate the sample proportion (p') and its complement, the estimated proportion of failures (q'). Using the formula p' = x/n, we find that p' = 162/195. Next, we determine q' by calculating q' = 1 - p'.
With the sample proportion and its complement, we can use the standard formula for a confidence interval for a population proportion: p' ± Z*sqrt((p'q')/n), where Z* is the Z-score corresponding to the given degree of confidence. For a 95% confidence level, the Z-score is approximately 1.96.
By substituting the values of p', q', n, and the Z-score into the formula, we calculate the margin of error (E) and then the lower and upper bounds of the 95 percent confidence interval.
Suppose p' is 0.83 and q' is 0.17 for n = 195 and the Z-score for a 95% confidence interval is 1.96. The margin of error (E) would then be 1.96 * sqrt((0.83*0.17)/195), and the confidence interval would be p' ± E, resulting in a specific numerical range which would constitute our 95% confidence interval for the true population proportion.
The 95% confidence interval for the population proportion [tex]\( p \)[/tex] is [tex]\( (0.7783, 0.8833) \)[/tex].
To construct a confidence interval for the population proportion [tex]\( p \),[/tex] we will use the given information: sample size [tex]\( n = 195 \)[/tex], number of successes [tex]\( x = 162 \),[/tex] and a confidence level of 95%.
The formula for the confidence interval for a population proportion [tex]\( p \)[/tex] is:
[tex]\[ \hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \][/tex]
where:
- [tex]\( \hat{p} \)[/tex] is the sample proportion [tex](\( \frac{x}{n} \)),[/tex]
- [tex]\( z^* \)[/tex] is the critical value from the standard normal distribution corresponding to the desired confidence level.
Calculate the sample proportion [tex]\( \hat{p} \):[/tex]
[tex]\[ \hat{p} = \frac{x}{n} = \frac{162}{195} \][/tex]
[tex]\[ \hat{p} \approx 0.8308 \][/tex]
For a 95% confidence level, the critical value [tex]\( z^* \)[/tex] can be found using the standard normal distribution table or a calculator. It corresponds to the middle 95% of the distribution, which leaves 2.5% in each tail.
The critical value [tex]\( z^* \)[/tex] for a 95% confidence level is approximately 1.96.
Calculate the standard error [tex]\( SE \):[/tex]
[tex]\[ SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \\ SE = \sqrt{\frac{0.8308 \cdot (1-0.8308)}{195}} \\ SE \approx \sqrt{\frac{0.8308 \cdot 0.1692}{195}} \\ SE \approx \sqrt{\frac{0.1405}{195}} \\ SE \approx \sqrt{0.0007205} \\ SE \approx 0.0268 \][/tex]
Now, we can construct the 95% confidence interval for [tex]\( p \):[/tex]
[tex]\[ \hat{p} \pm z^* \cdot SE \][/tex]
[tex]\[ 0.8308 \pm 1.96 \cdot 0.0268 \][/tex]
Calculate the margin of error:
[tex]\[ 1.96 \cdot 0.0268 \approx 0.0525 \][/tex]
So, the confidence interval is:
[tex]\[ 0.8308 \pm 0.0525 \][/tex]
Finalize the interval: [tex]\[ (0.7783, 0.8833) \][/tex]
The 95% confidence interval for the population proportion [tex]\( p \)[/tex] is approximately [tex]\( (0.7783, 0.8833) \)[/tex]. This means we are 95% confident that the true population proportion [tex]\( p \)[/tex] lies between 0.7783 and 0.8833.
simplify the expression the square root of 169
Ann is grouping 38 rocks. She can put them into groups of 10 rocks or as single rocks.what are different ways Ann can group the rocks?
14 1/6-10 11/24= what fraction
Sam used his calculator to find the answer and wrote down cos 1.75 = 0.999. Does this seem like a reasonable answer to you? What might he have done wrong on his calculator?
Since pi is equivalent to 3.14 and means 180°, therefore 1.75 which is about half of pi should be at almost 90°. The cos of a 90° angle which is located on the y axis should be zero, therefore cos 1.75 must be near zero. However in this case it is near one, clearly his calculator is in degree mode that is why he got an erroneous answer.
What Sam can do to get the correct answer would be to convert his calculator into radian mode or convert the given 1.75 rad into degrees:
θ = 1.75 rad (180° / π rad)
θ = 100.2676
cos 100.2676 = -0.1782 which is now correct since it is nearer to zero
Assume you have an infinite supply of these candy pieces from which to draw. if you draw two pieces one right after the other, are the events of getting a greena green on the first and a greena green on the second disjoint, independent, or neither?
First let us define what probability is. Probability is the measure of chance on getting a success or successful event out of all the total possible events. This can be expressed mathematically as:
Probability = Number of successful events / Total events
Which in this case can be written as:
Probability = Number of green candies / Total candies
So if we draw 1 candy in the 1st trial, therefore the value of the numerator and denominator would vary in the 2nd trial (since there is reduction of green candy & total candy). However the problem specifically states that there is an INFINITE supply of candies. Which means that the value do not change in the next trial.
Therefore the probability in the 2nd trial is not affected by the event of the 1st trial. So they are independent events. They cannot be disjoint since they are not mutually exclusive, both can occur at the same time (green and green).
Answer:
Independent
G prove that limx→∞(x 2 + 1)/x2 = 1 using the definition of limit at ∞.
Write and solve an equation to answer the question.
What number a is 24% of 25?
Final answer:
To find the number that is 24% of 25, you multiply 0.24 by 25, which equals 6.
Explanation:
To find what number a is 24% of 25, we can set up an equation using the concept that a percent is a way of expressing a fractional amount of something.
We know that 24% means 24 out of 100, which can be written as the fraction 24/100 or 0.24 in decimal form.
To translate the question into an equation, we write:
a = 0.24 × 25
Next, we calculate the value of a by multiplying 0.24 (the decimal form of 24%) by 25. This gives us:
a = 6
Therefore, the number a, i.e., 24% of 25 is 6.
Pentagon RSTUV is circumscribed about a circle. What is the value of x if RS = 6, ST = 9, TU = 7, UV = 15, and VR = 14?
Starting at vertex V going until vertex S, we make our calculations.
VR = 14
Rx = x
therefore, xS = VR – Rx
xS = 14 – x ---> 1
Now starting at vertex V going left to vertex S, we make our next set of calculations. Assuming x is on VR but nearer R
VR = 14 - x
at UV = 15 – (14 – x)
at UV = 1 + x
at TU = 7 – (1 + x)
at TU = 6 – x
at ST = 9 – (6 – x)
at ST = 3 + x ---> 2
Equate 1 and 2:
14 – x = 3 + x
2 x = 11
x = 5.5
To find the value of x in the circumscribed pentagon RSTUV with given sides, we set up an equation based on the sum of the lengths of the opposite sides. After solving the equation, we determined that x is 25.
Pentagon RSTUV and the Value of x
Given that pentagon RSTUV is circumscribed about a circle, we need to find the value of x,
where RS = 6, ST = 9, TU = 7, UV = 15, and VR = 14.
For a polygon circumscribed about a circle, the sum of the lengths of the pairs of opposite sides must be equal. Therefore, we can set up the equation:
RS + TU + x = ST + UV + VR
Substituting the given values:
6 + 7 + x = 9 + 15 + 14
Simplifying the right-hand side:
6 + 7 + x = 3813 + x = 38x = 38 - 13x = 25Thus, the value of x is 25.
Which of the following is the proper notation used to express a point in the three-dimensional coordinate plane?
A. (x)
B. (x, y, z)
C. (x, y)
D. (w, x, y, z)
Answer: Option 'B' is correct.
Step-by-step explanation:
Since we have given that
We need to express a point in the three-dimensional coordinate plane.
As we know that In 3- dimensional plane, there are 3 axes;
One for x-axis (say length)
One for y- axis (say breadth)
One for z-axis (say height)
So, the proper notation would be (x,y,z).
Hence, Option 'B' is correct.
The length of a rectangle is 5 cm longer than twice the length of the width. If the perimeter of the rectangle is 88 centimeters, what is the width?
Four students did a survey to find the soda flavor sixth-grade students prefer. The table below shows the method each student used to conduct the survey: Student Method Trey Asked 100 students at random from his seventh-grade class what their favorite soda flavor is Jesse Asked 100 sixth-grade students at random what their favorite soda flavor is Nita Asked 100 eighth-grade students at random what their favorite soda flavor is Ruben Asked 100 third-grade students at random what their favorite soda flavor is Which student's survey not biased?
Answer:
The answer is B
Step-by-step explanation:
3y^2-16y+16
could someone help me walk through this?
solve, explain why the algorithm works
3892
-2567
A company wants to save for a new piece of equipment that will cost 17,500. They have 15,000 to invest today in an account that earns 4.75% annual simple interest. How long must they invest the money in order to purchase the new equipment? Round to the tenth
If CD is the perpendicular bisector of AB, then the value of x is:
10.
5.
22.
11.
Ocean currents influence all of the following except
temperature
climate
radiation from the sun
weather
Use the zeros of the following quadratic to find the x-value of the vertex.
y = x^2 + 2x -8
A. 2
B. 1
C. -1
D. -8
PLEASE HELP ASAP
To determine whether the pipe welds in a nuclear power plant meet specifications, a random sample of welds is selected, and tests are conducted on each weld in the sample. weld strength is measured as the force required to break the weld. suppose the specifications state
Transmutation is the process of changing the substance, tangible or not, from one form to the other. It means the transformation of one element in the periodic table into another by one or a series of nuclear decays or reactions. One type of transmutation is nuclear transmutation. Nuclear transmutation is the conversion of one chemical element or isotope into another though nuclear reactions or nuclear decay. Second type of transmutation is artificial transmutation. Artificial transmutation occur in machinery that uses nough energy to cause changes in the nuclear structure of the elements.
Engineers test the weld strength of pipe welds in nuclear power plants to ensure they meet safety specifications. Tensile strength and pressure tolerance are key factors tested to maintain plant safety and functionality. Regulatory standards, like those from the NRC, guide acceptable performance levels for power plant components.
To ensure the safety and functionality of the nuclear power plant's infrastructure, engineers conduct tests on components such as pipe welds. These tests involve assessing weld strength, which is the force required to break a weld, and must meet specific specifications for the plant to operate safely. If not, the structural integrity of the power plant could be compromised, potentially leading to failures in electrical generation or even safety hazards.
The tensile strength of a material is a critical factor in this scenario. It is defined as the force per unit area (measured in Newtons per square meter or Pascals) needed to cause failure of the material by pulling it apart. This strength needs to be tested to prevent issues like the SS Schenectady ship's cracking under stress, which can have catastrophic consequences.
Regulatory standards, such as those set by the Nuclear Regulatory Commission (NRC), often determine the acceptable levels of component performance, including valves and tubing, within a power plant. These standards may change over time in response to new findings or incidents, reflecting the evolving understanding of material properties and failure risks under operational conditions.
In statistical terms, such tests might involve a hypothesis test to determine if the pressure tolerances of two valve types differ significantly, or if the structural components meet the required strength, safety, and operational standards.
A 102-inch length of ribbon is to be cut into three pieces. The longest piece is to be 38 inches longer than the shortest piece, and the third piece is to be half the length of the longest. Find the length of each ribbon.
A person wishes to mix coffee worth nine dollars per pound with coffee worth three dollars per pound to get 240 pounds of a mixture worth five dollars per pound how many pounds of the nine dollars and the three dollar coffee will be needed
Explain how to use the standard normal table to find the probability associated with the shaded area under the curve.
Answer:
The standard normal table gives areas under the curve to the left of z-scores.
Find the probability in the standard normal table that a value is to the left of 1.9.
Find the probability in the standard normal table that a value is to the left of 0.4.
Subtract the probability of a value being to the left of 0.4 from the probability of a value being to the left of 1.9.
Step-by-step explanation:
edge2020
In order to find the probability associated with the shaded area under the curve using the standard normal table, you should subtract the probability of a value left of 0.4 from the probability of a value left of 1.9.
How do you find the probability associated with the shaded area?We are trying to find the probability of the area between 1.9 and 0.4 so the first thing to do is to find the probability that a value is to the left of 1.9 from the Standard normal table.
After this is done, you then find the probability that the value is left of 0.4.
To find the probability of the middle area, subtract the value to the left of 0.4 from that of the value left of 1.9
Find out more on the standard normal table at https://brainly.com/question/7001627.
Donna rented a movie. She started the movie at 10:36 pm, and it was 2 hours 46. minutes long. When did the movie end?
Convert 85°to radians
1 degree = 0.0174533 radians
85* 0.0174533 = 1.4835305 radians
Round answer as needed
Answer:
17pi/36 (PLATO answer)
Step-by-step explanation:
..Simplify √x^100y^16
In a first aid kit the ratio of large bandages to small is 3 to 2 based on this ratio how many large bandages are in the kit if there are a total of 80 bandages
Mitch planted cabbage,squash and carrots on his 150 acre-farm. He planted half the farm with squash and 22% with carrots. How many acres did he plant with cabbagge?
Answer:
42 acres
Step-by-step explanation:
Mitch planted cabbage,squash and carrots on his 150 acre-farm.
He planted half the farm with squash. The acres planted with squash are:
1/2 × 150 acre = 75 acre
He planted 22% with carrots. The acres planted with carrots are:
22% × 150 acre = 33 acre
The sum of the acres planted with cabbage, squash and carrots is equal to the total number of acres.
cabbage + squash + carrots = 150 acre
cabbage = 150 acre - squash - carrots
cabbage = 150 acre - 75 acre - 33 acre = 42 acre
EASY 5 POINTS!!! Choose the equation that correctly uses the Pythagorean Theorem for the given right triangle.
Answer:
Option C
Step-by-step explanation:
In the given right angle triangle
Hypotenuse = 4.5 cm
Height = 2.5 cm
and base = g cm.
Pythagoras theorem says
(hypotenuse)² = (base)² + (height)²
So by putting the values in the formula
(4.5)² + (2.5)² + g²
Option C is the answer.
An acorn falls from the branch of a tree to the ground 25 feet below. The distance, S, that the acorn is from the ground as it falls is represented by the equation S(t) = –16t2 + 25, where t is the number of seconds. For which interval of time is the acorn moving through the air?
Answer: B or 0 < t < 5/4
Step-by-step explanation:
I got it wrong from the other answers and found out that this was the right one.
In triangle FE = 5 and angle D=43 . Find DE to the nearest tenth
Answer:
5.4? I don't really know I'm just going off other answers but clearing it up,,,hope it help :')
Step-by-step explanation:
if the area of triangle ABC= 25 square feet, then the area of the rectangle ABCD is:
25 square feet.
35 square feet.
40 square feet.
50 square feet.