Mathematics, 14.06.2021 21:50, aclm2001

A filling machine fills bottles of nail polish with amounts that are normallydistributed with mean 18 mL and standard deviation 0.6 mL. A random sample of 12 bottles of this nail polish is selected for inspection. Required:
a. What is the probability that one of the randomly selected bottles is filled with less than 17.5mL of nail polish?
b. What is the probability that the average fill of the twelve bottles is more than 17.5mL?

Answers: 2

Show answers

Answers

answered: needthehelp78

a. 0.2033 = 20.33% probability that one of the randomly selected bottles is filled with less than 17.5mL of nail polish

b. 0.0019 = 0.19% probability that the average fill of the twelve bottles is more than 17.5mL

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 normal distributions can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the z-score 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 p-value, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem establishes 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.

Mean 18 mL and standard deviation 0.6 mL.

This means that $\mu = 18, \sigma = 0.6$

a. What is the probability that one of the randomly selected bottles is filled with less than 17.5mL of nail polish?

This is the p-value of Z when X = 17.5. So

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

$Z = \frac{17.5 - 18}{0.6}$

Z = -0.83

Z = -0.83 has a p-value of 0.2033

0.2033 = 20.33% probability that one of the randomly selected bottles is filled with less than 17.5mL of nail polish.

b. What is the probability that the average fill of the twelve bottles is more than 17.5mL?

Twelve bottles, so now $n = 12, s = \frac{0.6}{\sqrt{12}}$

The probability is:

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

By the Central Limit Theorem

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

$Z = \frac{17.5 - 18}{\frac{0.6}{\sqrt{12}}}$

Z = -2.89

Z = -2.89 has a p-value of 0.0019

0.0019 = 0.19% probability that the average fill of the twelve bottles is more than 17.5mL

answered: Guest

answer: 1 and 3

step-by-step explanation:

Which would prove that abc xyz? check all that apply

answered: Guest

acd, jgvjgbgbgvhgbhjjn

Other questions on the subject: Mathematics

Mathematics, 21.06.2019 15:00, paovue48

What is the length of the segment joining the points at (4,5) and (6,-2) round to the nearest tenth if necessary

Answers: 1

continue

Mathematics, 21.06.2019 15:20, alaina3792

Which equation represents a circle with a center at (–3, –5) and a radius of 6 units? (x – 3)2 + (y – 5)2 = 6 (x – 3)2 + (y – 5)2 = 36 (x + 3)2 + (y + 5)2 = 6 (x + 3)2 + (y + 5)2 = 36

Answers: 3

continue

Mathematics, 21.06.2019 16:00, dayanawhite64

Which term best describes the association between variables a and b? no association a negative linear association a positive linear association a nonlinear association a scatterplot with an x axis labeled, variable a from zero to ten in increments of two and the y axis labeled, variable b from zero to one hundred forty in increments of twenty with fifteen points in a positive trend.

Answers: 2

continue

Mathematics, 21.06.2019 22:00, juhavyen

Determine the domain and range of the given function. the domain is all real numbers all real numbers greater than or equal to –2{x: x = –2, –1, 0, 1, 2}{y: y = –2, –1, 0, 1, 2}. the range is all real numbers all real numbers greater than or equal to –2{x: x = –2, –1, 0, 1, 2}{y: y = –2, –1, 0, 1, 2}.

Answers: 1

continue

Do you know the correct answer?

A filling machine fills bottles of nail polish with amounts that are normallydistributed with mean 1...