6 more than the product of 8 and a number x

Two catalysts may be used in a batch chemical process. Twelve batches were prepared using catalyst 1, resulting in an average yi

aalyn [17]

Answer:

a) $t=\frac{(91-85)-(0)}{2.490\sqrt{\frac{1}{12}}+\frac{1}{15}}=6.222$

$df=12+15-2=25$

$p_v =P(t_{25}>6.222) =8.26x10^{-7}$

So with the p value obtained and using the significance level given $\alpha=0.01$ we have $p_v$ so we can conclude that we have enough evidence to reject the null hypothesis.

b) $(91-85) -2.79 *2.490 \sqrt{\frac{1}{12} +\frac{1}{15}} =3.309$

$(91-85) +2.79 *2.490 \sqrt{\frac{1}{12} +\frac{1}{15}} =8.691$

Step-by-step explanation:

Notation and hypothesis

When we have two independent samples from two normal distributions with equal variances we are assuming that

$\sigma^2_1 =\sigma^2_2 =\sigma^2$

And the statistic is given by this formula:

$t=\frac{(\bar X_1 -\bar X_2)-(\mu_{1}-\mu_2)}{S_p\sqrt{\frac{1}{n_1}}+\frac{1}{n_2}}$

Where t follows a t distribution with $n_1+n_2 -2$ degrees of freedom and the pooled variance $S^2_p$ is given by this formula:

$\S^2_p =\frac{(n_1-1)S^2_1 +(n_2 -1)S^2_2}{n_1 +n_2 -2}$

This last one is an unbiased estimator of the common variance $\sigma^2$

Part a

The system of hypothesis on this case are:

Null hypothesis: $\mu_2 \leq \mu_1$

Alternative hypothesis: $\mu_2 > \mu_1$

Or equivalently:

Null hypothesis: $\mu_2 - \mu_1 \leq 0$

Alternative hypothesis: $\mu_2 -\mu_1 > 0$

Our notation on this case :

$n_1 =12$ represent the sample size for group 1

$n_2 =15$ represent the sample size for group 2

$\bar X_1 =85$ represent the sample mean for the group 1

$\bar X_2 =91$ represent the sample mean for the group 2

$s_1=3$ represent the sample standard deviation for group 1

$s_2=2$ represent the sample standard deviation for group 2

First we can begin finding the pooled variance:

$\S^2_p =\frac{(12-1)(3)^2 +(15 -1)(2)^2}{12 +15 -2}=6.2$

And the deviation would be just the square root of the variance:

$S_p=2.490$

Calculate the statistic

And now we can calculate the statistic:

$t=\frac{(91-85)-(0)}{2.490\sqrt{\frac{1}{12}}+\frac{1}{15}}=6.222$

Now we can calculate the degrees of freedom given by:

$df=12+15-2=25$

Calculate the p value

And now we can calculate the p value using the altenative hypothesis:

$p_v =P(t_{25}>6.222) =8.26x10^{-7}$

Conclusion

So with the p value obtained and using the significance level given $\alpha=0.01$ we have $p_v$ so we can conclude that we have enough evidence to reject the null hypothesis.

Part b

For this case the confidence interval is given by:

$(\bar X_1 -\bar X_2) \pm t_{\alpha/2} S_p \sqrt{\frac{1}{n_1} +\frac{1}{n_2}}$

For the 99% of confidence we have $\alpha=1-0.99 = 0.01$ and $\alpha/2 =0.005$ and the critical value with 25 degrees of freedom on the t distribution is $t_{\alpha/2}= 2.79$

And replacing we got:

$(91-85) -2.79 *2.490 \sqrt{\frac{1}{12} +\frac{1}{15}} =3.309$

$(91-85) +2.79 *2.490 \sqrt{\frac{1}{12} +\frac{1}{15}} =8.691$

7 0

3 years ago

F(x)=x^4-5x^2-6x-10 is divided by x-3

LiRa [457]

Answer:

Step-by-step explanation:

STEP

1

:

10

Simplify ——

x

Equation at the end of step

1

:

10

((((x4)-(5•(x2)))-6x)-——)-3

x

STEP

2

:

Equation at the end of step

2

:

10

((((x4) - 5x2) - 6x) - ——) - 3

x

STEP

3

:

Rewriting the whole as an Equivalent Fraction

3.1 Subtracting a fraction from a whole

Rewrite the whole as a fraction using x as the denominator :

x4 - 5x2 - 6x (x4 - 5x2 - 6x) • x

x4 - 5x2 - 6x = ————————————— = ———————————————————

1 x

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

STEP

4

:

Pulling out like terms

4.1 Pull out like factors :

x4 - 5x2 - 6x = x • (x3 - 5x - 6)

Polynomial Roots Calculator :

4.2 Find roots (zeroes) of : F(x) = x3 - 5x - 6

Polynomial Roots Calculator is a set of methods aimed at finding values of x for which F(x)=0

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers x which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient

In this case, the Leading Coefficient is 1 and the Trailing Constant is -6.

The factor(s) are:

of the Leading Coefficient : 1

of the Trailing Constant : 1 ,2 ,3 ,6

Let us test ....

P Q P/Q F(P/Q) Divisor

-1 1 -1.00 -2.00

-2 1 -2.00 -4.00

-3 1 -3.00 -18.00

-6 1 -6.00 -192.00

1 1 1.00 -10.00

2 1 2.00 -8.00

3 1 3.00 6.00

6 1 6.00 180.00

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

4.3 Adding up the two equivalent fractions

Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

x • (x3-5x-6) • x - (10) x5 - 5x3 - 6x2 - 10

———————————————————————— = ———————————————————

x x

Equation at the end of step

4

:

(x5 - 5x3 - 6x2 - 10)

————————————————————— - 3

x

STEP

5

:

Rewriting the whole as an Equivalent Fraction

5.1 Subtracting a whole from a fraction

Rewrite the whole as a fraction using x as the denominator :

3 3 • x

3 = — = —————

1 x

Checking for a perfect cube :

5.2 x5 - 5x3 - 6x2 - 10 is not a perfect cube

Trying to factor by pulling out :

5.3 Factoring: x5 - 5x3 - 6x2 - 10

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1: -6x2 - 10

Group 2: x5 - 5x3

Pull out from each group separately :

Group 1: (3x2 + 5) • (-2)

Group 2: (x2 - 5) • (x3)

Bad news !! Factoring by pulling out fails :

The groups have no common factor and can not be added up to form a multiplication.

Polynomial Roots Calculator :

5.4 Find roots (zeroes) of : F(x) = x5 - 5x3 - 6x2 - 10

See theory in step 4.2

In this case, the Leading Coefficient is 1 and the Trailing Constant is -10.

The factor(s) are:

of the Leading Coefficient : 1

of the Trailing Constant : 1 ,2 ,5 ,10

Let us test ....

P Q P/Q F(P/Q) Divisor

-1 1 -1.00 -12.00

-2 1 -2.00 -26.00

-5 1 -5.00 -2660.00

-10 1 -10.00 -95610.00

1 1 1.00 -20.00

2 1 2.00 -42.00

5 1 5.00 2340.00

10 1 10.00 94390.00

Polynomial Roots Calculator found no rational roots

Adding fractions that have a common denominator :

5.5 Adding up the two equivalent fractions

(x5-5x3-6x2-10) - (3 • x) x5 - 5x3 - 6x2 - 3x - 10

————————————————————————— = ————————————————————————

x x

Polynomial Roots Calculator :

5.6 Find roots (zeroes) of : F(x) = x5 - 5x3 - 6x2 - 3x - 10

See theory in step 4.2

In this case, the Leading Coefficient is 1 and the Trailing Constant is -10.

The factor(s) are:

of the Leading Coefficient : 1

of the Trailing Constant : 1 ,2 ,5 ,10

Let us test ....

P Q P/Q F(P/Q) Divisor

-1 1 -1.00 -9.00

-2 1 -2.00 -20.00

-5 1 -5.00 -2645.00

-10 1 -10.00 -95580.00

1 1 1.00 -23.00

2 1 2.00 -48.00

5 1 5.00 2325.00

10 1 10.00 94360.00

Polynomial Roots Calculator found no rational roots

Final result :

x5 - 5x3 - 6x2 - 3x - 10

————————————————————————

x

3 0

3 years ago

I need help please can someone help me !!!!!!

Valentin [98]

What are you trying to find? Area, Volume, Circumference, or perimeter?

7 0

3 years ago

The table shows the number of games a chess player won in professional competitions, based on the number of games played.

AfilCa [17]

slope = (25-4)/(30-10)

slope = 21/20

slope =21/20

using point slope form

y-y1 = m(x-x1)

y-4 = 21/20 (x-10)

y = 21/20x -21/2 +4

y = 21/20 x -21/2 +8/2

y = 21/20x -13/2

let x=40

y = 21/20 (40) -13/2

y = 42-13/2

y = 35.5 games

If we round the slope to 1

slope =1

using point slope form

y-y1 = m(x-x1)

y-4 = 1 (x-10)

y = 1x -10 +4

y = x -6

let x=40

y = 40-6

y = 34 games

4 0

3 years ago

Starting at 7:00 a.m., Lin spent a day hiking through a canyon. This graph shows her elevation (in meters) at some different tim

ollegr [7]

Answer:

Sorry for my bad handwriting it's hard write in a computer

Step-by-step explanation:

5 0

2 years ago