The mean diameter of the rim of Honda tires is 16 inches. Assume that the standard deviation of diameter of the rims is 0.3 inches. For quality control purposes, the diameter of the rims of 9 tires is measured every hour. The manager applies the rule that if the mean of diameter of a rim is greater or equal to 16.25, and lesser or equal to 15.75, the manufacturing should be stopped. If the diameter is between 15.75 and 16.25, the manufacturing process is not to be disturbed. a. Calculate the probability of stopping the manufacturing when the sample mean is 16 inches. b. Calculate the probability of stopping the manufacturing in case the mean is shifted to 16.05 inches. c. Calculate the probability of not disturbing the manufacturing if mean shifts to 16.25 inches.

Answer:

a. 0.0124 = 1.24% probability of stopping the manufacturing when the sample mean is 16 inches.

b. 0.0241 = 2.41% probability of stopping the manufacturing in case the mean is shifted to 16.05 inches.

c. 0.5 = 50% probability of not disturbing the manufacturing if mean shifts to 16.25 inches.

Step-by-step explanation:

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

Normal probability distribution

When the distribution is normal, we use the z-score formula.

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

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].

Assume that the standard deviation of diameter of the rims is 0.3 inches. Samples of 9.

This means that [tex]\sigma = 0.3, n = 9, s = \frac{0.3}{\sqrt{9}} = 0.1[/tex]

a. Calculate the probability of stopping the manufacturing when the sample mean is 16 inches.

Here we have [tex]\mu = 16[/tex]

Higher than 16.25:

This is 1 subtracted by the pvalue of Z when X = 16.25. So

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

By the Central Limit Theorem

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{16.25 - 16}{0.1}[/tex]

[tex]Z = 2.5[/tex]

[tex]Z = 2.5[/tex] has a pvalue of 0.9938

1 - 0.9938 = 0.0062

Lower than 15.75:

This is the pvalue of Z when X = 15.75. So

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{15.75 - 16}{0.1}[/tex]

[tex]Z = -2.5[/tex]

[tex]Z = -2.5[/tex] has a pvalue of 0.0062

Probability of stopping:

2*0.0062 = 0.0124

0.0124 = 1.24% probability of stopping the manufacturing when the sample mean is 16 inches.

b. Calculate the probability of stopping the manufacturing in case the mean is shifted to 16.05 inches.

Here we have [tex]\mu = 16.05[/tex]

Higher than 16.25:

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{16.25 - 16.05}{0.1}[/tex]

[tex]Z = 2[/tex]

[tex]Z = 2[/tex] has a pvalue of 0.9772

1 - 0.9772 = 0.0228

Lower than 15.75:

This is the pvalue of Z when X = 15.75. So

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{15.75 - 16.05}{0.1}[/tex]

[tex]Z = -3[/tex]

[tex]Z = -3[/tex] has a pvalue of 0.0013

Probability of stopping:

0.0228 + 0.0013 = 0.0241

0.0241 = 2.41% probability of stopping the manufacturing in case the mean is shifted to 16.05 inches.

c. Calculate the probability of not disturbing the manufacturing if mean shifts to 16.25 inches.

Between 16.25 and 15.75 with [tex]\mu = 16.25[/tex]. This is the pvalue of Z when X = 16.25 subtracted by the pvalue of Z when X = 15.75.

X = 16.25

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{16.25 - 16.25}{0.1}[/tex]

[tex]Z = 0[/tex]

[tex]Z = 0[/tex] has a pvalue of 0.5

X = 15.75

[tex]Z = \frac{X - \mu}{s}[/tex]

[tex]Z = \frac{15.75 - 16.25}{0.1}[/tex]

[tex]Z = -5[/tex]

[tex]Z = -5[/tex] has a pvalue of 0

0.5 - 0 = 0.5

0.5 = 50% probability of not disturbing the manufacturing if mean shifts to 16.25 inches.

9 × 2(3 + 5) – (10 × 12) = y

can somebody help? :(

What is the ratio and percent of the rectangle below is shaded

a bag contains 6 white marbles and 4 black marbles. a marble is drawn without replacing the first one. what is the probability of drawing a white marble on the first draw, followed by a white marble on the second?

Priya’s family exchanged 250 dollars for 4,250 pesos. Priya bought a sweater for 510 pesos. How many dollars did the sweater cost?

Translate the verbal phrase below to an inequality statement:
The opposite of 10 is greater than the sum of twice a number and 5.

agenthammerx Please help me on this I beg you

what is the answer to 25=9y+3=

The expression 42+57 shows how much money Nate spent at a store. Which expression also shows how much money Nate spent?
3(14+19)
3(14x19)
6(7+57)
6(7x57)

i need the selling price pls ty.​

12.1 times 17.6 what is the area

Adding (4-7i)+(-11+9i)​

Hello, happy Friday, I am just here with some geometry questions.

Please only answer this if you know the answer. If you have a comment, please add it to the comment box. Only answer the question if you can explain your answer and why you think it is accurate. Thanks, good luck!

Be sure to show your answer and work using a coordinate plane.

Help plz I’m suck at math

Pleeeeaase help me!! ;-;
what is the vertex of


f(x)= |x - 3| + 2


A. (2, 3)

B. (3, 2)

C. (2, -3)

D. (5, 5)

If you're given the points:
(4, 10) (8,9)
What are your y1, and y2?

A. y1= 9; y2= 10
B. y1= 4; y2= 8
C. y1= 10; y2= 9
D. y1= 8; y2= 4

A shopkeeper sold a dressing table for 7512.75 at a profit of 19*1/4% What was the cost price.​

The rectangle shown below has a diagonal of 15 cm and a width of 7 cm. To the nearest centimeter, what is the length, x, of the rectangle? Round your answer to the nearest tenth. 15 7 X​

The product of two consecutive integers is 23 more than seven times their sum. Find the integers.​

I really need help !!

The 24/7 Truck Rental Company has been charging an initial rental fee of $29 plus $2.50 per hour for rental of their small trucks. After a change in pricing policy, the new fee will be $19 up front plus $3.25 per hour. After how many hours of renting the truck will the new plan be more expensive than the old plan

pleased help meeeeeeeee

2x+3y=5×-y missing value

Which number line model represents the expression -4+7.25

Rewrite each of the following as equivalent expressions without the use of negative or zero exponents
Remember your order of operations

find the missing length indicated

Tanya wants to wrap the box
shown. The wrapping paper
costs $0.15 per square foot.
How much will the decorative
paper cost?
3 in
5 in
9 in
026 10

Can yall help me with the right answer

Graph the solution of the inequality -x+y > 2

PLEASE ANSWER ASAP FOR BRAINLEST!!!!!!!!!!!!!!!!!!!!!!!!!!!!!