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Reliability Engineer Certification CRE - ASQ > ⭐️ Statistical Concepts > Flashcards

⭐️ Statistical Concepts Flashcards

1
Q

The diameter of ball bearings with a mean of 75 and a standard deviation of 8.

What is the propability of the average diameter of 10 randommly selected ball bearings being greater than 77?

A

Standard deviation = Z-Value calculation

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2
Q

Laboratory tests of a new polymer prototype yielded the following proportions of a critical compound: 4.2 ; 4.1 ; 4.7 ; 4.9 ; 5.3 ;

What ist the sample standard deviation?

A

use calculator (formular) for sample standard deviation (IV - 8/9 ; 0,498)

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3
Q

A Process measurement has a mean of 758 and a standard devaition of 19.4. If the specification limits are 700 and 800

What precent of product can be expected to be outside the limits, assuming a normal distribution?

A

a) Calculate upper and lower Z-values P

b) use the standard normal table. (IV - 83/86 ; 1.66%)

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4
Q

A machined part has a mean thickness of 0.375 inches, with a standard deviation of 0.002 inches.

What is the propability that a part will be machined with a thickness of less than 0.372 inches?

A

a) Z value is calculated by:
Z = X-µ

b) Look in the standard normal table,
from the Appendix

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5
Q

A tire manufacturer has found that an average of 3 tires per 1000 produced.

What is the propability of finding exactly 3 defective tires in a sample of 1000 tires

A

This is a binominal Distribution P (x, n, p)

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6
Q

The spread of individual observations may be expressed numerically as:

A

6R/D

_
is the approximate population spread
IV -67, 80 and 87/89

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7
Q

An equipment has a hazard function h(t) defined by h(t) = 6 x 10^-8 x t^2 for t >0.

The equipment is required to operate for 100 hours. What is its reliability?

A

a) Equation = R(t) = e ^ - 0,02 ;

b) The hazard rate/function represents an instantaneous failure rate (t) and can be defined as the limit of the failure rate as the time interval length approach zero. The hazard function can be further defined as the conditional propability of failure in the interval (x + dx), given that there was NO failure by x.
(IV - 59 ; 0.98)

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8
Q

Which statement concerning statistical inference (Schlussfolgerung) is true?

A

The point estimate,
is a single value
used to estimate a population parameter.

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9
Q

A survey of 347 purchased laptop computer found that 6 were defective.

What is the propability of getting a defective laptop computer?

A

P = n/N

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10
Q

-

A

-

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11
Q

A total of 200 products are all tested to failure. In order to assess the failures,

which actions should the team take first?

A

Develop a Pareto chart of the failure types

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12
Q

A reliability engineer studies the repair times for biomedical equipment and calculates a mean time to repair (MTTR) of 1.5 hours, with a population standard deviation of 0.2 hours.

If the repair times are normally distributed 95% of all repair times will be less than?

A

a) Z-Value equation

b) take Z with 0.05 Unavailability

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13
Q

When designing a product for mechanical reliability, which factor must be considered?

A

Operations interruption

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14
Q

A sample of repair times in minutes has a lognormal distribution with a mean whose natural logarithm is 5 and a standard deviation with natural logarithm is 0.4.

Find the propability of making a rapair within 248 minutes.

A

a) find formular for R(x) lognormal
b) calculate Z value and look for p in table
c) put all together into formular R(x) lognormal

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15
Q

Which of the following continous distributions is symetrical and unskewed over its typical operating range?

A

Lognormal

IV - 20/40

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16
Q

A process is in control with:

x̄ = 50 ; r̄ = 4.0 ; n = 6

The three sigma limits of the process are approximately:

A

Requires estimation of the deviation:

(1) S = r̄/d2
Look up D2-value for the average range

(2) x̄ ± 3S
= Process Spread

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17
Q

Which of the following expressions is true only when A and B are independent events?

A

P(A∩B) = P(A) x P(B)

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18
Q

A manufacturer startet testing 10 items. Failure occure at 14, 15, 16 cycles. The test was stopped at 30 cycles.

This is an example of:

A

Right censoring

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19
Q

It the propability density function of part failures is:

t = 0 to 5	        f(t) = 0.05
t = 5 to 10	f(t) = 0.15

What is the reliability at x = 8?

A

a) the reliability function is the complement of the cumulative distribution function.
b) the cumulative distribution function represents the propability of failure at any time t.
c) Solution is best obteined by determining the area under the curve of pdf

  • From x = 0 to 5 the area equalts 0.05 x 5 = 0.25
  • From x = 5 to 8 the area equals 0.15 x 3 = 0.45
  • The cummulative distribution function for 8 equals 0.25 + 0.45 = 0.7
  • Since the CDF is for part failures, the R = 1 - 07 = 0.30 (IV - 53/56)
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20
Q

The diameter of a population is normally distributed with a mean of 75 and a standard deviation of 8.

What is the propability of the average diameter of 10 randomly selected ball bearings being greater than 77?

A

Z = X - μ / σ

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21
Q

A process measurement has a mean of 758 and a standard deviation of 19.4.

If the specification limits are 700 and 800, what precent of product can be expected to be outside the limit, assuming a normal distribution?

A

This questions requires a calculation of the upper and lower Z values,

the use of a standard nomal table and a

comparison with the answerts.

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22
Q

When using a P-chart to monitor proportion of nonconforming units in a sample, ..

A

The sample size can vary (IV - 71)

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23
Q

In 4 flips of a coin the propability of two heads and two tails (in any order) is:

A

This problem requires a calculation for propability using the binominal distribution formular/table (IV - 45) Parameters are:

n = the sample size 
y = the number of occurences (sometimes r) 
p = the propability of occurence
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24
Q

What is the general use of an R-chart?

A

To determine if the process variation is in control.

_
An R chart, or range chart is used to monitor the variation of a process.
It is generally used along with an X-bar chart, which monitors the process mean (IV - 66/69)

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25
Q

To determine a statistical tolerance limit from a sample with a normal distribution,
what must be specified? (5)

A

1) sample mean,
2) standard deviation,
3) sample size,
4) tolerance level
5) confidence level.
__
Statistical tolerance limits are similar to “process capability”, they show the practical boundries of process variability. Convidence level are used to determine a statistical tolerance limit. Population size ist NOT considered, (IV - 95)

.

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26
Q

What must be specified,

to determine a statistical tolerance limit from a sample with a normal distribution? (5)

A

1) Sample mean
2) standard deviation
3) sample size
4) tolerance level
5) confidence level

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27
Q

A plant operates 8 systems. There is a 0.3 propability that a machine fail.

What is the propability that at least 6 systems remain operating?

A

Use the binominal table with n = 8 ; r = 2 or fewer and p = 0.30

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28
Q

The variations,
which the normal distribution curve describes,
are due to: (4)

A

Variation that occurs within the normal distribution is called:

normal,
natural,
chance or
nonassignable.

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29
Q

Which tool uses logical analysis and starts with system failures?

A

FTA

30
Q

What is the survival probability?

A

Z = X - μ / σ

_
The cumulative failure probabilities are the likelihood of failing instead of surviving.

31
Q

What is the general use of an R-chart?

A

To determine if the process variation is in control.
__
An R chart, or range chart is used to monitor the variation of a process.
It is generally used along with an X-bar chart, which monitors the process mean (IV - 66/69)

32
Q

To determine a statistical tolerance limit from a sample with a normal distribution,
which of the following must be specified? (5)

A

(1) sample mean,
(2) standard deviation,
(3) sample size,
(4) tolerance level and
(5) confidence level

_
Statistical tolerance limits are similar to “process capability”, they show the practical boundries of process variability.

Population size ist NOT considered, (IV - 95)

33
Q

To determine a statistical tolerance limit from a sample with a normal distribution, which of the follwoing must be specified? (5)

A 
Sample mean and 
standard deviation, 
sample size, 
tolerance level and 
confidence level.
34
Q

A plant operates 8 systems. There is a 0.3 propability that a machine fail. What is the propability that at least 6 systems remain operating?

A

Use the binominal table with n = 8 ; r = 2 or fewer and p = 0.30

35
Q

The variations,
which the normal distribution curve describes,
are due to:

(4)

A

Variation that occurs within the normal distribution is called

normal,
natural,
chance or
nonassignable.

36
Q

If a product exhibits a constant failure rate, the R(t) at MTBF is

A

0.37

37
Q

If the mean-time-between failures is 200 hours, what is the propability of surviving for 2000 hours?

A

R = e ^ - λt

ACHTUNG: λ = 1 / MTBF ; R * 100 = %

38
Q

The diameter of a population of ball bearings is normally distributed with a mean of 75 and a standard deviation of 8.

What is the propability of the average diameter of 10 randomly selected ball bearings being breater than 77’

A

1) the standard deviation of the average of a sample of 10 is equal to the standard deviation of the individuals divided by the
square root of the sample size

2) Z = X - μ / σ (mean = 0 and standard deviation = 1)

39
Q

If 10 lots are inspected, with a total of 10 000 pieces and a combined 30 pieces are found to be defective, what is the upper control limit for the proportion defective (3)

A

1) n = total pieces / total lots
2) p = total defective / total inspected
3) UCL, LCL = p ± 3 √ (p(1-p) / n) (IV-72)

40
Q

For right censored, time-based life data - failures typically occur ..

A

beyond the present time

41
Q

When designing a product for mechanical reliability,

which factor must be considered?

A

Operations interruption

42
Q

When claculating confidence intervals for the population mean,
the confidence intervals are narrower …

A

when the sample size increases

43
Q

-

A

-

44
Q

Twelve eggs are in a carton of jumbo eggs. If the weights are xx-xx, for the entire data set, what is the standard deviation?

A

Using a calculator, the population standard deviation σ = xx (asked for) and the sample standard deviation s = xx

45
Q

-

A

-

46
Q

The propability density function of the exponential distribution is:

A

The exponential reliability function is defined as e^-λt, which is the integral of the exponentila propability density function λe^-λt. (IV-57/58)

47
Q

What indicates an “sigma” out-of-control condition?

A

2 of 3 consecutive points outside the 2-sigma warning limits
(IV-74)

48
Q

The diameter of a population of ball bearings is normally distirbuted with a mean of 75 and a standard devaition of 8.
What is the propability of the average diameter of 10 randomly selected ball bearinds beeing greater then 77.

A

1) Transform standard deviation Average (σ) to sample deviation: s = σ / √n
2) use equation: z = x - μ / σ

49
Q

A process measurement ha a mean of xx and a standard deviation of xx. If the specification limits are xx and xx, what precent of product can be
expected to be outside the limits, assuming a normal distribution?

A

Requires a calculation of the upper and lower Z values

Zu = USL - x̄ / s 
Zl = x̄ - LSL / s

=> The total failure rate is P(Z ≥ x) + P(Z ≤ x)

50
Q

Taking the natural logarithms of data which follows a lognormal distribution will yield values which approcimated which type of distribution?

A

Normal

51
Q

Seven apples were selected at random from a box of apples. The wights were xx-xx. What is the standard deviation?

A

Using a calculator with a standard deviation of the sample, s = xx

52
Q

If 10 lots are inspected, with a total of 10`000 pieces and a combined 30 pieces are found to be defective,
what is the upper control limit for the proportion devective?

A

see E&Q Attribute Data Control

	n bar = total inspected / total lots 
	p bar = total defective / total inspected
	UCL, LCL = p bar ± 3 √ p bar(1 - p bar) / n bar
53
Q

If the propability density function of failures is:

t = 0 to 5	 	f(t) = 0.05
t = 5 to 10 	f(t) = 0.15

What is the reliability at x = 8?

A

The cumulative distribution function at 8 equals 0.25 + 0.45 = 0.7

Since the CDF is for part failures, the R = 1 - 0.7 = 0.30

54
Q

Which statement is true aregarding Cp and Cpk?

A

A Cp of 1.2 is better than a Cp of 1.0

=> Cp and Cpk are process capability indicies
=> Cp = U - L / 6σ
55
Q

Which of the following is a nonparametric test?

A

Mood`s median test

_
Nonparametric tests are often called “distribution-free” since they make no assumption regarding the population distribution. (IV-10)

56
Q

Which of the following is a descriptor of the dispersion in a propability distribution?

A

Kurtosis and standard deviation

IV-4/8 and 19/20

57
Q

The Lottery draws 6 numbers from 50. How many combinations are possible?

A

NCR, order is not important. Enter 50 NCR 6 = 1,589 x 10^7

58
Q

What is the propability of getting an even number when rolling two normal dice?

A

The propability of an even total is 1 in 2.

59
Q

The length of a certain bushing are normally distributed, with mean X-bar.

How many standard deviation units symmetrical about X-bar will include 80% of the lengths?

A

The standard normal table must be consulted for the Z-value beyond which 10% of the readings will fail.

__
Thus the solution is ± 1.28

60
Q

In the binominal expression.

The sum of the exponents of each term after expansion is …

A

equal to the sample size

_
The coefficient term is: P(y/p) = n! / y!(n-y)!

61
Q

Which distribution has population mean equal to the population variance?

A

Poisson

=> It is an elementary that the mean and variance of Y are both equal to λ; E[Y] = Var[Y] = λ. In fact, this property characterizes the Poisson distribution.

62
Q

Which control chart is based on the Poisson distribution an is used for varying sample size?

A

u (number of defects per unit)

=> p (records fraction defective) ; np (records number of defectives) ; c (records number of defects)

63
Q

Which type of control chart should a team use

when monitoring defects
by drawing samples each of equivalent size for inspection or test?

A

c (records number of defects)

_
p (records fraction defective) ; np (records number of defectives) ; u (number of defects per unit)
64
Q

A small production lot of 50 parts is produced and 5 parts are taken as samples for testing. If there were 8 defective parts in the entire lot, what is the probability that no failed units will be found during the test?

A

Hypergeometric / Binominal Distribution p = 8/50 ; n = 5 ; r = 0

65
Q

Sample measurements were taken after an adjustment was made to a machine.

What range contains the true average to a 95% confidence?

A

Use the t-distribution value. Since the question concerns two tails.

_
Range = x̄ ± t x s /√n

66
Q

The MTBF for an automotive component is 6000 hrs.

What is the propability that the part fill have failed by 7000 hrs of operation?

A

1) Probability of survival: R = e ^ - λt

2) Propability of failure = 1 - R

67
Q

What is the propability that a train will arrive either early or late?

If the propability that a train will arrive early at a station is 0.23 and the propability that the train will arive late at the station is 0.18.

A

This is a mutuallly exclusive event since the train cannot arrive both early and late the same time.

0,41

68
Q

In dissecting the total breakdown of variation, the lot-to-lot and
stream-to-stream variatio have been determined and minimized.

The next logical step would be to reduce:

A

The time-to-time variation

69
Q

The propability density function (f(t)) of the exponential distribution is:

A

f(t) = λe ^ - λt

70
Q

For Right-censored, time-based life data - failures typically occur

A

beyond the present time

71
Q

Which table should be used to determine a confidence interval on the mean when σ is not known and the sample size is 10?

A

t

_
The F and X2 tests apply to variances. The Z table is used when the population standard deviation is known. The student`s t distribution is used for making inferences about a population mean when the population variance is unknown and the sample size is small (less than 30)

72
Q

If an average of 10 cars per hour arrive at a car wash, what is the propability of having exactly 8 cars in a given hour?

A

This is a Poisson distribution with mean = 10 and x = 8

Reliability Engineer Certification CRE - ASQ (10 decks)

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