The concentration of active ingredient in a liquid laundry detergent is thought to be affected by the type of catalyst used in the process. The standard deviation of active concentration is known to be 3 grams per liter regardless of the catalyst type. Ten observations on concentration are taken with each catalyst, and the data follow:

Catalyst 1: 57.9, 66.2, 65.4, 65.4, 65.2, 62.6, 67.6, 63.7, 67.2, 71.0
Catalyst 2: 66.4, 71.7, 70.3, 69.3, 64.8, 69.6, 68.6, 69.4, 65.3, 68.8.

a. Find a 95% confidence interval on the difference in mean active concentrations for the two catalysts. Find the P-value.
b. Is there any evidence to indicate that the mean active concentrations depend on the choice of catalyst? Base your answer on the results of part (a).
c. Suppose that the true mean difference in active concentration is 5 grams per liter. What is the power of the test to detect this difference if α = 0.05?
d. If this difference of 5 grams per liter is really important, do you consider the sample sizes used by the experimenter to be adequate? Does the assumption of normality seem reasonable for both samples?

Answer:

a. The 95% C.I. is approximately -5.83 < μ₁ - μ₂ < -0.57

b. Yes

c. The power of the test is 0.03836

d. The sample size can be considered adequate

Step-by-step explanation:

a. The given data in the study

Catalyst 1: 57.9, 66.2, 65.4, 65.4, 65.2, 62.6, 67.6, 63.7, 67.2, 71.0

Catalyst 2: 66.4 , 71.7, 70.3, 69.3, 64.8, 69.6, 68.6, 69.4, 65.3, 68.8

Using the Average, Standard Deviation function from Microsoft Excel, we have;

The mean for Catalyst 1, [tex]\overline x_1[/tex] = 65.22

The standard deviation for catalyst 1, σ₁ = 3.444416

The mean for Catalyst 2, [tex]\overline x_2[/tex] = 68.42

The standard deviation for catalyst 2, σ₂ = 2.22401

The 95% confidence interval on the difference in mean is given as follows;

[tex]\left (\bar{x}_1-\bar{x}_{2} \right ) \pm z_{c}\sqrt{\dfrac{\sigma _{1}^{2}}{n_{1}} + \dfrac{\sigma_{2}^{2}}{n_{2}}}[/tex]

The critical-z for a 95% confidence interval = 1.96

Therefore, we have;

[tex]\left (65.22-68.42 \right ) \pm 1.96 \times \sqrt{\dfrac{3^{2}}{10} + \dfrac{3^{2}}{10}}[/tex]

Therefore, we have;

The 95% C.I. is approximately -5.83 < μ₁ - μ₂ < -0.57

The test statistics is given as follows;

[tex]z=\dfrac{(\bar{x}_{1}-\bar{x}_{2})}{\sqrt{\dfrac{\sigma_{1}^{2} }{n_{1}}-\dfrac{\sigma _{2}^{2}}{n_{2}}}}[/tex]

Therefore we have;

[tex]z=\dfrac{(65.22-68.42)}{\sqrt{\dfrac{3^{2} }{10}+\dfrac{3^{2}}{10}}} = -2.385[/tex]

The p-value = P(Z<-2.39 or Z > 2.39) = 2P(Z<-2.39) = 2 × 0.00842 = 0.01684

b. From the confidence interval which range from approximately -5.83 to -0.57 and does not include 0, therefore, there is a difference in mean active concentration which depends on the choice of catalyst

c. The power of the test

 The sample mean difference is given as follows;

[tex]\left (\bar{x}_1-\bar{x}_{2} \right ) = \pm z_{c}\sqrt{\dfrac{\sigma _{1}^{2}}{n_{1}} + \dfrac{\sigma_{2}^{2}}{n_{2}}}[/tex]

Therefore, we have;

[tex]\left (\bar{x}_1-\bar{x}_{2} \right ) = \pm 1.96 \times \sqrt{\dfrac{3^{2}}{10} + \dfrac{3^{2}}{10}} = \pm 2.6296[/tex]

The z-value is given as follows;

[tex]z=\dfrac{(\bar{x}_{1}-\bar{x}_{2})-(\mu_{1}-\mu _{2} )}{\sqrt{\dfrac{\sigma_{1}^{2} }{n_{1}}-\dfrac{\sigma _{2}^{2}}{n_{2}}}}[/tex]

μ₁ - μ₂ = 5

When [tex]\bar{x}_1-\bar{x}_{2} = 2.6296[/tex]

[tex]z=\dfrac{2.6296-5}{\sqrt{\dfrac{3^{2} }{10}-\dfrac{3^{2}}{10}}} \approx -1.7667918[/tex]

When [tex]\bar{x}_1-\bar{x}_{2} = -2.6296[/tex]

[tex]z=\dfrac{-2.6296-5}{\sqrt{\dfrac{3^{2} }{10}-\dfrac{3^{2}}{10}}} \approx -5.686768[/tex]

The power of the test is given by P = P(Z<-5.69) + P(Z>-1.77) = 0 + 0.03836 = 0.03836

The power of the test = 0.03836

d. The sample size is statistically adequate because the confidence interval of -5.83 < μ₁ - μ₂ < -0.57 has a value of -5 as a possible population difference in mean

The assumption of normality seems adequate because the confidence interval obtained by using the sample standard deviation is given as follows;

(-5.74, -0.66) which also contains -5 which is the difference in the population mean

8. How tall is a stack of 26 coins each 0.03 m thick? ​

Please help me if you can!

Write a jingle to advertise
favorite dessert to the tune of
"Twinkle, Twinkle Little Star."
What is so special about this
dessert? How can you describe it
in a memorable, catchy way?

Y’all someone help no link ,will be reported

(-6, -7) and (3,-4)
Pleaseee helppppppp​

Zachary's rectangle measures 4 1 /4 units by 4 1 /4 units. What is its area?

On the midnight shift, the number of patients with head trauma in an emergency room has the probability distribution shown below. x 0 1 2 3 4 5 Total P(x) .02 .31 .25 .19 .13 .10 1.00 (a) Calculate the mean and standard deviation. (Round your mean value to 2 decimal places and standard deviation to 3 decimal places.) Mean Standard deviation (b) Describe the shape of this distribution. multiple choice Skewed to the right Skewed to the left Symmetric

Huh?! Its me DaBaby less gooo i need your help with this question. Yeah Yeah!

Please solve this system of equations.
5x-2y=9
12x-8y=4

What is the value of x in trapezoid ABCD?

Please help ASAP please help

Solve the Following Inequality Select the possible range of solution to make the Statement True.

-6m < -24

Group of answer choices

1, 2, 3

-1, -2, -3

4, 5, 6

-, 4, -5, -6

What type of triangle is this if lengths of its sides are 37m, 24m, and 5m *​

A university wants to investigate how much its students spend on textbooks each semester. Since all
students must purchase their materials through the campus bookstore, a random sample of 15 student
accounts is selected, and the averagą dollar amount spent on textbooks is calculated.
Which of these circumstances would make it inappropriate to use this sample to calculate a 95%
confidence interval for the mean amount spent on books by students at this university?
The standard deviation of the textbook costs is below $30.
The standard deviation of the textbook costs is larger than $100.
A stem-and-leaf plot of the textbook costs shows a roughly symmetric distribution
A histogram of the textbook costs is calculated and shows a slight skew to the right.
A boxplot of the textbook costs shows two students whose amounts would qualify as outliers,

Which of the following scenarios represents a possible impulse purchase?


Bryce researches interest rates on the purchase of a new car before deciding to make a purchase.

Bryce budgets for his first new car and decides to purchase a car through a dealership offering 0% interest to first-time buyers.

Bryce purchases the car of his dream by financing the entire cost of the car with a high interest bank loan.

I am not good at math, will give brainliest

Help Plis ...............

convert 2x + y = -1 to slope intercept form

Add 1/4+9/14+6/7. Simplify the answer and write it as a mixed number.

Cassandra works for a company that is based off her sales. If she is paid 9.5% of her total monthly sales, calculate her pay for the month if she sells $15,670 of product.

I need help on my trig can someone help me on the question please and thank you

3/2 inches of snow fell in 3/4 of an hour. Which two statements are correct?

Mary is having her living room and bedroom painted. Interior Designs USA charges $60.00 to evaluate the space plus $35.00 per hour of labor. Splash of Color charges $55.00 per hour with no initial fee. Which of the following are true? Select all that apply.

A) If it takes 7 hours to paint the two rooms, Interior Designs USA will charge the least.
B) Splash of Color will always charge the least.
C) If it takes more than 5 hours to paint the rooms, Splash of Color will be more cost effective.
D) If it takes 10 hours to paint the rooms, Splash of Color will charge $200 more than Interior Designs USA.
E) If it takes 3 hours to paint the rooms, both companies will charge the same amount.

"Jonas prepared himself to walk to the stage when the applause ended and the Chief Elder picked up the next folder and looked down to the group to call forward the next new Twelve. He was calm now that his turn had come. He took a deep breath and smoothed his hair with his hand." - The Giver *

1st person
2nd person
3rd person objective
3rd person limited
3rd person omniscient

S - 4= -8
What is S?

The calibration of a scale is to be checked by weighing a 13 kg test specimen 25 times. Suppose that the results of different weighings are independent of one another and that the weight on each trial is normally distributed with Sigma = 0.200 kg. Let µ denote the true average weight reading on the scale.

Required:
a. What hypotheses should be tested?
b. With the sample mean itself as the test statistic, what is the P-value?

Find the area of the regular octagon with radius of 14
feet.

write 98 as a product of its prime factor no links or photo or I will report you​

A warren of rabbits has a population of 350. The population is increasing at a rate of 3% per year. How can you write an exponential growth function to find the monthly growth rate? (I have other question pls help if you can)

(12x - 3)* 105*. What is the value
of x in the figure
at the right?

easton is paid $418.96 weekly. what is his average monthly net, or take-home pay?