David wants to buy a 6-foot long sandwich for a birthday party. If the sandwich costs $72.00, what is the cost per foot?

Answer:

The test statistic = 2.51

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 usually takes the form of the opposite of the theory to be tested. It usually contains the signs =, ≤ and ≥ depending on the directions of the test.

While, the alternative hypothesis usually confirms the the theory being tested by the experimental setup. It usually contains the signs ≠, < and > depending on the directions of the test.

For this question, Dr. Vegapunk wants to test the claim that watching anime decreases social skills in college students.

So, the null hypothesis would be that there isn't significant evidence to conclude that watching anime decreases social skills in college students. That is, watching anime does not decrease social skills in college students.

And the alternative hypothesis is that there is significant evidence to conclude that watching anime decreases social skills in college students.

If the previous known mean social skills of college students is 6.2 and the new population mean social skills of college students who watch anime is μ,

Mathematically,

The null hypothesis is represented as

H₀: μ ≥ 6.2

The alternative hypothesis is given as

Hₐ: μ < 6.2

To do this test, we will use the t-distribution because no information on the population standard deviation is known

So, we compute the t-test statistic

t = (x - μ)/σₓ

x = sample mean = 7.3

μ₀ = Standard to be compared against = 6.2

σₓ = standard error of sample mean = [σ/√n]

where n = Sample size = 30

σ = Sample standard deviation = 2.4

σₓ = (2.4/√30) = 0.4382

t = (7.3 - 6.2) ÷ 0.4382

t = 2.51

Hope this Helps!!!

Answer:

-13a+10b

Step-by-step explanation:

Answer:

3.14 miles but that doesn't sound right

Step-by-step explanation:

Assuming you have to find area of the circle its

A = πr² Radius is half of diameter so half of 2 = 1

A = 3.14 x 1²

A = 3.14 x 1

A = 3.14

If you have to find area than its 3.14 it just sounds odd

Sorry if its wrong

Answer:

The distribution is approximately normal.

The mean and standard deviation are 15 and 0.98 respectively.

Step-by-step explanation:

The population of the random variables X₁ and X₂ are distributed as follows:

[tex]X_{1}\sim (\mu_{1}=35, \sigma_{1}^{2}=3^{2})\\\\X_{2}\sim (\mu_{2}=20, \sigma_{2}^{2}=4^{2})[/tex]

Two independent random samples of sizes,

[tex]n_{1}=47\\\\n_{2}=54[/tex]

are selected form the two populations.

According to the Central Limit Theorem if we have an unknown population with mean μ and standard deviation σ and appropriately huge random samples (n > 30) are selected from the population with replacement, then the distribution of the sample mean will be approximately normally distributed.  

Then, the mean of the sample means is given by,

[tex]\mu_{\bar x}=\mu[/tex]

And the standard deviation of the sample means is given by,

[tex]\sigma_{\bar x}=\frac{\sigma}{\sqrt{n}}[/tex]

Both the sample selected from the two populations are quite large, i.e. [tex]n_{1}=47>30\\\\n_{2}=54>30[/tex]

So, according to the central limit theorem the sampling distribution of sample means [tex]\bar X_{1}\ \text{and}\ \bar X_{2}[/tex] can be approximated by the Normal distribution.

Then, the distribution of [tex]\bar X_{1}[/tex] is:

[tex]\bar X_{1}\sim N(35,\ 0.44)[/tex]

And the distribution of [tex]\bar X_{2}[/tex] is:

[tex]\bar X_{2}\sim N(20,\ 0.54)[/tex]

If two random variables are normally distributed then their linear function is also normally distributed.

So, the distribution of [tex]\bar X_{1}\ - \bar X_{2}[/tex] is Normal and the shape of the distribution is bell-shaped.

The mean of the distribution of [tex]\bar X_{1}\ - \bar X_{2}[/tex] is:

[tex]E(\bar X_{1}\ -\ \bar X_{2})=E(\bar X_{1})-E(\bar X_{2})\\=35-20\\=15[/tex]

The standard deviation of the distribution of [tex]\bar X_{1}\ - \bar X_{2}[/tex] is:

[tex]SD(\bar X_{1}-\bar X_{2})=SD(\bar X_{1})+SD(\bar X_{2})\\=0.44+0.54\\=0.98[/tex]

*X₁ and X₂ are independent.

Thus, the mean and standard deviation are 15 and 0.98 respectively.

Answer:

Step-by-step explanation:

first method is to try out all possible combinations and pick out the best one which has the minimum operations but that would be infeasible method if the no of matrices increases  

so the best method would be using the dynamic programming approach.

A1 = 100 x 50

A2 = 50 x 200

A3 = 200 x 50

A4 = 50 x 10

Table M can be filled using the following formula

Ai(m,n)

Aj(n,k)

M[i,j]=m*n*k

The matrix should be filled diagonally i.e., filled in this order

(1,1),(2,2)(3,3)(4,4)

(2,1)(3,2)(4,3)

(3,1)(4,2)

(4,1)

                 Table M[i, j]                                            

             1                      2                  3                    4

4    250000          200000        100000                0  

3      

750000        500000            0

2      1000000             0

1            

0

Table S can filled this way

Min(m[(Ai*Aj),(Ak)],m[(Ai)(Aj*Ak)])

The matrix which is divided to get the minimum calculation is selected.

Table S[i, j]

           1          2         3        

4

4          1           2         3

3          

1          2

2            1

1

After getting the S table the element which is present in (4,1) is key for dividing.

So the matrix multiplication chain will be (A1 (A2 * A3 * A4))

Now the element in (4,2) is 2 so it is the key for dividing the chain

So the matrix multiplication chain will be (A1 (A2 ( A3 * A4 )))

Min number of multiplications: 250000

Optimal multiplication order: (A1 (A2 ( A3 * A4 )))

to get these calculations perform automatically we can use java

code:

public class MatrixMult

{

public static int[][] m;

public static int[][] s;

public static void main(String[] args)

{

int[] p = getMatrixSizes(args);

int n = p.length-1;

if (n < 2 || n > 15)

{

System.out.println("Wrong input");

System.exit(0);

}

System.out.println("######Using a recursive non Dyn. Prog. method:");

int mm = RMC(p, 1, n);

System.out.println("Min number of multiplications: " + mm + "\n");

System.out.println("######Using bottom-top Dyn. Prog. method:");

MCO(p);

System.out.println("Table of m[i][j]:");

System.out.print("j\\i|");

for (int i=1; i<=n; i++)

System.out.printf("%5d ", i);

System.out.print("\n---+");

for (int i=1; i<=6*n-1; i++)

System.out.print("-");

System.out.println();

for (int j=n; j>=1; j--)

{

System.out.print(" " + j + " |");

for (int i=1; i<=j; i++)

System.out.printf("%5d ", m[i][j]);

System.out.println();

}

System.out.println("Min number of multiplications: " + m[1][n] + "\n");

System.out.println("Table of s[i][j]:");

System.out.print("j\\i|");

for (int i=1; i<=n; i++)

System.out.printf("%2d ", i);

System.out.print("\n---+");

for (int i=1; i<=3*n-1; i++)

System.out.print("-");

System.out.println();

for (int j=n; j>=2; j--)

{

System.out.print(" " + j + " |");

for (int i=1; i<=j-1; i++)

System.out.printf("%2d ", s[i][j]);

System.out.println();

}

System.out.print("Optimal multiplication order: ");

MCM(s, 1, n);

System.out.println("\n");

System.out.println("######Using top-bottom Dyn. Prog. method:");

mm = MMC(p);

System.out.println("Min number of multiplications: " + mm);

}

public static int RMC(int[] p, int i, int j)

{

if (i == j) return(0);

int m_ij = Integer.MAX_VALUE;

for (int k=i; k<j; k++)

{

int q = RMC(p, i, k) + RMC(p, k+1, j) + p[i-1]*p[k]*p[j];

if (q < m_ij)

m_ij = q;

}

return(m_ij);

}

public static void MCO(int[] p)

{

int n = p.length-1;     // # of matrices in the product

m    =    new    int[n+1][n+1];        //    create    and    automatically initialize array m

s = new int[n+1][n+1];

for (int l=2; l<=n; l++)

{

for (int i=1; i<=n-l+1; i++)

{

int j=i+l-1;

m[i][j] = Integer.MAX_VALUE;

for (int k=i; k<=j-1; k++)

{

int q = m[i][k] + m[k+1][j] + p[i-1]*p[k]*p[j];

if (q < m[i][j])

{

m[i][j] = q;

s[i][j] = k;

}

}

}

}

}

public static void MCM(int[][] s, int i, int j)

{

if (i == j) System.out.print("A_" + i);

else

{

System.out.print("(");

MCM(s, i, s[i][j]);

MCM(s, s[i][j]+1, j);

System.out.print(")");

}

}

public static int MMC(int[] p)

{

int n = p.length-1;

m = new int[n+1][n+1];

for (int i=0; i<=n; i++)

for (int j=i; j<=n; j++)

m[i][j] = Integer.MAX_VALUE;

return(LC(p, 1, n));

}

public static int LC(int[] p, int i, int j)

{

if (m[i][j] < Integer.MAX_VALUE) return(m[i][j]);

if (i == j) m[i][j] = 0;

else

{

for (int k=i; k<j; k++)

{

int   q   =   LC(p,   i,   k)   +   LC(p,   k+1,   j)   +   p[i-1]*p[k]*p[j];

if (q < m[i][j])

m[i][j] = q;

}

}

return(m[i][j]);

}

public static int[] getMatrixSizes(String[] ss)

{

int k = ss.length;

if (k == 0)

{

System.out.println("No        matrix        dimensions        entered");

System.exit(0);

}

int[] p = new int[k];

for (int i=0; i<k; i++)

{

try

{

p[i] = Integer.parseInt(ss[i]);

if (p[i] <= 0)

{

System.out.println("Illegal input number " + k);

System.exit(0);

}

}

catch(NumberFormatException e)

{

System.out.println("Illegal input token " + ss[i]);

System.exit(0);

}

}

return(p);

}

}

output:

Answer:

25% probability you will roll a prime number and spin a prime number

Step-by-step explanation:

If we have two events, A and B, and they are independent, we have that:

[tex]P(A \cap B) = P(A) \times P(B)[/tex]

In this question:

Event A: Rolling a prime number.

Event B: Spinning a prime number.

Both the cube and the spinner have four values, ranging from one to four.

2 and 3 are prime values, that is, 2 of those values. Then

[tex]P(A) = P(B) = \frac{2}{4} = \frac{1}{2}[/tex]

What is the probability you will roll a prime number and spin a prime number

The cube and the spinner are independent of each other. So

[tex]P(A \cap B) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} = 0.25[/tex]

25% probability you will roll a prime number and spin a prime number

Answer:

= 1/4

Step-by-step explanation:

In a  cube , we have  numbers labeled 1 - 6

the prime numbers we have is 2 , 3 and 5

The probability of selecting a prime number is

[tex]=\frac{3}{6} \\\\=\frac{1}{2}[/tex]

Now this means the probability of rolling a prime number here is 1/2

Now we calculate the probability of spinning a prime number

Prime numbers here are just 2 and 3

The probability of spinning a prime number is thus 2/4 = 1/2

Thus, the probability of rolling a prime number and spinning a prime number also becomes; 1/2 * 1/2

= 1/4

Question:

A small tie shop finds that at a sales level of x ties per day its marginal profit is MP(x )dollars per​ tie, where

MP(x) =1.85 + 0.12x - 0.0024x². ​Also, the shop will lose ​$65 per day at a sales level of x= 0. Find the profit from operating the shop at a sales level of x ties per day.

Answer:

P(x) = 1.85x + 0.06x^2 - 0.0008x^3 - 65

Step-by-step explanation:

MP(x) = 1.85 + 0.12(x) - 0.0024(x^2)

At sales level, x=0, loss = $65

P(0) = - $65

Integrating MP(x) with respect to x

P(x) = ∫MP(x) dx = ∫ 1.85 + 0.12(x) - 0.0024(x^2)

P(x) = 1.85x + 0.06x^2 - 0.0008x^3 + C

Therefore :

P(x) = 1.85x + 0.06x^2 - 0.0008x^3 - 65

Correct question:

A small tie shop finds that at a sales level of x ties per day its marginal profit is MP(x )dollars per​ tie, where

MP(x) =1.85 + 0.12x - 0.0024x². ​Also, the shop will lose ​$65 per day at a sales level of x= 0. Find the profit from operating the shop at a sales level of x ties per day.

Answer:

P(x) =1.85x +0.06x² - 0.0008x³ - 65

Step-by-step explanation:

Given the marginal profit function:

MP(x) =1.85 +0.12x - 0.0024x², P(0)= -65

We are to find P(x).

P(x) = ∫MP(x) dx

P(x) = ∫(1.85 + 0.12x - 0.0024x²) dx

= ∫1.85 dx+∫0.12x dx+∫(-0.0024x²)dx + C

= 1.85x + 0.06x² - 0.0008x³ + C

Initial condition at P(0) = - 65

where x(0), P(x) = -65

we have:

-65 = 1.85(0)+0.06(0)² - 0.0008x(0)³ + C

-65 = 0 + 0 - 0 + C

-65 = C

C = -65

P(x) =1.85x + 0.06x² - 0.0008x³ - 65

Answer:

The length of shadow of the woman is 14.375 ft.

Step-by-step explanation:

Here we are given that

Height of the woman = 5.75 ft

Height of the lamp post = 12 ft

Shadow of the lamp post = 30 ft

Let the angle of elevation of the light source be θ

By definition, in a right triangle, with respect to angle θ, we have;

tan(θ) = Opposite÷Adjacent

Therefore, tan(θ) = (Height of the lamp post)÷(Shadow of the lamp post)

and tan(θ) = 12 ft ÷ 30 ft = 0.4

θ = Actan(0.4) = tan⁻¹(0.4) = 21.8°

Since, angle of elevation of the light = Angle of depression of the light (alternate angles) = 21.8° we have

tan(θ) = (Height of the woman) ÷ (Length of shadow of the woman)

∴ Length of shadow of the woman = (Height of the woman) ÷ tan(θ)

Length of shadow of the woman = 5.75 ft ÷ tan(21.8) = 5.75 ft ÷ 0.4

Length of shadow of the woman = 5.75 ft ÷ 0.4 = 14.375 ft.

Answer:

19

Step-by-step explanation:

plug in 5 for x so f(5)=4(5)-1 = 20-1 = 19

The solution of expression f (x) = 4x - 1at x = 5 would be 19.

Given that the expression is,

f (x) = 4x - 1

Used the concept of substitution method to solve the expression.

Now, substitute x = 5 in the expression,

f (x) = 4x - 1

f (5) = 4(5) - 1

f (5) = 20 - 1

f (5) = 19

Therefore, the value of the expression is f (5) = 19.

To learn more about the mathematical expression visit:

brainly.com/question/1859113

#SPJ6

Answer:

Carlita's Speed or distance over time

Step-by-step explanation:

The rate of change of the data (her distance) is her speed.

Speed is the rate of change of distance, distance covered overtime.

Speed = distance/time

Unit = meter/seconds or miles per hour

Therefore, the rate of change for that data represent Carlita's Speed or distance over time

Answer:    x= 2/3

Step-by-step explanation:     For f(x) = g(x) we should equate the two functions to get the value of x.

f(x) = g(x)

−3x + 4 =  2

-3x + 4 - 4 = 2 - 4

-3x = -2

Dividing by -3 on both sides;

(-3x)/-3 = (-2)/-3

 x = 2/3

Answer:

[tex]\frac{2}{3}[/tex]

Step-by-step explanation:

If f(x)=g(x), should be :  -3x+4= 2

-3x + 4 = 2

-3x = -2

x= -2/-3

x = [tex]\frac{2}{3}[/tex]

Hope this helps ^-^

Answer:

We conclude that the reaction time is increased or remains same by frequent alcohol use.

Step-by-step explanation:

We are given that a substance abuse researcher wants to know whether reaction time is decreased by frequent alcohol use. He randomly selects 164 alcoholics and finds that their mean reaction time (in ms) to a stimulus equals 283.

Reaction times to the stimulus in the general population are distributed normally with a mean equal to 281 and a standard deviation equal to 28.

Let [tex]\mu[/tex] = mean reaction time after alcohol use.

So, Null Hypothesis, [tex]H_0[/tex] : [tex]\mu[/tex] [tex]\geq[/tex] 281      {means that the reaction time is increased or remains same by frequent alcohol use}

Alternate Hypothesis, [tex]H_A[/tex] : [tex]\mu[/tex] < 281      {means that the reaction time is decreased by frequent alcohol use}

The test statistics that would be used here One-sample z test statistics as we know about population standard deviation;

                    T.S. =  [tex]\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n}}}[/tex]  ~ N(0,1)

where, [tex]\bar X[/tex] = sample mean reaction time (in ms) = 283

            [tex]\sigma[/tex] = population standard deviation = 28

            n = sample of alcoholics = 164

So, test statistics  =  [tex]\frac{283-281}{\frac{28}{\sqrt{164}}}[/tex]  

                              =  0.915

The value of z test statistics is 0.915.

Now, at 0.01 significance level the z table gives critical value of -2.3263 for left-tailed test. Since our test statistics is more than the critical values of z, 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 the reaction time is increased or remains same by frequent alcohol use.

Answer:

from what I understand a 200% increase is tripling the number the increase is on

Step-by-step explanation:

Three m minus eight n divided by thirteen

Final answer:

Equations B (−x = y) and D (6x −3y = 12) represent linear functions because they can be written in the standard linear form y = mx + b, where m and b are constants.

Explanation:

When determining if an equation represents a linear function, we look for the standard form y = mx + b, where m represents the slope and b represents the y-intercept. In this form, both m and b must be constants, and the equation should not have variables with exponents other than 1. Given the options:

A. y = x2 - This equation is not linear because of the exponent on x.

B. −x = y - This can be represented in linear form as y = −x (or y = −0x + 0), hence it is linear.

C. y = 14x2 + 4 - This equation is not linear due to the exponent on x.

D. 6x −3y = 12 - This can be rewritten in linear form as y = 2x - 4, making it a linear equation.

E. y = −2x2 + 1 - This is not linear for the same reason as options A and C.

The equations that represent linear functions from the given options are B and D.

Answer:

Step-by-step explanation:

For a two tailed alternative hypothesis, it involves the "not equal to symbol,≠ ". This can never be in the null hypothesis.

The curve is symmetrical and the rejection regions would be in the left and right tails

In order to reject the null hypothesis, the test statistic must be smaller than the critical value on the left tail or greater than the critical value on the right tail.

Therefore, the following statements are false

B. When the alternative hypothesis is two-tailed, the null hypothesis is rejected for values of the test statistic located in either tail of that statistic's sampling distribution.

D. When the alternative hypothesis is two-tailed, the null hypothesis is rejected for values of the test statistic located in either tail of that statistic's sampling distribution and when the alternative hypothesis is two-tailed, the rejection region is split between the two tails of the test statistic's sampling distribution.

Final answer:

Statement d is false because it combines parts of two separate true statements, which could be misleading. A two-tailed test is indicated by a not equals symbol in the alternative hypothesis, and the rejection region is split between the two tails of the test statistic's sampling distribution.

Explanation:

The false statement is d. While it's true that when the alternative hypothesis is two-tailed, the null hypothesis is rejected for values of the test statistic located in either tail of that statistic's sampling distribution, statement d is false because it repeats the earlier part of the statement and combines it with a correct statement from option c, which can be misleading. The correct information is provided in separate statements b and c.

The alternative hypothesis indicates the type of test to be conducted. If the alternative hypothesis has a not equals symbol (≠), it indicates a two-tailed test. In a two-tailed test, the rejection region is split between the two tails of the test statistic's sampling distribution. In terms of p-value, it represents the probability of obtaining test results at least as extreme as the results actually observed, under the assumption that the null hypothesis is correct. A small p-value (< 0.05) typically indicates strong evidence against the null hypothesis, leading to its rejection.

Answer:

465

Step-by-step explanation:

424+41=465

Answer:

27.19% probability that a female driver was involved in an accident

Step-by-step explanation:

A probability is the number of desired outcomes divided by the number of total outcomes.

In this question:

74,250 car accidents. Of those, in 20,187, the driver was a female.

Based on this, what is the probability that a female driver was involved in an accident?

p = 20187/74250 = 0.2719

27.19% probability that a female driver was involved in an accident

The probability that a female driver was involved in one of the car accidents in 2005 is approximately 27.2%.

To determine the probability that a female driver was involved in an accident, we use the following formula:

In this case, the number of favorable outcomes is the number of accidents involving a female driver (20,187), and the total number of outcomes is the total number of car accidents (74,250).

Therefore, the probability that a female driver was involved in an accident is approximately 27.2%.

Answer:

.18 % increase

Step-by-step explanation:

First you would find what the difference is between the school enrollment

you would do 1456 - 1234 =  222

Then you would divide 1234/1456 = 84.75% to see the total percentage

The increase however has a percentage of .18%

To check your work you would do .18 x 1234 = 222.12 ( round to the nearest whole number so it would just be 222) then add 1234+ 222 = 1456!

Answer:

A) (1,456-1,234) ÷ 1,456

Answer: x = 0 or x = 2

Step-by-step explanation:

To solve this, we must note that the two equations speaks of a single function of y. So

y = 2x - 3 = x² - 3

So from.here we can equate the two together and solve for x.

x² - 3 = 2x - 3, biw convert to a quadratic expression

x² - 2x - 3 + 3 = 0

x² - 2x = 0

Now factorize

x( x - 2 ) = 0, so solving for x

x = 0 or x = 2. .I believed getting the 2 won't be a problem

When x - 2 = 0

Then x = 2.

Answer:

The answer is B

Step-by-step explanation:

Just did the assignment on Egn.

Answer:

(1, 1), (2.4), (0.5) (470)

Step-by-step explanation:

Honestly, I only know it's this one because the first one is a linear function. If you think about it on a graph, all of the points on the first one can connect with a straight line.

Answer:

Check the explanation

Step-by-step explanation:

Kindly check the attached image below to see the step by step explanation to the question above.

Answer:

360 degree

Step-by-step explanation:

The sum of all the exterior angles is 360 for any polygon.

This can be proved for polygon with 20 sides also.

We know that sum of all the angles of polygon is given by formula = (2*n -4) *90

where n is the no. of sides of polygon

for 20 sides polygon sum of  sum of all the angles = (2*20 -4) * 90

= (40-4)*90 = 36*90 = 3240 degree

measure of each angle of polygon = sum of total angles of polygon/ no of sides of polygon = 3240/20 = 162

We also know that sum of interior and exterior angle of triangle is 180 degree, as interior and exterior angle  lies on straight line and they  are supplementary

so

162 + value of exterior angle = 180

=> value of exterior angle = 180 - 162 = 18

Value of one exterior angle is 18 degree

in 20 sided polygon there are 20 exterior angle

therefore, value of 20 exterior angle is 18*20 degree which is 360 degree.

__________________________________________________________

As a general point one can memorize than sum of all the exterior angle for a polygon with any number of side is 360.

Answer:

The of the pentagon is A=232.48cm²

Step-by-step explanation:

This problem bothers on the mensuration of flat shapes, a pentagon

Now given the apothem the area can be expressed as

A=a²n tan (180/n)

where

a = the length of the apothem (in radius)

n = the number of sides= 5 for pentagon

tan is the tangent function calculated in degrees

Substituting our given data we can solve for the area A

A= 8²*5 tan (180/5)

A= 64*5 tan 36

A= 320tan 36

A= 320*0.7265

A=232.48cm²

1 meter = 100 cm

So a 800 cm roll can cover 8 meters

(800/100 = 8)

She has 5 rolls so 5 x 8 = 40 meters total.

Answer:

(B) all numbers less than 7

Step-by-step explanation:

Given the inequality:

x<7

It means that:

Therefore, x<7 consists of all numbers less than 7.

Answer:

B

Step-by-step explanation:

Answer:

5

Step-by-step explanation:

let a be the other number

so one number is [tex]\frac{2}{3}a - 3[/tex]

a + [tex]\frac{2}{3}a - 3[/tex] = 17

[tex]\frac{5}{3}a - 3 = 17[/tex]

[tex]\frac{5}{3}a[/tex] = 17 + 3 = 20

a = 20 x [tex]\frac{3}{5}[/tex] = 12

[tex]\frac{2}{3}(12) - 3[/tex] = 5

Answer:

The value of the limit is 12.

Step-by-step explanation:

Small typing mistake, the limit is of h tending to 0.

We have that:

[tex]f(x) = x^{3}[/tex]

Then

[tex]f(2+h) = (2+h)^{3} = 8 + 12h + 6h^{2} + h^{3}[/tex]

[tex]f(2) = 2^{3} = 8[/tex]

Calling the limit L

[tex]L = \frac{f(2+h) - f(2)}{h}[/tex]

[tex]L = \frac{8 + 12h + 6h^{2} + h^{3} - 8}{h}[/tex]

[tex]L = \frac{h^{3} + 6h^{2} + 12h}{h}[/tex]

h is the common term in the numerator, then

[tex]L = \frac{h(h^{2} + 6h + 12)}{h}[/tex]

Simplifying by h

[tex]L = h^{2} + 6h + 12[/tex]

Since h tends to 0.

[tex]L = 0^{2} + 6*0 + 12[/tex]

[tex]L = 12[/tex]

So the answer is 12.

Using the definition of a derivative, the expression lim h→0 [f(2+h)-f(2)]/h for f(x)=x^3 simplifies to 12, which is the derivative of the function at x=2.

The limit you are trying to find is representative of the definition of a derivative at a point. In this case, the function is f(x) = x^3 and we're trying to find the derivative at x = 2. This is the definition of the derivative at a point:

lim h→0 [f(x+h) - f(x)] / h

Now, we plug in the equation for f(x):

lim h→0 [(2+h)^3 - 2^3] / h = lim h→0 [8+12h+6h^2+h^3 - 8] / h

Simplify that to get:

lim h→0 h [12+6h+h^2] / h

Then, by cancelling the h's, you have:

lim h→0 [12+6h+h^2]

As h→0, 6h and h^2 become 0, so finally, you obtain 12 as the derivative of f at x=2.

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Answer:

[tex]t=\frac{6.5-7}{\frac{2.1}{\sqrt{57}}}=-1.798[/tex]    

Critical value

The degreed of freedom are given by:

[tex]df=n-1=57-1=56[/tex]  

We are looking for a critical value in the t distribution with 56 degrees of freedom who accumulates 0.10 of the area in the left and we got:

[tex] t_{\alpha}= -1.297[/tex]

And since the calculated value is lower than the critical value we have enough evidence to reject the null hypothesis for this case and makes sense conclude that the true mean is less than 7 minutes

P value

The p value for this case would be given by:

[tex]p_v =P(t_{(56)}<-1.798)=0.0388[/tex]  

Since the p value is lower than the significance level provided of 0.1 we have enough evidence to conclude that the true mean is significantly lower than 7 minutes

Step-by-step explanation:

Information provided

[tex]\bar X=6.5[/tex] represent the sample mean for tje waiting times

[tex]s=2.1[/tex] represent the sample standard deviation

[tex]n=57[/tex] sample size  

[tex]\mu_o =7[/tex] represent the value to verify

[tex]\alpha=0.1[/tex] represent the significance level

t would represent the statistic

[tex]p_v[/tex] represent the p value

System of hypothesis

We want to verify if the true mean for this case is less than 7 minutes, the system of hypothesis for this case sre:  

Null hypothesis:[tex]\mu \geq 7[/tex]  

Alternative hypothesis:[tex]\mu < 7[/tex]  

Since we don;t know the population deviation the statistic for the t test is given by:

[tex]t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}[/tex]  (1)  

If we replace the info given we got:

[tex]t=\frac{6.5-7}{\frac{2.1}{\sqrt{57}}}=-1.798[/tex]    

Critical value

The degreed of freedom are given by:

[tex]df=n-1=57-1=56[/tex]  

We are looking for a critical value in the t distribution with 56 degrees of freedom who accumulates 0.10 of the area in the left and we got:

[tex] t_{\alpha}= -1.297[/tex]

And since the calculated value is lower than the critical value we have enough evidence to reject the null hypothesis for this case and makes sense conclude that the true mean is less than 7 minutes

P value

The p value for this case would be given by:

[tex]p_v =P(t_{(56)}<-1.798)=0.0388[/tex]  

Since the p value is lower than the significance level provided of 0.1 we have enough evidence to conclude that the true mean is significantly lower than 7 minutes

Following are the solution to the given question:

Given:

size [tex](n) = 58[/tex]

mean [tex]\mu = 7.4 \ minutes[/tex]

standard deviation [tex]\sigma = 2.3 \ minutes[/tex]

null and alternative hypotheses:

[tex]H_0 : \mu = 8 \ minutes\\\\H_a : \mu < 8 \ minutes\\\\[/tex]

Calculating the test statistic:

So, the observed value

Calculating the degrees of freedom:

[tex]\to 58 - 1 = 57[/tex]

Calculating the range of p-value:

Therefore the t is the critical value at the significance level that is 0.05 with of freedom is,

Therefore,

[tex]\to test \ statistic = -1.987 < -1.672[/tex]  rejecting the null hypothesis.

Therefore, the conclusion of the manager's claim is "True".

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