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
rjkz [21]
3 years ago
6

Rockwell hardness of pins of a certain type is known to have a mean value of 50 and a standard deviation of 1.8. (Round your ans

wers to four decimal places.)
(a) If the distribution is normal, what is the probability that the sample mean hardness for a random sample of 17 pins is at least 51?
answer is 0.011

(b) What is the (approximate) probability that the sample mean hardness for a random sample of 45 pins is at least 51?
Mathematics
1 answer:
Alenkinab [10]3 years ago
8 0

Answer:

a) 0.011 = 1.1% probability that the sample mean hardness for a random sample of 17 pins is at least 51

b) 0.0001 = 0.1% probability that the sample mean hardness for a random sample of 45 pins is at least 51

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean \mu and standard deviation \sigma, the zscore of a measure X is given by:

Z = \frac{X - \mu}{\sigma}

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean \mu and standard deviation \sigma, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean \mu and standard deviation s = \frac{\sigma}{\sqrt{n}}.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this problem, we have that:

\mu = 50, \sigma = 1.8

(a) If the distribution is normal, what is the probability that the sample mean hardness for a random sample of 17 pins is at least 51?

Here n = 17, s = \frac{1.8}{\sqrt{17}} = 0.4366

This probability is 1 subtracted by the pvalue of Z when X = 51. So

Z = \frac{X - \mu}{\sigma}

By the Central Limit Theorem

Z = \frac{X - \mu}{s}

Z = \frac{51 - 50}{0.4366}

Z = 2.29

Z = 2.29 has a pvalue of 0.9890

1 - 0.989 = 0.011

0.011 = 1.1% probability that the sample mean hardness for a random sample of 17 pins is at least 51

(b) What is the (approximate) probability that the sample mean hardness for a random sample of 45 pins is at least 51?

Here n = 17, s = \frac{1.8}{\sqrt{45}} = 0.2683

Z = \frac{X - \mu}{s}

Z = \frac{51 - 50}{0.0.2683}

Z = 3.73

Z = 3.73 has a pvalue of 0.9999

1 - 0.9999 = 0.0001

0.0001 = 0.1% probability that the sample mean hardness for a random sample of 45 pins is at least 51

You might be interested in
Class A has 25 pupils and class B has 28 pupils. Both classes sit the same maths test. The mean score for class A is 69. The mea
il63 [147K]

Answer:

13

Step-by-step explanation:

Mean would mean average, which in this case is \frac{x+y}{2}

so your equation would look like \frac{69+x}{2}=41

Next multiply both sides by to clear the fraction 2*(\frac{69+x}{2})

to get 69+x=89

Subtract 69 from both sides 69+x=82\\-69   -69

you get x = 13

6 0
2 years ago
Type the equation for the graphbelow.Pi/3 2piy = [?] sin([ ]x)
Pie

To find the equation of the graph, what we do is to recognize some characteristics of a sine function:

- Amplitude: The amplitude of the sine function is the distance from the middle value or line running through the graph up to the highest point. In this case, the amplitude is 1.

- Period: The period of a sine function is defined as the length of one complete sine wave or one complete cycle of the curve. It can be found using the equation: P=2pi/B.

Why are these characteristics important?

Because the sine function has the following general form:

In this problem, we have A=1 and we know that the period equals 2pi/3. So,

Therefore, the equation of the graph is:

8 0
1 year ago
Your bill at a restaurant is $84.90. What will the tip be if you leave 20% for the server?*
nydimaria [60]

Answer:

$16.90

Step-by-step explanation:

20% of $84.80 is $16.90

3 0
2 years ago
What is the answer to 8= −6+x
Aliun [14]

Answer:

x = 14

Step-by-step explanation:

add 6 to both sides:

8+6 = 6-6+x

14 = x

x = 14

6 0
3 years ago
Help me fr this time
Aneli [31]

Answer:

228

Step-by-step explanation:

[Formulas Below]

Surface area means the sum of all the shapes on the figure, which means we have to calculate each area of the shapes on the triangular right prism and add them up. To calculate the triangles, we plug in 6 and 5 and get 15. Since there are 2 triangles, we multiply it by 2. Next, we have to find the areas of the rectangles. 11 * 6.5 = 71.5 (There are two of them so you multiply by 2), and 11 * 5 = 55. Finally, we add all of these up. 15 * 2 + 71.5 * 2 + 55 = 30 + 55 + 143 = 228. The surface area is <u><em>228</em></u>.

<u><em>Area of Triangle = (Length * Height)/2</em></u>

<u><em /></u>

<u><em>Area of Rectangle = Length * Width</em></u>

4 0
2 years ago
Read 2 more answers
Other questions:
  • Will the point (2,6) be on the graph of y=3x
    14·2 answers
  • Jeff has 2 dimes. Each day he gets 3 more. Which shows how many he has after (x) days?
    12·1 answer
  • Which values represent the independent variable?
    5·2 answers
  • Which expression is equivalent to
    10·1 answer
  • The Temple at the top of the pyramid is approximately 24 m above the ground and they are 91 steps leading up to the temple. How
    14·1 answer
  • Can someone help please!
    10·1 answer
  • File is below! thanks!!
    12·1 answer
  • Please help !!! I don’t know this brainliest?
    12·2 answers
  • Father's age is five times than the son's age. If the sum
    10·2 answers
  • Convert from degrees to radians<br><br> 250°
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!