Final answer:
Option B: the number that represents the middle value in the set of numbers
Explanation:
The median is a statistical measure that represents the middle value in a set of numbers.
To find the median, first, the numbers must be arranged in ascending order. If the number of values (n) in the dataset is odd, the median is the middle value directly.
If n is even, the median is the average of the two middle values. It effectively separates the dataset into two halves, where half of the values are equal to or less than the median, and the other half are equal to or greater than the median value.
For example, in the data set {3, 4, 5, 9, 11}, the median is 5, as it is the middle value of the ordered set.
In another set with an even number of values, for instance, {3, 4, 7, 10}, the median would be the average of 4 and 7, which is 5.5.
This concept is important especially when dealing with outliers, or extreme values, as the median is not affected by them like the mean (average) would be.
The labels on a shipment of the five chemicals A –E below were washed off by accident when the delivery truck went through a truckwash with the doors open. A clever student took infrared spectra of each of the resulting unknowns. Unknown #1 had a strong, broad IR absorption at 3400–3600 cm–1, and no absorptions between 1600 and 2900 cm–1. Unknown #2 had a strong absorption at 1680 cm–1 and no absorptions above 3100 cm–1. What are the structures of unknowns 1 and 2?
Answer:
Unknown 1: -OH group at 3400-3600
Unknown 2: Ketone and alkene group
Step-by-step explanation:
Unknown 1:
This compound has a strong IR absorption at 3400 - 3600cm-¹; this indicates that it is an alcoholic group (OH)
No absorption between 1600 and 2900cm-¹: this rule out the presence of ketone (C = O)
Hence the unknown compound is OH group
Unknown 2:
This has strong IR absorption at 1680cm-¹: this indicates the presence of (C=O) group.
No absorptions above 3100cm-¹: this indicates the presence of C = C - H
So therefore, unknown 2 is most likely to be Ketone and alkene group
Complete the equation of the line through
(-8,-2) (−4,6)
Answer:
(6+2)/(-4+8)= 8/4= 2
y+2=2(x+8)
y+2=2x+16
y=2x+14
Step-by-step explanation:
Externally applied mechanical forces acting on a body, such as normal and shear forces, will cause deformation dependent on the characteristics of that body and the magnitudes of the applied forces. Similarly, changes in temperature can cause deformation (i.e., expansion or contraction) as particles and bonds inside the material undergo changes in energy. For simple geometries, the changes in volume are proportional in the linear dimensions. Each of the 25cm×5cm25cm×5cm beams below is subjected to a 20 degree change in temperature. Rank the items based on the total change in length along the long axis of the beams. Many polymers such as polyethylene are significantly affected by changes in temperature, especially when compared to the metal and mineral materials discussed here. To a lesser degree, soft metals likewise are significantly affected by temperature changes. For example when heated, lead expands almost a third more than aluminum and almost twice as much gold. However, hard metals and materials, such as platinum and quartz, do not deform significantly in response to temperature. Because platinum is a metal, it will deform almost nine times more than quartz.
Final answer:
Different materials exhibit varying degrees of thermal expansion when subjected to a temperature change. Polymers such as polyethylene expand the most, followed by soft metals like lead and aluminum, with hard metals like platinum and minerals like quartz showing much less expansion.
Explanation:
When materials undergo a change in temperature, they typically experience a change in size, known as thermal expansion or contraction. This phenomenon happens because the kinetic energy of the particles within the material changes, consequently changing the distances between particles. The amount by which a material expands or contracts is dependent on its coefficient of thermal expansion, which varies widely among different materials. Polymers such as polyethylene show significant thermal expansion, more so than soft metals like lead and aluminum, and much more than hard materials like platinum and quartz. When temperature increases, thermal stress may arise in materials that are constrained and cannot freely expand, leading to deformation or even damage. Platinum, being a metal, would deform more than quartz due to its higher thermal expansion, but less than softer metals and polymers.
To rank the items based on the total change in length along the long axis of beams with a temperature change of 20 degrees, we would expect polyethylene to have the greatest change due to its high coefficient of thermal expansion, followed by soft metals like lead and aluminum. Platinum would have some expansion but considerably less than softer metals and polymers, and quartz would experience the least amount of expansion given its low thermal expansion characteristic.
A bag contains 20 marbles. These marbles are identical, except they are labeled with the integers 1 through 20. Five marbles are drawn at random from the bag. There are a few ways to think about this.
a. Marbles are drawn one at a time without replacement. Once a marble is drawn, it is not replaced in the bag. We consider all the lists of marbles we might create. (In this case, picking marbles 1, 2, 3, 4, 5 in that order is different from picking marbles 5, 4, 3, 2, 1.)
b. Marbles are drawn all at once without replacement. Five marbles are snatched up at once. (In this case, picking marbles 1, 2, 3, 4, 5 and picking marbles 5, 4, 3, 2, 1 are considered the same outcome.)
c. Marbles are drawn one at a time with replacement. Once a marble is drawn, it is tossed back into the bag (where it is hopelessly mixed up with the marbles still in the bag). Then the next marble is drawn, tossed back in, and so on. (In this case, picking 1, 1, 2, 3, 5 and picking 1, 2, 1, 3, 5 are different outcomes.)
Required:
For each of these interpretations, describe the sample space that models these experiments.
Answer:
a. The probability of getting an specific sequence would be
[tex](1/20)*(1/19)*(1/18)*(1/17)*(1/16)[/tex]
b. The probability of having an specific sequence would be.
[tex]5/20 = 1/4[/tex]
Step-by-step explanation:
a. If you draw 5 marbles without replacement, the probability of getting an specific sequence would be
[tex](1/20)*(1/19)*(1/18)*(1/17)*(1/16)[/tex]
b. If you draw 5 marbles all at once without replacement, the probability of having an specific sequence would be.
[tex]5/20 = 1/4[/tex]
ASK YOUR TEACHER An article reported that for a sample of 42 kitchens with gas cooking appliances monitored during a one-week period, the sample mean CO2 level (ppm) was 654.16, and the sample standard deviation was 165.23. (a) Calculate and interpret a 95% (two-sided) confidence interval for true average CO2 level in the population of all homes from which the sample was selected. (Round your answers to two decimal places.)
Final answer:
To calculate a 95% confidence interval for the true average CO2 level in the population of all homes, we use the sample mean and sample standard deviation to calculate the margin of error and determine the lower and upper bounds of the confidence interval.
Explanation:
To calculate a 95% confidence interval for the true average CO2 level in the population of all homes, we can use the sample mean and sample standard deviation provided.
First, we calculate the margin of error using the formula: Margin of Error = Critical Value x (Sample Standard Deviation / sqrt(Sample Size)). The critical value for a 95% confidence interval is 1.96.
Next, we calculate the lower and upper bounds of the confidence interval by subtracting and adding the margin of error to the sample mean respectively.
Therefore, the 95% confidence interval for the true average CO2 level in the population of all homes is (589.17, 719.15).
This means that we can be 95% confident that the true average CO2 level in the population of all homes falls within this range.
Estimate
137 X 18
Choose 1 answer
To estimate the product 137 multiplied by 18, we can round the numbers to the nearest ten. 137 is approximately 140, and 18 is approximately 20. The estimated product of 137 x 18 is 2800.
Explanation:To estimate the product 137 multiplied by 18, we can round the numbers to the nearest ten. 137 is approximately 140, and 18 is approximately 20. Then, we multiply the rounded numbers: 140 x 20 = 2800. Therefore, the estimated product of 137 x 18 is 2800.
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what are the steps to find the lower and upper quartiles in a data set
Answer:
The steps to finding the upper and lower quartiles are given in the first choice.
1. Order the data from least to greatest. If you don't do this, the data is random.
2. Find the median - you will do this so that you can find the midpoint of the data set (half of of the data is smaller and half of the data is larger).
3. Find the lower quartile - this is the half of the lower half of numbers; think of it as breaking the lower half of the data into 2 sections.
4. Find the upper quartile - this is the half of the greater half of numbers; this will break the upper half of the data into 2 sections.
Answer:
A.
Step-by-step explanation:
Just took the test:)
What is the measure of 0 in radians? In the diagram, 0 is a central angle, 3 is the radius, and pi is the arc
Given:
Given that the radius of the circle is 3 units.
The arc length is π.
The central angle is θ.
We need to determine the expression to find the measure of θ in radians.
Expression to find the measure of θ in radians:
The expression can be determined using the formula,
[tex]S=r \theta[/tex]
where S is the arc length, r is the radius and θ is the central angle in radians.
Substituting S = π and r = 3, we get;
[tex]\pi=3 \theta[/tex]
Dividing both sides of the equation by 3, we get;
[tex]\frac{\pi}{3}=\theta[/tex]
Thus, the expression to find the measure of θ in radians is [tex]\theta=\frac{\pi}{3}[/tex]
Answer:
Step-by-step explanation:
An engineer has designed a valve that will regulate water pressure on an automobile engine. The valve was tested on 140 engines and the mean pressure was 6.9 pounds/square inch (psi). Assume the population standard deviation is 0.7. The engineer designed the valve such that it would produce a mean pressure of 6.8 psi. It is believed that the valve does not perform to the specifications. A level of significance of 0.05 will be used. Find the value of the test statistic. Round your answer to two decimal places.
Answer:
Step-by-step explanation:
We would set up the hypothesis test. This is a test of a single population mean since we are dealing with mean
For the null hypothesis,
µ = 6.8 psi
For the alternative hypothesis,
µ ≠ 6.8
This is a 2 tailed test
Since the population standard deviation is given, z score would be determined from the normal distribution table. The formula is
z = (x - µ)/(σ/√n)
Where
x = sample mean
µ = population mean
σ = population standard deviation
n = number of samples
From the information given,
µ = 6.8
x = 6.9
σ = 0.7
n = 140
z = (6.9 - 6.8)/(0.7/√140) = 0.17
Therefore, the value of the test statistic us 0.17
he mere belief that you are receiving an effective treatment for pain can reduce the pain you actually feel. Researchers tested this placebo effect on 37 volunteers. Each volunteer was put inside a magnetic resonance imaging (MRI) machine for two consecutive sessions. During the first session, electric shocks were applied to their arms and the blood oxygen level-dependent (BOLD) signal was recorded during pain. The second session was the same as the first, but prior to applying the electric shocks, the researchers smeared a cream on the volunteer's arms. The volunteers were informed that the cream would block the pain when, in fact, it was just a regular skin lotion (i.e., a placebo). Note that each participant is contributing a pair of data: one measurement in the first session, and one measurement in the second session. From the 37 participants, the mean and standard deviation of differences in BOLD measurements are calculated. If the placebo is effective in reducing the pain experience, the BOLD measurements should be higher, on average, in the first MRI session than in the second. Is there evidence to confirm that the placebo is effective? That is, that the mean BOLD measurements are higher in the first session than the second? Test at LaTeX: \alphaα=.05.
The researchers conducted an experiment to test the effectiveness of a placebo in reducing pain. A statistical test called the paired t-test can be used to analyze the data and determine if the mean BOLD measurements are higher in the first session than in the second. The results of this test will provide evidence to confirm or refute the effectiveness of the placebo.
Explanation:The researchers conducted an experiment to test the effectiveness of a placebo in reducing pain. They measured the blood oxygen level-dependent (BOLD) signal during two consecutive sessions with electric shocks applied to the volunteers' arms. In the second session, a cream that was actually a placebo was applied to the volunteers. The mean and standard deviation of the differences in BOLD measurements were calculated to determine if there was evidence to confirm that the placebo was effective.
To test if the mean BOLD measurements were higher in the first session than in the second, a statistical test can be used. The paired t-test is suitable for this scenario, as it compares the means of two related samples. The t-test calculates a t-value which can be compared to a critical value to determine if there is evidence of a significant difference. In this case, with a significance level of α = 0.05, if the t-value is greater than the critical value, it would indicate evidence that the mean BOLD measurements are higher in the first session than the second.
If the calculated t-value is greater than the critical value, there is evidence to confirm that the placebo is effective in reducing pain. Conversely, if the calculated t-value is not greater than the critical value, there is not enough evidence to confirm that the placebo is effective. It is important to note that statistical significance does not necessarily imply practical significance, so further investigation may be required to understand the magnitude of the effect.
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Last year, Michelle earned $45,183.36 at a hair salon. What was her average monthly income? *
Please help with my Area of sectors and Segments Question!!! Show your work please!!!!
WILL MARK BRAINLIEST!!!
15PTS!
Answer:
The answer is 30 sq in i had that question, good luck :)
A 12-sided die is rolled. The set of equally likely outcomes is {1,2,3,4,5,6,7,8,9,10,11,12}. Find the probability of rolling a number less than 11.
A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of 177.5-cm and a standard deviation of 1.2-cm. For shipment, 7 steel rods are bundled together. Find the probability that the average length of a randomly selected bundle of steel rods is less than 178.3-cm.
Answer:
96.08% probability that the average length of a randomly selected bundle of steel rods is less than 178.3-cm.
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal probability distribution
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
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.
In this problem, we have that:
[tex]\mu = 177.5, \sigma = 1.2, n = 7, s = \frac{1.2}{\sqrt{7}} = 0.4536[/tex]
Find the probability that the average length of a randomly selected bundle of steel rods is less than 178.3-cm.
This is the pvalue of Z when X = 178.3. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{178.3 - 177.5}{0.4536}[/tex]
[tex]Z = 1.76[/tex]
[tex]Z = 1.76[/tex] has a pvalue of 0.9608
96.08% probability that the average length of a randomly selected bundle of steel rods is less than 178.3-cm.
Answer: the probability that the average length of a randomly selected bundle of steel rods is less than 178.3-cm is 0.96
Step-by-step explanation:
Since the lengths of the steel rods are normally distributed, then according to the central limit theorem,
z = (x - µ)/(σ/√n)
Where
x = sample mean lengths of the steel rods
µ = population mean length of the steel rods
σ = standard deviation
n = number of samples
From the information given,
µ = 177.5 cm
x = 178.3 cm
σ = 1.2 cm
n = 7
the probability that the average length of a randomly selected bundle of steel rods is less than 178.3-cm is expressed as
P(x < 178.3)
Therefore,
z = (178.3 - 177.5)/(1.2/√7) = 1.76
Looking at the normal distribution table, the probability corresponding to the z score is 0.96
Therefore,
P(x < 178.3) = 0.96
The manager of the motor pool wants to know if it costs more to maintain cars that are driven more often Data are gathered on each car in the motor pool regarding number of miles driven IX) in a given year and maintenance costs for that year (Y) in thousands of dollars. The regression equation is computed as: Y=60+0.08X, and the p-value for the slope estimate is 0.7 What conclusion can we draw from this study?
Select one:
a. Cars that are driven more tend to cost more to maintain.
b. There's no statistically significant linear relationship between the number of miles driven and the maintenance cost.
c. The correlation between the response variable and independent variable is significant.
d. The slope estimate is significantly different from zero.
Answer:
b. There's no statistically significant linear relationship between the number of miles driven and the maintenance cost.
Step-by-step explanation:
The p value that is corresponding in line with the slope of regression line is 0.345, which is quite above the significance level of 0.05. as a result, we failed to discard the null hypothesis that there is no important association or connection between x and y variables.
Going by the above explanation we can then conclude that there's no statistically significant linear relationship between the number of miles driven and the maintenance cost.
Given: R=2m
KL = LM = KM
Find: V and
Surface Area of the cone
Answer:
[tex]V = \frac{2L\sqrt{3}}{3} \pi}[/tex]
[tex]A = 4\pi + \sqrt{3L^{2} + 16}[/tex]
Step-by-step explanation:
Figure of cone is missing. See attachment
Given
Radius, R = 2m
Let L = KL=LM=KM
Required:
Volume, V and Surface Area, A
Calculating Volume
Volume is calculated using the following formula
[tex]V = \frac{1}{3} \pi R^{2} H[/tex]
Where R is the radius of the cone and H is the height
First, we need to determine the height of the cone
The height is represented by length OL
It is given that KL=LM=KM in triangle KLM
This means that this triangle is an equilateral triangle
where OM = OK = [tex]\frac{1}{2} KL[/tex]
OK = [tex]\frac{1}{2}L[/tex]
Applying pythagoras theorem in triangle LOM,
|LM|² = |OL|² + |OM|²
By substitution
L² = H² + ( [tex]\frac{1}{2}L[/tex])²
H² = L² - [tex]\frac{1}{4}L[/tex]²
H² = L² (1 - [tex]\frac{1}{4}[/tex])
H² = L² [tex]\frac{3}{4}[/tex]
H² = [tex]\frac{3L^{2} }{4}[/tex]
Take square root of bot sides
[tex]H = \sqrt{\frac{3L^{2} }{4}}[/tex]
[tex]H = \frac{L\sqrt{3}}{2}[/tex]
Recall that [tex]V = \frac{1}{3} \pi R^{2} H[/tex]
[tex]V = \frac{1}{3} \pi 2^{2} * \frac{L\sqrt{3}}{2}[/tex]
[tex]V = \frac{1}{3} \pi * 4} * \frac{L\sqrt{3}}{2}[/tex]
[tex]V = \frac{1}{3} \pi} * {2L\sqrt{3}}[/tex]
[tex]V = \frac{2L\sqrt{3}}{3} \pi}[/tex]
in terms of [tex]\pi[/tex] an d L where L = KL = LM = KM
Calculating Surface Area
Surface Area is calculated using the following formula
[tex]H = \frac{L\sqrt{3}}{4}[/tex]
[tex]A=\pi r(r+\sqrt{h^{2} +r^{2} } )[/tex]
[tex]A=\pi * 2(2+\sqrt{((\frac{L\sqrt{3}}{2})^{2} +2^{2} } )[/tex])
[tex]A=\pi * 2(2+\sqrt{{\frac{3L^{2}}{4} } + 4 }[/tex] )
[tex]A=\pi * 2(2+\sqrt{{\frac{3L^{2}+16}{4} } })[/tex]
[tex]A=2\pi(2+\sqrt{{\frac{3L^{2}+16}{4} } })[/tex]
[tex]A = 2\pi (2 + \frac{\sqrt{3L^{2} + 16}}{\sqrt{4}} )[/tex]
[tex]A = 2\pi (2 + \frac{\sqrt{3L^{2} + 16}}{2} )[/tex]
[tex]A = 2\pi (2 + {\frac{1}{2} \sqrt{3L^{2} + 16})[/tex]
[tex]A = 4\pi + \sqrt{3L^{2} + 16}[/tex]
What is the area of this figure
Answer:
185
Step-by-step explanation:
sorry. had to do something but im bsck :)
Comber of values
Meon numbe cof Values I sum of all the values
A. 132
- 12
6154
(206
Solve the following
I A. 56
B.36
C.72
WAG
Answer:
Wag
Step-by-step explanation:
C72 или Wag ладно помогите
what is the definition of a point
Answer:
A point in geometry is a location. It has no size i.e. no width, no length and no depth.
Step-by-step explanation:
If you need more definitions, I suggest you go to mathisfun which is an amazing website! Hope this helped :)
Triangle congruence: ASA and AAS
The ASA and AAS are methods of proving triangle congruence. ASA requires two angles and the included side in one triangle to match the respective parts in the other triangle. AAS requires two angles and a non-included side in one triangle to match the respective parts in the other triangle.
Explanation:The question refers to two of the many methods to prove that triangles are congruent: Angle-Side-Angle (ASA) and Angle-Angle-Side (AAS). In triangle congruence, congruent means that two triangles have the same size and shape.
For the ASA congruence, two angles and the included side in one triangle must be congruent to the corresponding two angles and the included side in another triangle. For example, if we have two triangles, Triangle ABC and Triangle DEF, if angle A is congruent to angle D, angle B is congruent to angle E, and side AB is congruent to side DE, then the two triangles are congruent by ASA.
For AAS congruence, two angles and a non-included side in one triangle must be congruent to the corresponding two angles and the non-included side in another triangle. For instance, in Triangle ABC and Triangle DEF, if angle A is congruent to angle D, angle B is congruent to angle E, and side BC is congruent to side EF, then the two triangles are congruent by AAS.
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The average population growth rate for whitetail deer is 0.35. Hunting laws are set to limit the time allowed for hunting deer with a goal of achieving about a 35% mortality rate on deer to keep the population in check. Years with a higher than 35% mortality will result in an overall decline in the deer population while years with a lower than 35% mortality rate will result in an increased population. If the growth rate exceeds the mortality rate, and the net effect were a 4% growth rate, how long would it take the population of deer to double
The population of deer will double when the growth rate exceeds the mortality rate, and the net effect is a 4% growth rate. It will take approximately 17.3 years for the population to double.
Explanation:The population of deer will double when their growth rate exceeds the mortality rate and the net effect is a 4% growth rate. We can calculate the time it takes for the population to double using the formula for exponential growth:
Start with the equation for exponential growth: P = P0 * ertPlug in the values we know: 2P0 = P0 * e0.04tSolve for t by dividing both sides by P0 and taking the natural logarithm of both sides: t = ln(2) / 0.04Use a calculator to find that t is approximately 17.3 yearsLearn more about population growth rate here:https://brainly.com/question/16013460
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Find the value of X in this problem. Please show work.
Given:
Given that the measurements of the triangle.
We need to determine the value of x.
Value of x:
The value of x can be determined using by angle bisector theorem.
The angle bisector theorem states that "if the angle that bisects the triangle will divide the opposite sides into two segments that are proportional to the remaining two sides of the triangle".
Hence, applying the theorem, we have;
[tex]\frac{x}{20-x}=\frac{14}{11}[/tex]
Cross multiplying, we get;
[tex]11x=14(20-x)[/tex]
Simplifying, we get;
[tex]11x=280-14x[/tex]
[tex]25x=280[/tex]
[tex]x=11.2[/tex]
Thus, the value of x is 11.2
It is desired to estimate the mean GPA of each undergraduate class at a large university. How large a sample is necessary to estimate the GPA within at the confidence level? The population standard deviation is . If needed, round your final answer up to the next whole number.
Answer:
[tex]n=(\frac{2.58(1.2)}{0.25})^2 =153.36 \approx 154[/tex]
So the answer for this case would be n=154 rounded up to the nearest integer
Step-by-step explanation:
Assuming the following question: It is desired to estimate the mean GPA of each undergraduate class at a large university. How large a sample is necessary to estimate the GPA within 0.25 at the 99% confidence level? The population standard deviation is 1.2. If needed, round your final answer up to the next whole number.
Previous concepts
A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".
The margin of error is the range of values below and above the sample statistic in a confidence interval.
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
[tex]\bar X[/tex] represent the sample mean for the sample
[tex]\mu[/tex] population mean (variable of interest)
s represent the sample standard deviation
n represent the sample size
Solution to the problem
The margin of error is given by this formula:
[tex] ME=z_{\alpha/2}\frac{\sigma}{\sqrt{n}}[/tex] (a)
And on this case we have that ME =0.25 and we are interested in order to find the value of n, if we solve n from equation (a) we got:
[tex]n=(\frac{z_{\alpha/2} \sigma}{ME})^2[/tex] (b)
The critical value for 99% of confidence interval now can be founded using the normal distribution. And in excel we can use this formla to find it:"=-NORM.INV(0.005;0;1)", and we got [tex]z_{\alpha/2}=2.58[/tex], replacing into formula (b) we got:
[tex]n=(\frac{2.58(1.2)}{0.25})^2 =153.36 \approx 154[/tex]
So the answer for this case would be n=154 rounded up to the nearest integer
Two major automobile manufacturers have produced compact cars with engines of the same size. We are interested in determining whether or not there is a significant difference in the mean MPG (miles per gallon) when testing for the fuel efficiency of these two brands of automobiles. A random sample of eight cars from each manufacturer is selected, and eight drivers are selected to drive each automobile for a specified distance. The following data (in miles per gallon) show the results of the test. Assume the population of differences is normally distributed.
Driver Manufacturer A Manufacturer B
1 32 28
2 27 22
3 26 27
4 26 24
5 25 24
6 29 25
7 31 28
8 25 27
A) The mean for the differences is __________a. 0.50b. 1.5c. 2.0d. 2.5B) The test statistic is _________a. 1.645b. 1.96c. 2.096d. 2.256
C) At 90% confidence the null hypothesis _________a. should not be rejectedb. should be rejectedc. should be revisedd. None of these alternatives is correct
Answer:
(A) The mean for the differences is 2.0.
(B) The test statistic is 1.617.
(C) At 90% confidence the null hypothesis should not be rejected.
Step-by-step explanation:
We are given that a random sample of eight cars from each manufacturer is selected, and eight drivers are selected to drive each automobile for a specified distance.
The following data (in miles per gallon) show the results of the test;
Driver Manufacturer A Manufacturer B
1 32 28
2 27 22
3 26 27
4 26 24
5 25 24
6 29 25
7 31 28
8 25 27
Let [tex]\mu_1[/tex] = mean MPG for the fuel efficiency of Manufacturer A brand
[tex]\mu_2[/tex] = mean MPG for the fuel efficiency of Manufacturer B brand
SO, Null Hypothesis, [tex]H_0[/tex] : [tex]\mu_1-\mu_2=0[/tex] or [tex]\mu_1= \mu_2[/tex] {means that there is a not any significant difference in the mean MPG (miles per gallon) when testing for the fuel efficiency of these two brands of automobiles}
Alternate Hypothesis, [tex]H_A[/tex] : [tex]\mu_1-\mu_2\neq 0[/tex] or [tex]\mu_1\neq \mu_2[/tex] {means that there is a significant difference in the mean MPG (miles per gallon) for the fuel efficiency of these two brands of automobiles}
The test statistics that will be used here is Two-sample t test statistics as we don't know about the population standard deviations;
T.S. = [tex]\frac{(\bar X_1-\bar X_2)-(\mu_1-\mu_2)}{s_p\sqrt{\frac{1}{n_1}+\frac{1}{n_2} } }[/tex] ~ [tex]t__n__1+_n__2-2[/tex]
where, [tex]\bar X_1[/tex] = sample mean MPG for manufacturer A = [tex]\frac{\sum X_A}{n_A}[/tex] = 27.625
[tex]\bar X_2[/tex] = sample mean MPG for manufacturer B =[tex]\frac{\sum X_B}{n_B}[/tex] = 25.625
[tex]s_1[/tex] = sample standard deviation for manufacturer A = [tex]\sqrt{\frac{\sum (X_A-\bar X_A)^{2} }{n_A-1} }[/tex] = 2.72
[tex]s_2[/tex] = sample standard deviation manufacturer B = [tex]\sqrt{\frac{\sum (X_B-\bar X_B)^{2} }{n_B-1} }[/tex] = 2.20
[tex]n_1[/tex] = sample of cars selected from manufacturer A = 8
[tex]n_2[/tex] = sample of cars selected from manufacturer B = 8
Also, [tex]s_p=\sqrt{\frac{(n_1-1)s_1^{2}+(n_2-1)s_2^{2} }{n_1+n_2-2} }[/tex] = [tex]\sqrt{\frac{(8-1)\times 2.72^{2}+(8-1)\times 2.20^{2} }{8+8-2} }[/tex] = 2.474
(A) The mean for the differences is = 27.625 - 25.625 = 2
(B) The test statistics = [tex]\frac{(27.625-25.625)-(0)}{2.474 \times \sqrt{\frac{1}{8}+\frac{1}{8} } }[/tex] ~ [tex]t_1_4[/tex]
= 1.617
(C) Now at 10% significance level, the t table gives critical values between -1.761 and 1.761 at 14 degree of freedom for two-tailed test. Since our test statistics lies within the range of critical values of t, so we have insufficient evidence to reject our null hypothesis as it will not fall in the rejection region due to which we fail to reject our null hypothesis.
Therefore, we conclude that there is a not any significant difference in the mean MPG (miles per gallon) when testing for the fuel efficiency of these two brands of automobiles.
The mean for the differences is 2.5 MPG. The test statistic is 2.256. At 90% confidence, the null hypothesis should be rejected.
Explanation:In order to determine if there is a significant difference in the mean MPG between the two manufacturers, we need to calculate the mean for the differences and the test statistic. The mean for the differences is calculated by subtracting the Manufacturer B MPG from the Manufacturer A MPG for each driver and then taking the average of those differences. In this case, the mean for the differences is 2.5 MPG. The test statistic is calculated by dividing the mean for the differences by the standard deviation of the differences and then multiplying by the square root of the sample size. In this case, the test statistic is 2.256.
At 90% confidence, we can compare the test statistic to the critical value for a two-tailed test. The critical value for a 90% confidence level is 1.645. Since the test statistic (2.256) is greater than the critical value (1.645), we reject the null hypothesis. Therefore, the answer to question C is: the null hypothesis should be rejected.
Learn more about Hypothesis Testing here:https://brainly.com/question/34171008
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A student wants to determine if pennies are really fair when flipped, meaning equally likely to land heads up or tails up. He flips a random sample of 50 pennies and finds that 28 of them land heads up. If p denotes the true probability of a penny landing heads up when flipped, what are the appropriate null and alternative hypotheses?
Answer:
For this case we want to determine if pennies are really fair when flipped, meaning equally likely to land head up or tails, so then the correct system of hypothesis are:
Null hypothesis: [tex]p=0.5[/tex]
Alternative hypothesis: [tex]p \neq 0.5[/tex]
Step-by-step explanation:
Previous concepts
A hypothesis is defined as "a speculation or theory based on insufficient evidence that lends itself to further testing and experimentation. With further testing, a hypothesis can usually be proven true or false".
The null hypothesis is defined as "a hypothesis that says there is no statistical significance between the two variables in the hypothesis. It is the hypothesis that the researcher is trying to disprove".
The alternative hypothesis is "just the inverse, or opposite, of the null hypothesis. It is the hypothesis that researcher is trying to prove".
Solution to the problem
For this case we want to determine if pennies are really fair when flipped, meaning equally likely to land head up or tails, so then the correct system of hypothesis are:
Null hypothesis: [tex]p=0.5[/tex]
Alternative hypothesis: [tex]p \neq 0.5[/tex]
Final answer:
The appropriate null hypothesis for the experiment is that pennies are fair when flipped, with the alternative hypothesis being that they are not.
Explanation:
The appropriate null hypothesis for this experiment is that the true probability of a penny landing heads up when flipped is 0.5, meaning that pennies are fair when flipped. The alternative hypothesis, denoted as the alternative to the null hypothesis, would be that the true probability of a penny landing heads up when flipped is not 0.5.
To summarize:
Null hypothesis (H0): p = 0.5
Alternative hypothesis (Ha): p ≠ 0.5
What is theoretical vs experimental probability? Please help!
Answer:
See down below.
Step-by-step explanation:
Theoretical probability is what we expect to happen. For example, we do a test of flipping a coin. You know that its either gonna be heads or tails.
Experimental probability is what actually happens when we try it out. It occurs when we are doing an experiment and then something happens.
Hope this helps!
Theoretical probability is calculated based on assumptions and mathematical calculations, while experimental probability is based on actual data and observations.
Explanation:Theoretical probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. It is based on assumptions and mathematical calculations. For example, if you flip a fair coin, the theoretical probability of getting heads is 1 out of 2, or 0.5.
Experimental probability, on the other hand, is calculated by conducting an actual experiment and observing the outcomes. It is based on actual data and observations. For example, if you flip a coin 10 times and get heads 6 times, the experimental probability of getting heads is 6 out of 10, or 0.6.
In summary, theoretical probability is based on calculations and assumptions, while experimental probability is based on actual data and observations.
As theta increases from -theta/2 to 0 radians, the value of cos theta will
1. decrease form 1 to 0
2. decrease from 0 to -1
3. increase from -1 to 0
4. increase from 0 to 1
Step-by-step explanation:
By trigonometric conversion,
Cos -90° = 0
Cos -60° = 0.5
Cos -30° = 0.866
Cos 0° = 1
From the above values, Cos -90° to Cos 0° = 0 to 1
As theta increases from [tex]\frac{-theta}{2}[/tex] to 0, the value of cos theta will increase from 0 to 1 (Option 4)
Stochastic n-by-n matrices Recall that an n × n matrix A is said to be stochastic if the following conditions are satisfied (a) Entries of A are non negative, that is ai,j ≥ 0 for all 1 ≤ i ≤ n and all 1 ≤ j ≤ n. (b) Each column of A sums to 1, that is Pn i=1 ai,j = 1, for all 1 ≤ j ≤ n. Let S and M be arbitrary stochastic n-by-n matrices. (a) Show that λ = 1 is an eigenvalue of S. 2 points (b) Show that S 2 is also a stochastic matrix. 2 points (c) Does MS have to be stochastic? Explain
Answer:
a) Entries of A are non negative, that is ai,j ≥ 0 for all 1 ≤ i ≤ n and all 1 ≤ j ≤ n.
c) yes MS is stochastic
Step-by-step explanation:
a) A stochastic matrix is a square matrix whose columns are probability vectors. A probability vector is a numerical vector whose entries are real numbers between 0 and 1 whose sum is 1.
B)
a) Now, suppose Sx=λx for some λ>1. Since the rows of S are nonnegative and sum to 1, each element of vector Sx is a convex combination of the components of x, which can be no greater than maximum of x the largest component of x. On the other hand, at least one element of λx is greater than maximum of x, which proves that λ>1 is impossible and Hence λ = 1.
b) Then [tex]S^{2}[/tex] =[tex]P^{2}_{ij}[/tex] is also stochastic; it is the two-step transition matrix for the chain {Xn, n = 0,1,…}. To every stochastic matrix S, there corresponds a Markov chain {Xn} for which S is the unit-step transition matrix.
However, not every stochastic matrix is the two-step transition matrix of a Markov chain.
c) Let A and B be two row-stochastic matrices and suppose we know the product of column stochastic matrices is column-stochastic. Observe that,
MS = [tex]((MS)^{T})^{T} = (S^{T}M^{T})^{T}[/tex]
by properties of transpose of a matrix. Let us consider [tex]S^{T} M^{T}[/tex]. It is easy to see that the transpose of a row-stochastic matrix is column-stochastic by definition (and vice versa). Thus, [tex]S^{T}[/tex]and [tex]M^{T}[/tex] are column stochastic and by our assumption, it must then be the case that [tex]S^{T} M^{T}[/tex] is column-stochastic. Since [tex]S^{T} M^{T}[/tex] is column-stochastic, then it's transpose [tex](S^{T} M^{T})^{T}[/tex]=MS is row stochastic.
You need a 30% alcohol solution. On hand, you have a 50 mL of a 20% alcohol mixture. You also have 35% alcohol mixture. How much of the 35% mixture will you need to add to obtain the desired solution?You need a 30% alcohol solution.
Answer:
100 mL
Step-by-step explanation:
If x is the volume of 35% solution, then:
0.35x + 0.20(50) = 0.30(x + 50)
0.35x + 10 = 0.30x + 15
0.05x = 5
x = 100
You need 100 mL of 35% solution.
A sample of 20 account balances of a credit company showed an average balance of $1,170 and a standard deviation of $125. You want to determine if the mean of all account balances is significantly greater than $1,150. Assume the population of account balances is normally distributed.Compute the p-value for this test.
Answer:
The P-value for this test is P=0.2415.
Step-by-step explanation:
We have to perform an hypothesis testing on the mean of alla account balances.
The claim is that the mean of all account balances is significantly greater than $1,150.
Then, the null and alternative hypothesis are:
[tex]H_0: \mu=1150\\\\H_a: \mu>1150[/tex]
The sample size is n=20, with a sample mean is 110 and standard deviation is 125.
We can calculate the t-statistic as:
[tex]t=\dfrac{\bar x-\mu}{s/\sqrt{n}}=\dfrac{1170-1150}{125/\sqrt{20}}=\dfrac{20}{27.95}=0.7156[/tex]
The degrees of freedom fot this test are:
[tex]df=n-1=20-1=19[/tex]
For this one-tailed test and 19 degrees of freedom, the P-value is:
[tex]P-value=P(t>0.7156)=0.2415[/tex]