How Do Standard Deviations Affect Tolerance in Normal Distributions?

To do that, multiply both sides of the equation by 0.12. x- 16.29= 0.12z and add 16.29 to both sides. x= 0.12z+ 16.29.
  • #1
brad sue
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Hi, I have 2 problems I would like some help. It is about normal distribution(probability)

PROBLEM 1: Extruded plastic rods are automatically cut into lenghts of 6 inches. Actual lengths are normally distributed about a mean of 6 inches and their standard deviation is 0.06 inch.

1- what proportion of the rods have lenghts that are outside the tolerance limits of 5.9 and 6.1 inches?


Here I did:
p=F((6.1-6)/0.06)- F((5.9-6)/0.06)= F(1.67)-F(-1.67)=0.9525-0.0475=0.905

P(outside tolerance)=1-0.905=0.095

2- To what value does the standard deviation needs to be reduce if 99% of the rods must be within the tolerance?
I can not fin this question.

PROBLEM 2:
In a photographic process, the developping time of prints may be looked upon as a random variable having the normal distribution with a mean of 16.28 seconds and a standard deviation of 0.12 second.
- For which value is the probability 0.95 that it will be exceeded by the time it takes to develop one of the prints?
I don't get this one.

Can I have some help please?
 
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  • #2
Well, for the second half of problem one, substitute all the values in except those given to you:

[tex] P(5.9<Z<6.1) = \phi (\frac{x_1 - \mu}{\sigma}) - \phi (\frac{x_2 - \mu}{\sigma}) [/tex]
[tex] 0.99 = \phi (\frac{6.1 - 6}{\sigma}) - \phi (\frac{5.9 - 6}{\sigma}) [/tex]
[tex] 0.99 = \phi (\frac{0.1}{\sigma}) + \phi (\frac{-0.1}{\sigma})[/tex]
[tex] 0.99 = \phi (\frac{0.1}{\sigma}) + \phi (\frac{0.1}{\sigma}) - 1[/tex]
[tex] 1.99 = 2 \phi (\frac{0.1}{\sigma}) [/tex]
[tex] 0.995 = \phi (\frac{0.1}{\sigma}) [/tex]

Use the inverse normal distribution function on 0.995, and the answer should be apparent.
 
Last edited:
  • #3
I got the first problem.
Thanks lapo3399
 
  • #4
brad sue said:
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Hi, I have 2 problems I would like some help. It is about normal distribution(probability)

PROBLEM 1: Extruded plastic rods are automatically cut into lenghts of 6 inches. Actual lengths are normally distributed about a mean of 6 inches and their standard deviation is 0.06 inch.

1- what proportion of the rods have lenghts that are outside the tolerance limits of 5.9 and 6.1 inches?


Here I did:
p=F((6.1-6)/0.06)- F((5.9-6)/0.06)= F(1.67)-F(-1.67)=0.9525-0.0475=0.905

P(outside tolerance)=1-0.905=0.095

2- To what value does the standard deviation needs to be reduce if 99% of the rods must be within the tolerance?
I can not fin this question.

PROBLEM 2:
In a photographic process, the developping time of prints may be looked upon as a random variable having the normal distribution with a mean of 16.28 seconds and a standard deviation of 0.12 second.
- For which value is the probability 0.95 that it will be exceeded by the time it takes to develop one of the prints?
I don't get this one.
In other words, Find x so that P(X> x)= 0.95. Look up the z that gives P(z)= 0.95 in the standard normal distribution and solve (x- 16.29)/0.12= z.
 

FAQ: How Do Standard Deviations Affect Tolerance in Normal Distributions?

1. What is a normal distribution of rods?

A normal distribution of rods refers to a statistical concept where the values of a particular variable tend to cluster around the mean in a symmetrical bell-shaped curve. This distribution is often used to represent natural phenomena, such as the height or weight of individuals in a population.

2. How is a normal distribution of rods calculated?

The calculation for a normal distribution of rods involves using the mean, standard deviation, and a mathematical formula known as the Gaussian function. This formula takes into account the average value, the spread of the data, and the probability of a particular value occurring.

3. What is the purpose of studying the normal distribution of rods?

Studying the normal distribution of rods can help scientists better understand the characteristics and behavior of a particular variable in a population. It can also be used to make predictions and determine the likelihood of certain values occurring.

4. What are the assumptions of a normal distribution of rods?

The assumptions of a normal distribution of rods include that the data is continuous, the mean, median, and mode are equal, and the data is normally distributed. Additionally, the data should not be skewed and should follow the 68-95-99.7 rule, where 68% of the data falls within one standard deviation from the mean, 95% within two standard deviations, and 99.7% within three standard deviations.

5. How is a normal distribution of rods used in scientific research?

A normal distribution of rods is commonly used in scientific research to analyze and interpret data, make comparisons between groups, and determine the probability of certain outcomes. It is also used in hypothesis testing and in the development of statistical models.

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