1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Nastasia [14]
3 years ago
6

A consumer products company is formulating a new shampoo and is interested in foam height (in mm). Foam height is approximately

normally distributed and has a standard deviation of 20 mm. The company wishes to test H0: μ = 175 mm versus H1: μ > 175 mm, using a random sample of n = 10 samples.(a) Find P-value if the sample average is = 185? Round your final answer to 3 decimal places.(b) What is the probability of type II error if the true mean foam height is 200 mm and we assume that α = 0.05? Round your intermediate answer to 1 decimal place. Round the final answer to 4 decimal places.(c) What is the power of the test from part (b)? Round your final answer to 4 decimal places.
Mathematics
1 answer:
Genrish500 [490]3 years ago
3 0

Answer:

a) 0.057

b) 0.5234

c) 0.4766

Step-by-step explanation:

a)

To find the p-value if the sample average is 185, we first compute the z-score associated to this value, we use the formula

z=\frac{\bar x-\mu}{\sigma/\sqrt N}

where

\bar x=mean\; of\;the \;sample

\mu=mean\; established\; in\; H_0

\sigma=standard \; deviation

N = size of the sample.

So,

z=\frac{185-175}{20/\sqrt {10}}=1.5811

\boxed {z=1.5811}

As the sample suggests that the real mean could be greater than the established in the null hypothesis, then we are interested in the area under the normal curve to the right of  1.5811 and this would be your p-value.

We compute the area of the normal curve for values to the right of  1.5811 either with a table or with a computer and find that this area is equal to 0.0569 = 0.057 rounded to 3 decimals.

So the p-value is  

\boxed {p=0.057}

b)

Since the z-score associated to an α value of 0.05 is 1.64 and the z-score of the alternative hypothesis is 1.5811 which is less than 1.64 (z critical), we cannot reject the null, so we are making a Type II error since 175 is not the true mean.

We can compute the probability of such an error following the next steps:

<u>Step 1 </u>

Compute \bar x_{critical}

1.64=z_{critical}=\frac{\bar x_{critical}-\mu_0}{\sigma/\sqrt{n}}

\frac{\bar x_{critical}-\mu_0}{\sigma/\sqrt{n}}=\frac{\bar x_{critical}-175}{6.3245}=1.64\Rightarrow \bar x_{critical}=185.3721

So <em>we would make a Type II error if our sample mean is less than 185.3721</em>.  

<u>Step 2</u>

Compute the probability that your sample mean is less than 185.3711  

P(\bar x < 185.3711)=P(z< \frac{185.3711-185}{6.3245})=P(z

So, <em>the probability of making a Type II error is 0.5234 = 52.34% </em>

c)

<em>The power of a hypothesis test is 1 minus the probability of a Type II error</em>. So, the power of the test is

1 - 0.5234 = 0.4766

You might be interested in
Which statement about the value of x is true?
Anarel [89]
The best way to find the answer is to solve for x. To start you would add 38 and 39, getting 77, then subtract that from 180 to get the last angle in the triangle, 103. Finally, you would subtract 103 from 180, getting 77, to get x. The only thing you need to do now is to look at all of the answers and figure out which one makes sense for the answer you got. You know it can't be x<77 because it doesn't have the or equal to sign. You know it can't be x>103 because 77 is lower than 103. You know it can't be x<39 because 77 is greater than 39, so the answer has to be x > 38.
4 0
3 years ago
Read 2 more answers
Factor the polynomial, x2 + 5x + 6
patriot [66]

Answer:

Choice b.

x^{2} + 5\, x + 6 = (x + 3)\, (x + 2).

Step-by-step explanation:

The highest power of the variable x in this polynomial is 2. In other words, this polynomial is quadratic.

It is thus possible to apply the quadratic formula to find the "roots" of this polynomial. (A root of a polynomial is a value of the variable that would set the polynomial to 0.)

After finding these roots, it would be possible to factorize this polynomial using the Factor Theorem.

Apply the quadratic formula to find the two roots that would set this quadratic polynomial to 0. The discriminant of this polynomial is (5^{2} - 4 \times 1 \times 6) = 1.

\begin{aligned}x_{1} &= \frac{-5 + \sqrt{1}}{2\times 1} \\ &= \frac{-5 + 1}{2} \\ &= -2\end{aligned}.

Similarly:

\begin{aligned}x_{2} &= \frac{-5 - \sqrt{1}}{2\times 1} \\ &= \frac{-5 - 1}{2} \\ &= -3\end{aligned}.

By the Factor Theorem, if x = x_{0} is a root of a polynomial, then (x - x_0) would be a factor of that polynomial. Note the minus sign between x and x_{0}.

  • The root x = -2 corresponds to the factor (x - (-2)), which simplifies to (x + 2).
  • The root x = -3 corresponds to the factor (x - (-3)), which simplifies to (x + 3).

Verify that (x + 2)\, (x + 3) indeed expands to the original polynomial:

\begin{aligned}& (x + 2)\, (x + 3) \\ =\; & x^{2} + 2\, x + 3\, x + 6 \\ =\; & x^{2} + 5\, x + 6\end{aligned}.

4 0
2 years ago
Eugene went to the ice cream shop after school. The large dish of ice-
KIM [24]

Answer:

the answer would be 4

Step-by-step explanation:

10  -  4 =  6 divided by 1.25 = 4

hope this helps!

8 0
3 years ago
Read 2 more answers
Twenty-one fish are swimming in an aquarium. Eleven are removed, leaving 10 fish. Which number is the additive inverse of the nu
agasfer [191]

The additive inverse: The usage of addition with a negative number.

The original equation is 21 - 11 = 10

To use the additive inverse, change it so that there is an addition sign.

21 + (-11) = 10

B) -11 is your answer

hope this helps

7 0
3 years ago
Read 2 more answers
Find the slope and the y-intercept of the graph of the linear equation.<br><br> y = 6x
Snowcat [4.5K]
Y-intercept=0
slope=6/1

hope this helps
4 0
3 years ago
Other questions:
  • What is the quotient of 3/5 and 1/10<br> A.1/6<br> B.1/3<br> C.3<br> D.6
    7·2 answers
  • Please answer all :3
    5·1 answer
  • One number is five times another number. Their sum is 42. What are the two numbers?
    11·2 answers
  • How does math help with science
    14·2 answers
  • What is the probability annie does not choose a girl who is a junior
    5·2 answers
  • A trainer takes a survey of all the athletes in a school about their height, rounded to the nearest inch, and their grade level.
    15·1 answer
  • Doug purchased land for $8,000 in 1995. The value of the land depreciated by 4% each year after 1995. Write an exponential funct
    11·2 answers
  • What is the value of x in the figure? Enter your answer in the box.<br> 1450<br> (2x + 15)<br> x=
    13·1 answer
  • You have a rectangular prism as shown below. Its height is 7 cm, its width is 4 cm, the length is 3 cm. How would we calculate t
    15·1 answer
  • A store buys 8 sweaters for $72 and sells them for $370. How much profit does the store make per sweater?
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!