Deck 7: Sampling Distributions

Question 1

Small haphazard changes in a process variable due to alteration in the production environment that is not controllable are said to be assignable causes.

Answer 1

False

Question 2

For a control chart, the lower and upper control limits are usually set at:

A) one standard deviation from the centerline
B) two standard deviations from the centerline
C) three standard deviations from the centerline
D) four standard deviations from the centerline
E) five standard deviations from the centerline

Answer 2

three standard deviations from the centerline

Question 3

If a process is in control, we expect all the data values to fall within three standard deviations of the mean.

Answer 3

True

Question 4

Once a process is in control and is producing a satisfactory product, the process variables are monitored by use of control charts.

Question 5

The mean of the sample means and the standard deviation of 50 samples of size 5 taken from a production process under control are found to be 300 and 25, respectively. The lower control limit for the chart is located at:

A) 333.54
B) 289.39
C) 310.61
D) 266.46
E) 342.88

Question 6

The first objective in statistical process control is to eliminate assignable causes of variation in the process variable and then get the process in control. The next step is to reduce variation and get the measurements on the process variable within specification limits, the limits within which the measurements on usable items or services must fall.

Question 7

If a process is in control, we expect all the data values to fall within two standard deviations of the mean.

Question 8

Assignable cause variation refers to variation in the output of a process that is naturally occurring, expected, and that may be the result of random causes.

Question 9

Random cause variation refers to variation n, the output of a process that is unexpected and has an assignable cause.

Question 10

Fifty samples of size 500 were drawn from a manufacturing process and the number of defectives in each sample was counted. The average of the sample proportion was 0.032. The centerline for the p chart is:

A) 0.032
B) 16.0
C) 0.512
D) 0.968
E) 4.2

Question 11

Twenty five samples of size 1000 each were drawn from a manufacturing process and the number of defectives in each sample was counted. The average sample proportion was 0.05. The upper control limit for the p chart is:

A) 0.0475
B) 0.0206
C) 0.0293
D) 0.0707
E) 0.0635

Question 12

Random variation in a process can be eliminated.

Question 13

Process control charts, such as the and p-charts, are used to provide signals to indicate when the output of a process is out of control.

Question 14

Statistical process control (SPC) methodology was developed to monitor, control, and improve products and services.

Question 15

If a process is in control, we expect all the data values to fall within one standard deviation of the mean.

Question 16

In most processes, the process control limits are set to correspond with the specification limits on the process.

Question 17

The cause of a change in a process variable being monitored is regarded as random variation if it can be found and corrected.

Question 18

One of the most common sources of random cause variation is the people who are working in the process.

Question 19

In control process is typically defined as a process in which all output is operating within 3 standard deviation of the centerline of the process.

Question 20

If the variation in a process variable is solely random, the process is said to be in control.

Question 21

On a particular freeway in Michigan, it is reported that the proportion of cars that exceed the speed limit is 0.15. Given this information, the probability that a sample of 250 cars will have a sample proportion below 0.12 is approximately .0918.

Question 22

The standard error of the sampling distribution of sample proportion, SE ( ), depends on the value of the population proportion p, and the closer the value of p to .50, the larger SE ( ) will be for a given sample size n.

Question 23

Standard error of the sample proportion is another term for the variance of the sampling distribution of the sample proportion .

Question 24

A producer of brass rivets randomly samples 500 rivets each hour and calculates the proportion of defectives in the sample. The mean sample proportion calculated from 250 samples was equal to 0.025. Calculate the upper and lower control limits for a control chart for the proportion of defectives in samples of 500 rivets.
UCL = ______________
LCL = ______________

Question 25

Assignable cause variation is also called:

A) random variation
B) special cause variation
C) common cause variation
D) within-group variation
E) none of these

Question 26

Among the most common sources of random variation is:

A) people
B) materials
C) equipment
D) all of these
E) none of these

Question 27

Sixty samples of size 600 each were drawn from a manufacturing process and the number of defectives in each sample was counted. The average sample proportion was 0.04. The lower control limit for the p chart is:

A) 0.008
B) 0.016
C) 0.064
D) 0.048
E) 0.032

Question 28

Samples of n = 100 items were selected hourly over a 100-hour period, and the sample proportion of defectives was calculated each hour. The mean of the 100 sample proportions was 0.042.
Calculate the upper and lower control limits for a p chart.
UCL = ______________
LCL = ______________
Explain how to construct a p chart for the process and how it can be used.
________________________________________________________

Question 29

The Central Limit Theorem applies to the sampling distribution of sample proportion but not for the sampling distribution of sample mean .

Question 30

As the sample size increases, the standard error of the sample proportion decreases.

Question 31

The expression SE ( ) represents the standard error sampling distribution of the sample proportion, .

Question 32

A candy bar factory is pouring chocolate into molds to cool. The finished bars are sold as 1.25 ounce bars. The company will lose money if the molds are over-filled. If the molds are under-filled, the weight of the candy bars will be less than the wrapper label says, and the Food and Drug Administration will fine the company for misrepresenting the size of its product. The company wants to create an chart to monitor the weight of the candy bars. Suppose there are 5 samples of size 30 each taken, and their sample means are 1.27, 1.22, 1.26, 1.23, and 1.26 ounces.
What is the centerline value?
______________
The calculated value of s, the sample standard deviation of all nk = (30)(5) = 150 observations, is 0.12 ounces. What is the standard error of the mean of 30 observations?
______________
What are the control limits?
UCL = ______________
LCL = ______________
How is the chart used in this situation?
________________________________________________________

Question 33

The mean of the sampling distribution of the sample proportion when n = 100 and p = 0.5 is 5.0.

Question 34

As a general rule, the normal distribution is used to approximate the sampling distribution of the sample proportion only if the sample size n is greater than or equal to 30.

Question 35

Suppose that 56% of all registered voters in Michigan are supporting John Kerry for president. A sample of 1500 voters is selected at random from Michigan. Based on the concept of sampling distribution of sample proportion , it can be assumed that = .56.

Question 36

Recall the rule of thumb used to indicate when the normal distribution is a good approximation of the sampling distribution for the sample proportion . For the combination n = 25; p = 0.05, the rule is satisfied.

Question 37

The expression SE ( ) represents the standard deviation of the sampling distribution of the sample proportion, .

Question 38

The sampling distribution of the sample proportion can be approximated by a normal distribution as long as the population proportion p is very close to .50.

Question 39

Suppose an actual census showed that 20% of the households in Michigan have incomes in excess of $60,000. Assume that a random sample of 500 households in Michigan is taken. Then, the standard error of the sampling distribution of sample proportion of households who have incomes in excess of $60,000 will be:

A) 0.0179
B) 0.0003
C) 0.1600
D) 0.0256
E) 0.1482

Question 40

The standard error of the sampling distribution of the sample proportion when n = 100 and p = 0.15 is 0.001275.

Question 41

A student government representative at a local university claims that 60% of the undergraduate students favor a move to Division I in college football. A random sample of 250 undergraduate students is selected.
Is the sampling distribution of the sample proportion approximately normal?
______________
What is the mean?
______________
What is the standard deviation?
______________
Find the probability that the sample proportion exceeds 0.65.
______________

Question 42

If the standard error of the sampling distribution of the sample proportion is 0.02049 for samples of size 500, then the population proportion must be either:

A) 0.2 or 0.8
B) 0.5 or 0.5
C) 0.3 or 0.7
D) 0.6 or 0.4
E) 0.2 or 0.5

Question 43

A candy bar company is interested in reducing the percent defective candy bars made, where a defective candy bar has too few almonds by weight. The company randomly samples 100 candy bars a day for 5 days and finds the percent defective to be 0.0200, 0.0125, 0.0225, 0.0100, and 0.0150. The company wants to construct a control chart for the proportion defective in samples of size n = 100.
Estimate the process fraction defective p.
______________
Estimate the standard deviation of the sample proportions.
______________
What are the control limits?
UCL = ______________
LCL = ______________
How are these control limits used?
________________________________________________________

Question 44

As a general rule, the normal distribution provides a good approximation to the sampling distribution of the sample proportion only if:

A) the sample size n is greater than 30
B) the sample size n is greater than 100
C) the population proportion p is greater than 0.50
D) the underlying population has a small standard deviation and n is large
E) np and n(1-p) are both greater than 5

Question 45

Which of the following statements is true regarding the sampling distribution of sample proportion ?

A) The mean of the sampling distribution will equal the population proportion p.
B) The sampling distribution will be approximately normal provided that np > 5 and nq > 5.
C) An increase in the sample size n will result in a decrease in the standard error of.
D) All of these.
E) None of these.

Question 46

A statistics professor has stated that 90% of his students pass the class. To check this claim, a random sample of 150 students who have taken the class indicated that 129 passed the class. If the professor's claim is correct, what is the probability that 129 or fewer will pass the class this semester?

A) .9484
B) .0516
C) .5516
D) .4484
E) .1624

Question 47

Completely describe the distribution of the sample proportion for samples of size n = 500 from a population with p = 0.4.
Is the sampling distribution of the sample proportion approximately normal?
______________
What is the mean?
______________
What is the standard deviation?
______________

Question 48

A recent nationwide survey by the American Cancer Society found the percentage of women who smoke has increased to 30%. That seems a little low for your state, so you sample 500 women from your state and find 180 of them smoke.
Find the sample proportion of women who smoke in your state.
______________
Describe sampling distribution of ; the proportion of women who smoke.
Is the sampling distribution approximately normal?
______________
What is the mean?
______________
What is the standard deviation?
______________
What is the probability that at least 36% of women in your state are smokers if the true population proportion p = 0.30?
______________
Does your answer to the previous question cast doubt on the American Cancer Society's claim?
______________

Question 49

Suppose public opinion is split 65% against and 35% for increasing taxes to help balance the federal budget. 500 people from the population are selected randomly and interviewed.
Is the sampling distribution of the sample proportion of people who are in favor of increasing taxes to help balance the federal budget approximately normal?
______________
What is the mean?
______________
What is the standard deviation?
______________
What is the probability the proportion favoring a tax increase is more than 30%?
______________

Question 50

The proportion of individuals with an Rh-positive blood type is 88%. You have a random sample of n = 500 individuals.
What is the mean of , the sample proportion with Rh-positive blood type?
______________
What is the standard deviation?
______________
Is the distribution approximately normal?
______________
What is the probability that the sample proportion exceeds 85%.
______________
What is the probability that the sample proportion lies between 86% and 91%?
______________
99% of the time, the sample proportion would lie between what two limits?
Lower Limit = ______________
Upper Limit = ______________

Question 51

Assume that the proportion of defective items in a sample of 400 items is 0.20.
Is the normal approximation to the sampling distribution of appropriate in this situation?
______________
Find the probability that is greater than 0.23.
______________
Find the probability that lies between 0.16 and 0.24.
______________

Question 52

A well-known juice manufacturer claims that its citrus punch contains 15% real orange juice. A random sample of 150 cans of the citrus punch is selected and analyzed for content composition.
Is the sampling distribution of the sample proportion approximately normal?
______________
What is the mean?
______________
What is the standard deviation?
______________
Find the probability that the sample proportion will be less than 0.10.
______________
Would a value of = 0.25 be considered unusual?
______________
Explain.
________________________________________________________

Question 53

A TV pollster believed that 70% of all TV households would be tuned in to Game 6 of the 1997 NBA Championship series between the Chicago Bulls and the Utah Jazz. A random sample of 500 TV households is selected.
Is the sampling distribution of the sample proportion approximately normal?
______________
What is the mean?
______________
What is the standard deviation?
______________
Find the probability that the sample proportion on will be between 0.65 and 0.75.
______________

Question 54

Which of the following statements is false regarding the Central Limit Theorem (CLT) of the sample proportion ?

A) CLT indicates that the mean of the sampling distribution ofwill be equal to the population proportion p.
B) CLT indicates that the sampling distribution ofcan be approximated by a normal distribution if np > 5 and nq > 5.
C) CLT is applied regardless of the sample size n and the population proportion p.
D) CLT is applied to non-normal populations.
E) CLT indicates that the mean of the sampling distribution ofwill be equal to the population proportion p and CLT indicates that the sampling distribution ofcan be approximated by a normal distribution if np > 5 and nq > 5.

Question 55

Given a population proportion of p = .8 and a sample size of n = 100, these imply a standard deviation of the sampling distribution of the sample proportion of:

A) .0258
B) .0355
C) .0400
D) .0538
E) none of these

Question 56

Suppose a sample of 120 items is drawn from a population of manufactured products and the number of defective items is recorded. Prior experience has shown that the proportion of defectives is 0.05.
Is the sampling distribution of the proportion of defectives approximately normal?
______________
What is the mean?
______________
What is the standard deviation?
______________
What is the probability that the sample proportion is less than 0.10?
______________
How would the sampling distribution of the sample proportion change if the sample size were raised to 200?
________________________________________________________

Question 57

A machine which manufactures a part for a car engine was observed over a period of time before a random sample of 300 parts was selected from those produced by this machine. Out of the 300 parts, 15 were defective.
Find the proportion of defective parts in the sample.
______________
Is the sampling distribution of the proportion of defectives approximately normal?
______________
What is the mean?
______________
What is the standard deviation?
______________
What is the probability that the sample proportion will lie within 0.02 of the true population proportion of defective parts?
______________

Question 58

In a recent study, it was reported that the proportion of employees who miss work on Fridays is 0.15, and that the standard deviation of the sampling distribution of sample proportion is 0.025. However, the report did not indicate what the sample size n was. What was it?

A) 204
B) 26
C) 108
D) 58
E) More information needed to determine the value of n.

Question 59

Which of the following statement is true?

A) The mean of all possible sample proportionsand the expected value of the sampling distribution ofare the same.
B) The standard deviation of the sampling distribution of the proportionis denoted by SE ().
C) The standard deviation of the sampling distribution of the proportionis denoted by().
D) The mean of all possible sample proportionsand the expected value of the sampling distribution ofare the same and The standard deviation of the sampling distribution of the proportionis denoted by SE ().
E) The mean of all possible sample proportionsand the expected value of the sampling distribution ofare the same and The standard deviation of the sampling distribution of the proportionis denoted by().

Question 60

A southern state has an unemployment rate of 6%. The state conducts monthly surveys in order to track the unemployment rate. In a recent month, a random sample of 700 people showed that 35 were unemployed.
If the true unemployment rate is 6%, describe the sampling distribution of . Is the sampling distribution of the sample proportion approximately normal?
______________
What is the mean?
______________
What is the standard deviation?
______________
Find the probability that the sample unemployment rate is at most 5%.
______________
Assume the population proportion, p, is unknown. Describe the sampling distribution of based on the most recent sample. Is the sampling distribution of the sample proportion approximately normal?
______________
What is the mean?
______________
What is the standard deviation?
______________
Based on the most recent sample, find the probability that the sample proportion will lie within 0.005 of the true proportion p of people who are unemployed.
______________

Question 61

The total area under a probability density function curve is a number between 0.5 and 1.0.

Question 62

The sampling distribution of the sample mean is exactly normally distributed, regardless of the sample size n.

Question 63

The most important contribution of the Central Limit Theorem is in statistical inference.

Question 64

According to one study, 50% of Americans admit to overeating sweet foods when stressed. Suppose that the 50% figure is correct and that a random sample of n = 100 Americans is selected.
Does the distribution of , the sample proportion of Americans who relieve stress by overeating sweet foods, have an approximately normal distribution?
______________
What is the mean?
______________
What is the standard deviation?
______________
What is the probability that the sample proportion, , exceeds 0.54?
______________
What is the probability that lies within the interval 0.39 to 0.59?
______________
Would it be unusual if the sample proportion were as small as 34%?
______________

Question 65

The standard deviation of a statistic that is used to estimate an unknown parameter is called the standard error of the statistic.

Question 66

The sample standard deviations measures the variability of all possible sample mean values that might be obtained.

Question 67

According to the Central Limit Theorem, for large samples the standard error of the sample mean is the population standard deviation divided by the square root of the sample size.

Question 68

The Central Limit Theorem does not apply to the sample means of large samples drawn from a discrete distribution.

Question 69

The total area under a probability density function curve is equal to 1.0.

Question 70

If random samples of size n = 36 are drawn from a nonnormal population with finite mean and standard deviation , then the sampling distribution of the sample mean is approximately normally distributed with mean and standard deviation

Question 71

When all possible simple random samples of size n are drawn from a population that is normally distributed, the sampling distribution of the sample means will be normal regardless of sample size n.

Question 72

The spread of the distribution of sample means is considerably less than the spread of the sampled population.

Question 73

As the sample size increases, the standard error of the sample mean decreases.

Question 74

A summary measure calculated for a population is called a parameter and is designated by Greek letters (such as for mean or for proportion).

Question 75

If is the mean of a simple random sample taken from a large population and if the N population values are normally distributed, the sampling distribution of is also normally distributed, regardless of sample size, n.

Question 76

A sampling distribution is a probability distribution that shows the likelihood of occurrence associated with all the possible values of a parameter, which values would be obtained when drawing all possible samples of a given size from a population.

Question 77

If random samples of size n = 50 are drawn from a nonnormal population with finite mean =100 and standard deviation = 20, then, the sampling distribution of the sum of sample measurements is approximately normally distributed with mean and standard deviation .

Question 78

A production process produces 10 percent defective items.
Describe the sampling distribution of the proportion of defectives for a simple random sample of n = 50.
Is the distribution approximately normal?
______________
What is the mean?
______________
What is the standard deviation?
______________
What is the likelihood of encountering a sample proportion within 0.03 of the population proportion?
______________
If the production process is shut down whenever a sample proportion of defectives exceeds 0.05, what is the likelihood of this happening?
______________

Question 79

A sampling distribution is defined as a sample chosen in such a way that every possible subset of like size has an equal chance of being selected.

Question 80

The Central Limit Theorem describes the distribution of the sample mean except for populations that are normal.