Statistics Ch.8: Sampling Methods & the Central Limit Theorem

The average height of American males (in 2016) is 5 ft. 9 1/2 inches (69.5 inches) with a standard deviation of 3 inches. What is the probability that the average height of a random group of 16 American men would be over five feet ten inches? Assume that the heights of American men are normally distributed.

0.2514 (see pic)

What are the Reasons for Sampling?

1.) To Contact the whole population would be time consuming 2.) The cost of studying all the items in a population may be prohibitive 3.) Physically Impossible to sample everyone in population 4.) Destructive Nature of Some Testing 5.) Sample is adequate

A marketing firm is polling 60 students at a college using a stratified sample. If two thirds of the students are women, and one quarter of the students are from out of state, how many out-of-state students should be polled?

15 (60*(1/4))

Cluster Sampling

A population is divided into clusters using naturally occurring geographic or other boundaries. Then, clusters are randomly selected and a sample is collected by randomly selecting from each cluster.

Choose the statement that best defines the Sampling Distribution of the Sample Mean.

A probability distribution of all possible sample means of a given sample size.

What is a simple random sample?

A sample selected so that each member of the population has the same likelihood of being included.

Standard Error of the Mean

Actual Name: Standard Deviation of the sampling distribution of the sample mean. -Formula: see picture. σ = standard deviation in the population n = the number of observations in each sample -REMEMBER: -the mean of the distribution of sample means will be exactly equal to the population mean -There will be less dispersion in the sampling distribution of the sample mean than in the population

Which of the populations listed below is a natural candidate for cluster sampling?

The voters in a state, who are grouped into political districts.

The Central Limit Theorem describes an expected distribution shape. Which of these statements is correct?

The sampling distribution of the sample mean is nearly normal.

Pick the statement that describes the formula for the standard error of the mean in ordinary language.

The standard error is equal to the population standard deviation divided by the square root of the sample size.

Stratified Random Sampling

-A population is divided into subgroups, called strata, and a sample is randomly selected from each strata. -used when population can clearly be divided into groups -guarantees that each group is represented -multiply number in each strata by the relative frequency to determine number to be sampled from each strata

Sampling Distribution of the Sample Mean

-A probability distribution of all possible sample means of a given sample size. STEPS TO FIND: 1.) Find population mean 2.) Use combination formula to find number of possible samples, given population size and sample size. (NCn) 3.) List out all of the possible samples & Find the mean for each possible sample. 4.) Sum all sample means and divide by the number of samples (formula).

Systematic Random Sampling

-A random starting point is selected, and then every kth member of the population is selected. -(K= population size / sample size) or (N/n) -round K down to the nearest whole number -Start with a random number between 1 and k-count from there

Which of the following statements describe valid reasons to use a sample instead of evaluating a much larger population? Select all that apply.

-Contacting the entire population would be time consuming. -Contacting the whole population would be only marginally more accurate than a sample.

"Destructive Testing" destroys a sample in the course of measuring it. Which of the following are examples of this process? Select all that apply.

-Determining a table's maximum load capacity by increasing its load until it breaks. -Evaluating coffee for taste and aroma.

Simple Random Sample

-Each item or person in the population has an equal chance of being selected for sample -Most widely used (EX: drawing numbers from a hat, assigning numbers and using table of random numbers

Which of the following statements are true features of the Central Limit Theorem?

-For large sample size, the distribution of sample means is independent of the shape of the population. -The distribution shape is approximately Normal. -It describes the sampling distribution of the sample mean.

Choose the two statements that are correct descriptions of the sampling distribution of the sample mean.

-It is a distribution of means from samples of all one size. -It is a probability distribution of all possible sample means.

Economics plays a role in the sampling process. Which statements correctly describe this relationship?

-Larger samples cost more, and increasing size gives diminishing marginal returns in accuracy. -The cost of studying an entire population may be prohibitive.

Identify the steps involved in taking a cluster sample. Select all that apply.

-Randomly select a subset of clusters. -Divide the population into groups using naturally occurring boundaries. -Select a random sample from each sub group.

Which of the following statements correctly describe the relationship between a population and a sample?

-Samples are used to estimate population characteristics. -A sample is a subset of the population. -A sample statistic is probably not exactly the same as the corresponding population characteristic

Identify the steps required in taking a systematic random sample

-Select a random starting point. -Select every kth member of the population from the starting point.

Which of the following distinguish systematic random sampling from simple random sampling? Select all that apply.

-Systematic random sampling uses only one random choice, instead of several. -Systematic random sampling is quicker and easier.

Which of the following statements correctly describe characteristics of the set of sample means?

-The mean of the sample means equals the mean of the population. -The dispersion of the sample means is narrower than the population dispersion.

Other reasons besides cost and size may make it difficult or impossible to evaluate the entire population. Which of the following describe such reasons? Select all that apply.

-The population may be changing too fast to allow complete sampling. -A large part of the population may be physically inaccessible.

In using a random number table to select a simple random sample, where does "randomness" occur in the process? Select all that apply.

-The table itself is the one source of randomness. -The manner in which numbers are taken from the table.

There are two conditions under which we can assume that the sample means follow a normal distribution. What are they?

-We don't know the population distribution, but the sample size is 30 or larger. -We know that the population is normally distributed.

The average height of American women (in 2016) is 5 ft. 4 inches (64 inches) with a standard deviation of 3 inches. What is the probability that the average height of a random group of nine American women would be less than five feet three inches? Assume that the heights of American women are normally distributed.

0.1587

Which one of the following statements is true about the dispersion of the distribution of sample means?

As the sample size increases, the variability in the sample means decreases.

random

Because each sample is different, each sample will have a different mean and standard deviation. Therefore, sample statistics are ____________ variables that can be described with distributions.

Which of the following populations is a good candidate for Systematic random sampling?

Companies listed on the New York Stock Exchange. (because there is no obvious intrinsic order in this population).

Finding the z value of bar x when the population standard deviation is known

Formula: see picture x̄ = sample mean μ= population mean σ = standard deviation of n observations n = number of observations in sample bar x - mean / standard error of the mean

Central Limit Theorem

If all samples of a particular size are selected from any population, the sampling distribution of the sample mean is approximately a normal distribution. -The approximation improves with larger samples.

In cluster sampling the clusters are chosen from the population using simple random sampling. What kind of sampling is used within the individual clusters?

Random Sampling

Sampling "Error"

Sampling error is the difference between a sample statistic and its corresponding population parameter. -Due to chance (value based on random selection) -Example: Difference of sample mean and population mean.

What characterizes a "strata" within a population?

Some identifiable characteristic that clearly divides the population into groups.

Why is systematic random sampling sometimes used in place of simple random sampling?

Sometimes it is difficult to assign random numbers.

Choose the statement that best describes sampling error.

The difference between a sample mean and the population mean.

What characteristic of a population requires the use of stratified random sampling to avoid bias?

The population is clearly divided into groups based on some characteristic.

What characteristic of a population makes it a good candidate for cluster sampling?

The population is widely scattered over a large geographical area.

There are many reasons why it may be undesirable to sample an entire population. Which of the following is a reason that a modest size sample may be adequate?

Very few problems require 100% accuracy.

When is it inappropriate to use systematic random sampling?

When the order of items in the population is related to some particular characteristic.

When to Convert to z values

When we take a sample from a population and we need to know if the difference between the population and the sample (sampling error) is due to chance -Convert to z values: -when samples are taken from populations that follow normal distributions (sample size doesn't matter) -When the shape of the population distribution is not known or abnormal, but the sample has at least 30 observations.

Finding the z value of bar x when the population standard deviation is known (EXAMPLE): A normal population has a mean of 60 and a standard deviation of 12. You select a random sample of 9. Compute the probability that the sample mean is: -Greater than 63 -Less than 56 -Between 56 and 63

a.) Greater than 63= Z= (63-60)/(12/√9) Z= 0.75 = 0.2634 .50-.2634 = 0.2366 Probability that the value is greater than 63 = 0.2366 b.) Less than 56= Z= (56-60)/(12/√9) = 1 = 0.3413 (0.50-0.3413)= 0.1587 below 56 c.) Between 56 and 63= 0.2634 + 0.3413 = 0.6047

Sampling Distribution of the Sample Mean (EXAMPLE): A population consists of the following four values: 12a, 12b, 14, and 16 a.) Find the population mean b.) List the possible samples of size 2 and find the mean for each sample c.) Compute the mean of the distribution of sample mean and compare it to the population mean. d.) Compare the dispersion in the population and that of the sample mean

a.) population mean =(12+12+14+16) / 4 = 13.5 b.) 4C2 (combination) = 6 possible samples All Possible Samples: (12a, 12b), (12a, 14), (12a, 16), (12b, 14), (12b, 16), (14, 16) Mean of each sample= 12, 13, 14, 13, 14, 15 c.) Sample mean =(12+13+14+13+14+15) / 6 = 13.5 (same as population mean). d.) sample has a smaller dispersion and the sampling distribution of the sample mean looks like a normal distribution

Which of the following is an expression that represents sampling error?

bar x − μ (difference between sample mean and population mean)

The bigger the sample size, the smaller the standard error of the mean

true

the sampling distribution of the sample mean will be equal to the population mean

true