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sampling distribution of the sample means

The probability distribution of the population of all possible sample means that could be obtained from all possible samples of the sample size.

sampling distribution assumption

has a normal distribution if the sampled population has a normal distribution; has the same mean μ, denoted with "x bar"; has a standard deviation of: σ ÷ (√n)

Central Limit Theorem

If the sampled population is N( μ , σ ) , then the population of the sampling distribution of the sample means is N(μ, [σ ÷ √n]) regardless of sample size

if n is sufficiently large...

No matter what the distribution of the sampling population is, if the sample size n is sufficently large (n ≥ 30) , then the population of all possible sample means is approximately normally distributed. The larger the sample size n is, the more nearly normally distributed is the population of all possible sample means.

when to use sampling distribution

asks for mean ____ (ie mean payment time)

Sampling Distribution of the sample proportion

The proportion of all possible sample proportions is approximately normally

distributed with mean "μ denoted with p" = p and standard deviation "σ denoted with p" = √(pq ÷ n) if (np ≥ 5 and n(1-p) ≥ 5).

distributed with mean "μ denoted with p" = p and standard deviation "σ denoted with p" = √(pq ÷ n) if (np ≥ 5 and n(1-p) ≥ 5).

"p hat"

probability ÷ sample size, then solve for z-value

Stratified Random Sampling

First the population is divided into nonoverlapping groups of similar elements (people, objects, etc.) then a random sample is selected from each stratum, and these samples are combined to form the full sample.

Systematic Sampling

Systematically select a sample of n elements without replacement from a frame of N elements. We ÷ N by n & round the result down to the nearest whole number. Let the rounded result l , we then randomly select one element from the first l elements in the frame. This is the first element in the systematic sample. The remaining elements in the sample are obtained by selecting every l the element following the first element.

sampling error

not a mistake, just a variation due to sampling

The sampling distribution of a sample statistic is the probability distribution of the population of all possible values of the sample statistic.

true

A sample statistic is an unbiased point estimate of a population parameter if the mean of the populations of all possible values of the sample statistic equals the population parameter.

true

A minimum variance, unbiased point estimate has a variance that is as small or smaller than the variances of any other unbiased point estimate.

true

The reason sample variance has a divisor of n-1 rather than n is that it makes the variance an unbiased estimate of the population variance.

true

The central limit theorem states that as sample size increases, the population distribution more closely approximates a normal distribution.

false

If the sampled population is exactly normal distribution, then the sampling distribution of "x bar" is also expected to be normal regardless of the sample size.

true

If a population is known to be normally distributed, then it follows that the sample mean must equal the population mean.

false

If the sampled population distribution is skewed, then in most cases the sampling distribution of the mean can be approximated by the normal distribution if the sample size n is at least 30.

true

As the sample size increases, the standard deviation of the sampling distribution increases

false

The mean of the sampling distribution of "x bar" is always equal to the mean of the sampled population.

true

The sampling distribution of the sample mean is developed by repeatedly taking samples of size n and computing the sample means and reporting the resulting sample means in the form of a probability distribution.

true

The central limit theorem states that as the sample size increases the distribution of the sample _____ approach the normal distribution.

means

Consider two population distributions labeled A and B. Distribution A is highly skewed and non-normal, while the distribution B is slightly skewed and near normal. In order for the sampling distributions of A and B to achieve the same degree of normality:

population A will require a larger sample size

As the sample size ______ the variation of the sampling distribution of "x bar" ______.

increases, decreases

If the sampled population has a mean 48 and standard deviation 16, then the mean and the standard deviation for the sampling distribution of "x bar" for n= 16 are:

48 and 4

16A manufacturing company measures the weight of boxes before shipping them to the customers. If the box weights have a population mean and standard deviation of 90 lbs. and 24 lbs. respectively, then based on a sample size of 36 boxes, the probability that the average weight of the boxes will exceed 94 lbs. is:

(check notebook)

A manufacturing company measures the weight of boxes before shipping them to the customers. If the box weights have a population mean and standard deviation of 90 lbs. and 24 lbs. respectively, then based on a sample size of 36 boxes, the probability that the average weight of the boxes will be less than 84 lbs. is:

6.68%

Whenever the population has a normal distribution, the sampling distribution of "x bar" is normal or near normal distribution:

for any sample size

If a population distribution is known to be normal, then it follows that:

"none of the above"

For non-normal populations, as the sample size (n) ____, the distribution of sample means approaches a(n) _____ distribution.

increases, normal

In a manufacturing process a machine produces bolts that have an average length of 3 inches with a variance of .03. If we randomly select three bolts from this process process the standard deviation of the sample mean is___.

.1732

In a manufacturing process a machine produces bolts that have an average length of 3 inches with a variance of .03. If we randomly select three bolts from this process, the probability the mean length of the bolt is at least 3.16 inches is ________.

5.48%

The population of lengths of aluminum-coated steel sheets is normally distributed with a mean of 30.05 inches and a standard deviation of 0.2 inches. What is the probability that randomly selected 4 sheets will have an average length of less than 29.9 inches long?

.0668

The population of lengths of aluminum-coated steel sheets is normally distributed with a mean of 30.05 inches and a standard deviation of 0.2 inches. What is the probability that randomly selected 4 sheets will have an average length between 30.25 and 30.35 inches long?

.02145

The population of lengths of aluminum-coated steel sheets is normally distributed with a mean of 30.05 inches and a standard deviation of 0.2 inches. What is the probability that average length of a steel sheet from a sample of 9 units is more than 29.95 inches long?

.9332

If we have a sample size of 100 and the estimate of the population proportion is .10,

we can estimate the sampling distribution of "p hat" with a normal distribution.

we can estimate the sampling distribution of "p hat" with a normal distribution.

true

30If p = .9 and n = 40, then we can conclude that the sampling distribution of "p hat" is approximately a normal distribution.

false

31The population of all sample proportions has a normal distribution if the sample size (n) is sufficiently large. The rule of thumb for ensuring that n is sufficiently large is:

np ≥ 5 and n(1-p) ≥ 5

32If we have a sample size of 100 and the estimate of the population proportion is .10, the standard deviation of the sampling distribution of the sample proportion is:

.03

33The internal auditing staff of a local manufacturing company performs a sample audit each quarter to estimate the proportion of accounts that are delinquent more than 90 days overdue. The historical records of the company show that over the past 8 years 13 percent of the accounts are delinquent. For this quarter, the auditing staff randomly selected 250 customer accounts. What is the probability that no more than 40 will be classified as delinquent?

.9207

34 The internal auditing staff of a local manufacturing company performs a sample audit each quarter to estimate the proportion of accounts that are delinquent more than 90 days overdue. The historical records of the company show that over the past 8 years 13 percent of the accounts are delinquent. For this quarter, the auditing staff randomly selected 250 customer accounts. What is the probability that at least 30 will be classified as delinquent?

.6808

35According to a hospital administrator, historical records over the past 10 years have shown that 20% of the major surgery patients are dissatisfied with after-surgery care in the hospital. A scientific poll based on 400 hospital patients has just been conducted. What is the probability that at least 70 patients will not be satisfied with the after-surgery care?

98.44%