Determine whether the statement (p∧(p⟶q))⟶q is a tautology one time by using truth table and the other time without using truth table

Answers

Answer 1

It is.........................................................

Determine Whether The Statement (p(pq))q Is A Tautology One Time By Using Truth Table And The Other Time
Answer 2

One way to prove this without a truth table is to use a conditional proof. We assume the portion p ^ (p --> q). If that's true, then so is p and p-->q

Using p and p-->q, the modus ponens rule allows us to derive q. It says that if p is true and p --> q, then q must be true as well.

Since we arrive at q, we have found the conclusion we're after. The assumption (p ^ (p-->q)) leads to q, and therefore the entire statement (p ^ (p-->q)) --> q is true for any combination of p,q.


Related Questions

An Internet service provider is implementing a new program based on the number of connected devices in each household currently,customers are charged a flat rate of $175 per month.the new plan would charge a flat rate of $94 plus an additional $4.50 per device connected to the network.find the number of devices,x,for which the cost of the new plan is less than the cost of the current plan.

Answers

Answer:

(x=6) is less than 18 which would give you the cost of the current plan

Step-by-step explanation:

If you take six, first you must multiply 4.50 by 6, ($27) then add it, to $94, giving you $121. Now we have to find which phone will give us the same cost, for this I choose 18. if you do 18 x 4.50, you get $81, and if you add this to 94, it gives you 175.

A bike wheel is 26 inches in diameter. What is the bike wheel's diameter in millimeters (1 inch = 25.4 millimeters)?

Answers

Answer:

Diameter of wheel in millimetres is 660.4

Step-by-step explanation:

Diameter of wheel in inches = 26

given

1 inch = 25.4 millimeters

multiplying RHS and LHS by 26

26*1 inch = 26*25.4 millimeters

=>26 inch = 660.4 mm.

Thus, diameter of wheel in millimetres is 660.4

Nisha is looking out the window of her apartment building at a sculpture in a park across the street. The top of Nisha's window is 80 feet from the ground. The angle of depression from the top of Nisha's window to the bottom of the sculpture is 20°. How far away from the building is the sculpture? Round your answer to the nearest hundredth.

Answers

Answer:

219.80 feet

Step-by-step explanation:

Tan 20= 80/b

Tan 20= 0.363970234266

(0.363970234266)b=80

b= 219.80 feet

The distance between the sculpture and the bottom of the building is required.

The distance between the building and sculpture is 219.80 feet.

Trigonometry

[tex]\theta[/tex] = Angle of depression = Angle of elevation = [tex]20^{\circ}[/tex]

p = Height of building = 80 feet

b = Required length

From the trigonometric ratios we have

[tex]\tan\theta=\dfrac{p}{b}\\\Rightarrow b=\dfrac{p}{\tan\theta}\\\Rightarrow b=\dfrac{80}{\tan 20}\\\Rightarrow b=219.80\ \text{feet}[/tex]

Learn more about trigonometry:

https://brainly.com/question/23899312

What is the measure of FEG?

A. 30 degrees

B. 40 degrees

C. 50 degrees

D. 70 degrees

Please include ALL work!! <3

Answers

Answer:

C. 50 degrees

Step-by-step explanation:

Because 6x + 5x = 110° and x = 10

5×10 = FEG 50°

What value does the 2 in the number 0.826?

Answers

Answer:

.02

Step-by-step explanation:

2 is in "Hundredths' place in .826

So, the number is multiplied with 1/100 or .01

=> 2 x 1/100

=> 2/100

=> .02

=> 2 x .01

=> .02

The value of 2 in .826 is .02

What is the solution to the following system of equations? 3x-2y=12 6x - 4y = 24

Answers

Answer:

D question,somewhat confusing, itsit's like simultaneous equation,but values are different

Answer:

x = 4 + 2y/3

Step-by-step explanation:

Angela took a general aptitude test and scored in the 90th percentile for aptitude in accounting. (a) What percentage of the scores were at or below her score? % (b) What percentage were above?

Answers

Answer:

Step-by-step explanation:

From the given information;

Angela took a general aptitude test and scored in the 90th percentile for aptitude in accounting.

What percentage of the scores were at or below her score?

The percentage of scores that were at or below her score is

Percentage P = 90%

This benchmark is more than average, so we can typically say more of the scores were at or below her score

What percentage were above?

Since The percentage of scores that were at or below her score is

Percentage P = 90%

Then the percentage of the scores that were above will be : (100 -90)%

= 10 %

We can see here that  less percentage of score were above Angela's Score.

The points (0,3) and (1,12) are solutions of an exponential function. What is the equation of the exponential function?

Answers

Answer:

[tex]f(x) =3\,*\,\,4^x[/tex]

Step-by-step explanation:

to find the equation of an exponential function, just points on the function's graph  are needed.

Recall that the exponential function has a general expression given by:

[tex]f(x) = a \,e^{b\,x}[/tex]

so we impose the condition for the function going through the first point (0,3) as:

[tex]f(0) = a \,e^{b\,(0)}= 3\\a\,e^0=3\\a\,(1)=3\\a = 3[/tex]

Now,knowing the parameter a, we can find the parameter b using the other point:

[tex]f(1) = 3 \,e^{b\,x}= 12\\3\,e^{b\,(1)}=12\\e^b=12/3\\e^b=4\\b=ln(4)[/tex]

Therefore, the function can be written as:

[tex]f(x) = 3 \,e^{ln(4)\,x}=3\,\,\,4^x[/tex]

Answer:

C)

h(x) = 3(4)x

Reducing scrap of 4-foot planks of hardwood is an important factor in reducing cost at a wood-floor manufacturing company. Accordingly, engineers at Lumberworks are investigating a potential new cutting method involving lateral sawing that may reduce the scrap rate. To examine its viability, two independent, random, representative samples of planks were examined. One sample contained 200 planks which were sawed using the old method. The other sample contained 400 planks which were sawed using the new method. Sixty-two of the 200 planks were scrapped under the old method of sawing, whereas 36 of the 400 planks were scrapped under the new method.

Required:
a. Construct the 90% confidence interval for the difference between the population scrap rates between the old and new methods, respectively.
b. Write the null and alternative hypotheses to test for differences in the population scrap rates between the old and new cutting methods, respectively.
c. Using the part a results, can we conclude at the 10% significance level that the scrap rate of the new method is different than the old method?

Answers

Answer:

The critical value for two tailed test at alpha=0.1 is ± 1.645

The calculated  z= 9.406

Step-by-step explanation:

Formulate the hypotheses as

H0: p1= p2 there is no difference between the population scrap rates between the old and new cutting methods

Ha : p1≠ p2

Choose the significance level ∝= 0.1

The critical value for two tailed test at alpha=0.1 is ± 1.645

The test statistic is

Z = [tex]\frac{p_1- p_2}\sqrt pq(\frac{1}{n_1} + \frac{1}{n_2})[/tex]

p1= scrap rate of old method = 62/200=0.31

p2= scrap rate of new method = 36/400= 0.09

p = an estimate of the common scrap rate on the assumption that the two rates are same.

p = n1p1+ n2p2/ n1 + n2

p =200 (0.31) + 400 (0.09) / 600

p= 62+ 36/600= 98/600 =0.1633

now q = 1-p= 1- 0.1633= 0.8367

Thus

z= 0.31- 0.09/ √0.1633*0.8367( 1/200 + 1/400)

z= 0.301/√ 0.13663( 3/400)

z= 0.301/0.0320

z= 9.406

The calculated value of z falls in the critical region therefore we reject the null hypothesis and conclude that the 10% significance level that the scrap rate of the new method is different from the old method.

Determine the value of x in the figure. Question 1 options: A) x = 90 B) x = 85 C) x = 45 D) x = 135

Answers

Answer:

A.) x=90°

Step-by-step explanation:

Note:

The triangle shown is an isosceles triangle, which means that it has 2 congruent sides (as shown by the small intersecting lines), and this also means that it has two congruent angles.

We are given an angle measure adjacent to one of the missing angles. These two form supplementary angles, which means that they're sum is equal to 180°, or a straight line. So, to find:

[tex]180=135+y[/tex]

y is the unknown angle. Solve for y:

[tex]180-135=y\\\\y=45[/tex]

y is 45°. Since this and the other angle are congruent, add:

[tex]45+45=90[/tex]

Note:

Triangles angles will always add up to a total of 180°.

To find the missing angle x°, use:

[tex]180=a+b+c[/tex]

These are the angles in a triangle. Substitute any known values and solve:

[tex]180=45+45+x\\\\180=90+x\\\\180-90=x\\\\x=90[/tex]

The missing angle x° is 90°.

:Done

Which equations has no solution?​

Answers

Answer: I think it is C

Step-by-step explanation:

There is no answer because A can be many solutions, B is x = -25, you just cannot solve C, and D is y = 7/6

Combine like terms to create an equivalent expression. 1/7 - 3 (3/7n - 2/7)

Answers

━━━━━━━☆☆━━━━━━━

▹ Answer

1 - 9/7n

▹ Step-by-Step Explanation

1/7 - 3(3/7n - 2/7)

Remove the parentheses (Distribute -3 among the parentheses):

1/7 - 9/7n + 6/7

Calculate:

1 - 9/7n

Hope this helps!

CloutAnswers ❁

━━━━━━━☆☆━━━━━━━

Answer:

1-9/7n

Step-by-step explanation:

[tex]\frac{1}{7}-3(\frac{3}{7}n-\frac{2}{7} ) \\=\frac{1}{7}-\frac{9}{7}n +\frac{6}{7} \\=\frac{1-9n+6}{7} \\=\frac{7-9n}{7}\\=1-\frac{9}{7}n[/tex]

A random variable is not normally distributed, but it is mound shaped. It has a mean of 14 and a standard deviation of 3. If you take a sample of size 10, can you say what the shape of the sampling distribution for the sample mean is

Answers

Answer:

Step-by-step explanation:

from the question,

the mean 14

the standard deviation is 3

and sample size is 10.

since the n which is the sample size is 10, then the distribution is mound shaped.

why?

this is due to the fact that the random variable from which we took the sample is mound shaped.

The sampling distribution of the mean is normally distributed although the question says the random variable is not normally distributed. so the shape is bell shaped and normally distributed.

the standard deviation of the mean is

3/√10

= 0.948

The mathematics teacher proposes to his students that whoever determines their years of Experience as a teacher will have an extra point, for this they will have to solve the following expression

-5 + {4 * 6 + 3 + 1 + (3- (4-8) + (3-2)]}

How many years of experience does the teacher have?

Answers

Answer:

29 years of experience.

Step-by-step explanation:

So let's take the expression step by step.  Remember that you need to follow the order of precedence here for the operations.  Parentheses, exponentials, multiplication, and addition.

-5 + { 4 * 6 + 3 + 1 + [ 3 - ( 4 - 8 ) + ( 3 - 2 ) ] }

-5 + { 4 * 6 + 3 + 1 + [ 3 - ( -4 ) + ( 1 ) ] }

-5 + { 4 * 6 + 3 + 1 + [ 3 + 4 + 1 ] }

-5 + { 4 * 6 + 3 + 1 + [ 8 ] }

-5 + { 24 + 3 + 1 + 8 }

-5 + { 36 }

29

So the teacher has 29 years of experience.

Cheers.

Stephanie is twice as old as her sister Rosa. If Stephanie is 18 years old, how old is Rosa?

Answers

Answer:

rose. is. 18/2=9 years old

Answer:

Stephanie is 18years old and she is twice older than her sister

so rosa is 18÷2(since stephanie is twice older than rosa

so rosa is 9 years old

A random sample of 149 recent donations at a certain blood bank reveals that 76 were type A blood. Does this suggest that the actual percentage of type A donations differs from 40%, the percentage of the population having type A blood? Carry out a test of appropriate hypotheses using a significance level of 0.01. Would your conclusion have been different if a significance level of 0.05 has been used?

Answers

Answer:

Yes it suggest that the actual percentage of type A donations differs from 40%, the percentage of the population having type A blood.

Well if a significance level of 0.05 is used it will not affect the conclusion

Step-by-step explanation:

From the question we are told that

     The  sample size is   [tex]n = 149[/tex]

     The  number that where  type A blood is  k =  76

       The population proportion is   [tex]p = 0.40[/tex]

       The  significance level is  [tex]\alpha = 0.01[/tex]

Generally the sample proportion is mathematically represented as

      [tex]\r p = \frac{k}{n}[/tex]

=>    [tex]\r p = \frac{76}{149}[/tex]

=>    [tex]\r p = 0.51[/tex]

The  Null hypothesis is   [tex]H_o : p = 0.41[/tex]

The  Alternative hypothesis is  [tex]H_a : p \ne 0.40[/tex]

Next we obtain the critical value of [tex]\alpha[/tex] from the z-table.The value is  

       [tex]Z_{\alpha } = Z_{0.01} = 1.28[/tex]

Generally the test statistics is mathematically evaluated as

           [tex]t = \frac{\r p - p }{ \sqrt{ \frac{p(1-p)}{n} } }[/tex]    

substituting values

           [tex]t = \frac{0.51 - 0.40 }{ \sqrt{ \frac{0.40 (1-0.40 )}{149} } }[/tex]    

           [tex]t =2.74[/tex]

So looking at the values for t  and  [tex]Z_{0.01}[/tex] we see that  [tex]t > Z_{0.01}[/tex] so we reject the null hypothesis. Which means that there is no sufficient evidence to support the claim

Now if [tex]\alpha = 0.05[/tex] , the from the z-table the critical value for [tex]\alpha = 0.05[/tex] is  [tex]Z_{0.05} = 1.645[/tex]

So comparing the value of  t and  [tex]Z_{0.05} = 1.645[/tex]  we see that [tex]t > Z_{0.05}[/tex] hence the conclusion would not be different.

If A and B are independent events with P( A) = 0.35 and P( B) = 0.55, then P( A| B) is:_________.
a. .19
b. 1.57
c. .64
d. .91

Answers

Answer:

P( A| B)= 0.35. None of the options are correct

Step-by-step explanation:

Two events A and B are said to be independent if the occurrence of one of the events does not affect the other occurring. For example, the event of tossing two coins is an independent event since they occur simultaneously. Two events are therefore independent if the following are true.

P(A|B) = P(A)

P(B|A) = P(B)

P(A and B) = P(A)P(B)

If A and B are independent events with P( A) = 0.35 and P( B) = 0.55,

then P( A| B) is a probability of A occurring provided that B has occurred. This is known as conditional probability for an independent event.

From the condition above for independent events, P(A|B) = P(A) and since P(A) =  0.35, hence P(A|B) =0.35

The base of a right rectangular prism has an area of 173.6 square centimeters and a height of 9 centimeters. What is the volume, in cubic centimeters, of the right rectangular prism?

Answers

Answer:

D) 1562.4 cubic centimeters

Step-by-step explanation:

volume = area of the base × height

volume = 173.6cm² × 9 cm

volume = 1562.4 cm³

Calculate the nominal rate of interest convertible once every four years that is equivalent to a nominal rate of discount convertible quarterly. Let d^(4) be the nominal rate of discount convertible quarterly.

Answers

Answer:

i am having issues using the math editor and my time is almost running out. i added an attachment.

Step-by-step explanation:

I need help please, m bda =

And m bca =

Answers

Step-by-step explanation:

Exterior angle BOA = 250°

Interior angle BOA = 360°- 250° = 110°

Now,

(A) BDA = interior angle BOA / 2 = 55°( Property of circles)

(B) From the figure, we observe that AOBC is a cyclic quadrilateral (i.e. sum of opposite angles is 180°).

Therefore, BCA + BOA = 180°

BCA = 180° - 110° = 70°

(3) In a group of 60 seldiers have enough food for
20 days - How many soldiers should leave the group
so that the food is enough for 100 days ? Find it.​

Answers

Answer:

48 soldiers

Step-by-step explanation:

60 soldiers                    20 days

x soldiers                       100 days

60 x 20 = 1200 = 100x

Therefore x = 1200/100 = 12

60 - 12 = 48

Hope that helped!!! k

Answer:

30 Soldiers

Step-by-step explanation:

Given:

1) No of soldiers=60

No of days=20

2) No of days=100

No of soldiers=?

No of soldiers to leave=?

Solution:

Let us use the cross multiplication method.

Let x be the no of soldiers.

No of days                                     No of  soldiers

1) 20                                                     60

2) 100                                                    x

by cross multiplying,

20x=100 x 60

20x=600

x=600/20

x=30 soldiers

Therefore, No of soldiers to leave =60-30=30 soldiers

What is f(0) given f(x) = 5(x + 2)2 – 10?

Answers

Answer:

10

Step-by-step explanation:

f(o) is given when x= 0 in f(x)

f(0) = 5 ( 0 + 2 ) 2 - 10

= 20 - 10

= 10

Answer:

[tex] \boxed{ \bold{ \huge{ \sf{f{(0) = 10}}}}}[/tex]

Step-by-step explanation:

Given, f ( x ) = 5 ( x + 2 )² - 10

Let's find f ( 0 ) :

[tex] \sf{f(0) = 5( {0 + 2)}^{2} - 10}[/tex]

Add the numbers

⇒[tex] \sf{f(0) = 5( {2)}^{2} - 10}[/tex]

Evaluate the power

⇒[tex] \sf{f(0) = 5 \times 4 - 10}[/tex]

Multiply the numbers

⇒[tex] \sf{ 20 - 10}[/tex]

Subtract 10 from 20

⇒[tex] \sf{10}[/tex]

Hope I helped !

Best regards !!

Andrea is told that the means of two groups in a study were statistically significant. She knows the means and standard deviations of the two groups and is interested in calculating an estimate of effect size. Given this information, which effect size estimate should she calculate

Answers

Answer:

Cohen's D

Step-by-step explanation:

Cohen's D is a statistic that measures effect size. It shows standardised difference between 2 means.

Effect size is defined as how large the effect of a something is or its magnitude.

Cohen's D works effectively when the sample is >50 (that is for large samples). However a correction factor can be used to make results from small samples more accurate

The formular for Cohen's D is:

D = (mean1 - mean2) ÷ (√({standard deviation1}^2 + {standard deviation 2}^2)/2)

This is the most appropriate method in the given scenario

Please help me solve for the median !!!

Answers

Answer:

50.93

Step-by-step explanation:

Add up the frequencies:

2 + 5 + 14 + 15 + 21 + 18 + 15 + 9 + 2 = 101

Divide by 2: 101/2 = 50.5

So the median is the 51st number, with 50 below and 50 above.

Add up the frequencies until you find the interval that contains the 51st number.

2 + 5 + 14 + 15 = 36

2 + 5 + 14 + 15 + 21 = 57

So the median is in the group 49.5 − 51.5.  To estimate the median, we use interpolation.  Find the slope of the line from (36, 49.5) to (57, 51.5).

m = (51.5 − 49.5) / (57 − 36)

m = 2/21

So at x = 51:

2/21 = (y − 49.5) / (51 − 36)

y = 50.93

What is the perimeter of the image attached?? (PLEASE HELP I WILL MARK BRAINLIEST)

Answers

Answer:

[tex] Perimeter = 3x + 3 [/tex]

Step-by-step explanation:

Perimeter of the given triangle in the figure is the sum of all three sides.

The expressions for the 3 sides are given as, [tex] x, (x - 3), (x + 6) [/tex].

Therefore,

[tex] Perimeter = x + (x - 3) + (x + 6) [/tex]

Simplify,

[tex] Perimeter = x + x - 3 + x + 6 [/tex]

Collect like terms

[tex] Perimeter = x + x + x - 3 + 6 [/tex]

[tex] Perimeter = 3x + 3 [/tex]

When traveling with the wind, it takes an airplane 3 hours to travel 1800 miles. It takes the same airplane 3.6 hours to travel the same 1800 miles when traveling against the wind. Assuming the airplane travels at a constant speed during both trips, what is the speed of the airplane and the speed of the wind?

Answers

Answer:

The speed of the airplane in still air is 550 mph.

The speed of the wind is 50 mph.

Step-by-step explanation:

speed = distance / time

distance = speed * time

or simply

d = st

Let v be the speed of the airplane with no wind.

Let w = speed of wind

With the wind:

d = 1800; s = v + w; t = 3

1800 = 3(v + w)

Against the wind:

d = 1800; s = v - w; t = 3.6

1800 = 3.6(v - w)

We have a system of two equations:

3(v + w) = 1800

3.6(v - w) = 1800

Divide both sides of the first equation by 3. Divide both sides of the second equation by 3.6.

v + w = 600

v - w = 500

Add the equations.

2v = 1100

v = 550

The speed of the airplane in still air is 550 mph.

v + w = 600

550 + w = 600

w = 50

The speed of the wind is 50 mph.

qaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

Answers

Answer:

32.8 miles

Step-by-step explanation:

Amy is driving to Seattle. Suppose that the remaining distance to drive (in miles) is a linear function of her driving time (in minutes). When graphed, the function gives a line with a slope of -0.95. See the figure below. Amy has 48 miles remaining after 31 minutes of driving. How many miles will be remaining after 47 minutes of driving?

Answer: The general equation of a line is given as y = mx + c, where m is the slope of the line and c is the intercept on the y axis. Given that the slope is -0.95, substituting in the general equation :

y = -0.95x + c

Amy has 48 miles remaining after 31 minutes of driving, to find c, we substitute y = 48 and x = 31. Therefore:

48 = -0.95(31) + c

c = 48 + 0.95(31)

c = 48 + 29.45

c = 77.45

The equation of the line is

y = -0.95x + 77.45

After 47 minutes of driving, the miles remaining can be gotten by substituting x = 47 and finding y.

y = -0.95(47) + 77.45

y = -44.65 + 77.45

y = 32.8 miles

PLEASE HELP ME I DONT HAVE THAT MANY POINTS AND ITS DUE TODAY I NEED HELP ASAP
The table contains the data for your first weeks sales. Complete the table by calculating your commission and earnings for each day of the week

Answers

Answer with explanation:

Sales    Commission(10% of sales)

$2,200      0.1×$2,200= $220  

$2,000       0.1× $2,000= $200

$3,134      0.1×$3,134=$313.4

$2,417         0.1×$2,417=$241.7                  

$2,579        0.1×$2,579 =$257.9

The completed table is given as follows

Day   Sales    Commission      Non-Sales pay        Earning

                    (10% of sales)                           (Commission +Non Sales pay)

Mon  $2,200      $220             $9.50                $220+ $9.50=$229.50

Tue   $2,000         $200                 $9.50                $200 +$9.50=$209.50

Thurs  $3,134      $313.4               $9.50                $313.4+ $9.50=$322.9  

Fri     $2,417         $241.7               $9.50                $241.7+$9.50= $251.2

Sat    $2,579        $257.9                 $9.50                $257.9+$9.50=$267.4

A manufacturer of chocolate chips would like to know whether its bag filling machine works correctly at the 407 gram setting. It is believed that the machine is underfilling the bags. A 20 bag sample had a mean of 397 grams with a standard deviation of 13. A level of significance of 0.05 will be used. Assume the population distribution is approximately normal. State the null and alternative hypotheses.

Answers

Answer:

Null hypothesis: μ = 407

Alternative hypothesis: μ < 407.

Step-by-step explanation:

In this case, the machine is SUPPOSED to fill the bag so that the bag weighs 407 grams. So, the null hypothesis will be that the machine is doing what it is supposed to be doing. And so, μ = 407 grams would be the null.

The worker thinks the machine is filling the bags to LESS THAN what it is supposed to. So, the alternative hypothesis is that the machine is NOT doing what it is supposed to and μ < 407 grams.

Hope this helps!

In a study of treatments for very painful​ "cluster" headaches, 140 patients were treated with oxygen and 158 other patients were given a placebo consisting of ordinary air. Among the 140 patients in the oxygen treatment​ group, 113 were free from headaches 15 minutes after treatment. Among the 158 patients given the​ placebo, 35 were free from headaches 15 minutes after treatment. Use a significance level to test the claim that the oxygen treatment is effective. A) Find test statistic z B) Find the P-value C) Construct the appropriate confidence interval D) determine if the oxygen treatment is effective

Answers

Answer:

A

  [tex]t = 10.1[/tex]

B

  [tex]p-value = p(t > 10.1)= 0.000[/tex]

C

  [tex]0.4714 < p_1 - p_2 <0.6988[/tex]

D

  The  oxygen treatment  is effective because

1 the is no 0 in the interval telling us that the treatments are different

 2 the upper and the lower limit are positive tell us that the proportion of those treated by the oxygen treatment is greater

Step-by-step explanation:

From the question we are told that

    The  first sample size is  [tex]n_1 = 140[/tex]

     The  number of patient which the oxygen cured is k = 113  

     The second sample size is  [tex]n_2 = 158[/tex]

     The number of patient that  placebo cured is  l =  35

The first sample proportion is

           [tex]\r p_1 = \frac{ 113}{140 }[/tex]

           [tex]\r p_1 = 0.8071[/tex]

The second sample proportion is  

           [tex]\r p_2 = \frac{ 35}{ 158 }[/tex]

          [tex]\r p_2 = 0.222[/tex]

The null hypothesis is  [tex]H_o : p_1 = p_2[/tex]

 The  alternative hypothesis is  [tex]H_a : p_1 > p_2[/tex]

Let assume the level of significance be[tex]\alpha = 0.05[/tex]

Generally the pooled proportion is mathematically evaluated as

    [tex]p = \frac{p1 * n1 + p2 * n2}{n1 + n2}[/tex]

substituting values

     [tex]p = \frac{0.8071 * 140 + 0.222 * 158}{140 + 158}[/tex]

      [tex]p = 0.4969[/tex]

Generally the standard error is mathematically represented

      [tex]SE = \sqrt{ p(1- p ) * [ \frac{1}{n_1} + \frac{1}{n_1}] }[/tex]

      substituting values

     [tex]SE = \sqrt{ 0.4969(1- 0.4969 ) * [ \frac{1}{140} + \frac{1}{158}] }[/tex]

     [tex]SE = 0.0580[/tex]

Generally the test statistics is mathematically represented as

       [tex]t = \frac{ \r p_1 - \r p_2}{ SE}[/tex]

       [tex]t = \frac{ 0.8071 -0.222}{ 0.0580}[/tex]

       [tex]t = 10.1[/tex]

The  p-value is from the normal distribution table as  

      [tex]p-value = p(t > 10.1)= 0.000[/tex]

given that [tex]t< \alpha[/tex] the null hypothesis is rejected

From the normal distribution table the critical value of  [tex]\frac{\alpha }{2}[/tex]  is  

     [tex]Z_{\frac{\alpha }{2} } = 1.96[/tex]

Generally the margin of error is mathematically the represented as

         [tex]E = Z_{\frac{\alpha }{2} } * SE[/tex]      

         [tex]E = 1.96 * 0.0580[/tex]      

          [tex]E = 0.1137[/tex]

The 95% confidence interval is mathematically  represented as

           [tex](\r p_1 - \r p_2) - E < p_1 - p_2 < (\r p_1 - \r p_2) + E[/tex]

substituting value

             [tex](0.8071 - 0.222) - 0.1137 < p_1 - p_2 < (0.8071 - 0.222) + 0.1137[/tex]

             [tex]0.4714 < p_1 - p_2 <0.6988[/tex]

The  oxygen treatment  is effective because

1 the is no 0 in the interval telling us that the treatments are different

 2 the upper and the lower limit are positive tell us that the proportion of those treated by the oxygen treatment is greater

             

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