Chapter 7
As sample size (n) increase, the size of the SE decrease
*Researchers can greatly reduce error by increasing sample size up to around n = 30
Purpose of the standard error?
1) Describes the distribution of sample means *How much difference is expected from one sample to another 2) How well an individual sample mean represents its population mean how much distance between M and mu
By the time the sample size reaches n = ____, the distribution is perfectly normal
30
If all the possible random samples, each with n = 9 scores, are selected from a normal population with mu = 80 and standard deviation = 18, and the mean is calculated for each sample, then what is the average value for all of the sample means?
80
For large populations and samples, the number of possible samples increases dramatically and it is virtually impossible to actually obtain every possible random sample Fortunately, it is possible to determine exactly what the distribution of sample means looks like without taking hundreds or thousands of samples ____
Central limit theorem
Primary use for the distribution of sample means
Find the probability (i.e., proportion) of selecting a sample with a specific mean We can use proportions of the distribution to determine the probability of obtaining a sample with a specific mean
According to the Central Limit Theorem, the standard error is equal to
Thus, the magnitude of the standard error is determined by two factors The size of n The standard deviation of the population from which the sample is selected
A sample is obtained from a population with mu = 100 and sigma = 20. Which of the following samples would produce the z-score closest to 0? a. A sample of n = 25 scores with M = 102 b. A sample of n = 100 scores with M = 102 c. A sample of n = 25 scores with M = 104 d. A sample of n =100 scores with M = 104
a. 2/4 = .5
We are now finding a location for a sample mean (M) within the distribution of sample means (z-score) The sign (+/-) tells whether the sample mean is located ... The number tells the distance between ....
above or below the population mean the sample mean and population mean in terms of the number of standard errorsq
Any probability question requires that you have complete information about the population from which the sample is being collected Thus, to find the probability for any specific sample mean, you must first know ____
all the possible sample means
The size of a sample influences how accurately the sample represents its population As sample size increases, the error between sample mean and population mean should ____ Known as the ____
decrease law of large numbers
Two separate samples will probably be ____ even though they are taken from the same population
different
The ability to predict sample characteristics is based on the _____
distribution of sample means The collection of sample means for all of the possible random samples of a particular size (n) that can be obtained from that population
Although a sample mean should be representative of the population mean, there typically is some ____ between the sample and the population
error
As n gets ____, the distribution of sample means will closely approximate a normal distribution When n > ____, the distribution is almost normal regardless of the shape of the original population
larger 30
A random sample is selected from a population with mu = 80 and sigma = 20. How large must the sample be to ensure a SE of 2 points or less?
n > 100
Standard error provides a method for defining and measuring ____
sampling error There will be some error, and the standard error tells exactly how much error
Single rule to remember with standard error: Whenever you are working with a sample mean, you must use the
standard error
If random samples, each with n = 4 scores are selected from a population with mu = 80 and sigma = 10, then what is the expected value of the mean for the distribution of sample means?
80
A random sample of n = 4 scores is obtained from a normal population with mu = 20 and sigma = 4. What is the probability of obtaining a mean greater than M = 22 for this sample?
SE = 4/2 = 2 (1 SE) 22 is 2 points above mean (1 SE) z-score= 2/2 = 1 Tail of z = +1.00 = 0.1587
A random sample of n = 4 scores is obtained from a normal population mu = 40 and sigma = 6. What is the probability of obtaining a mean greater than M = 46 for this sample?
SE = 6/2 = 3 z= 6/3 = 2 Tail of z = +2.00 = 0.0228
In this chapter, we are moving on from single scores to ____
Sample means
A sample typically will not provide a perfectly accurate representation of its population Discrepancy (or error) between a statistic computed for a sample and a population
Sampling Error
The distribution of sample means is an example of a sampling distribution as sample means are statistics Often called the _____
Sampling distribution of M
In general, the difficulty of working with samples is that a sample provides an incomplete picture of the population ____: the natural discrepancy, or amount of error, between a sample statistic and its corresponding population parameter
Sampling error
The central limit theorem describes the distribution sample means by identifying the 3 basic characteristics that describe any distribution
Shape Central tendency Variability
____ SEM = sample means are close together and have similar values ____ SEM = sample means are scattered over a wide range and there are big differences from one sample to another
Small Large
Why is the standard error valuable?
Specifies precisely how well a sample mean estimates its population mean i.e., how much error between M and m
How accurate of a representation of the population is the sample mean? Close to population mean = fairly accurate representation For each sample mean, you can measure the error (or distance) between sample mean and population mean
Standard error
The Standard Error of M (Variability) The value we will be working with is the standard deviation for the distribution of sample means Known as... _____ Identified by the symbol sigmaM
Standard error of M
In a normal distribution, any score located in the tail of the distribution beyond z = + 2.00 is an ____, and a score this large has a probability of only p = 0.0228
extreme value
In commonsense terms, a sample mean is "expected" to be near its population mean When all of the sample means are obtained, the average value is equal to ____
mu
A sample obtained from a population with sigma = 10 has a standard error of 2 points. How many scores are in the sample?
n = 25
The Mean of the Distribution of Sample Means: The Expected Value of M The average value of all the sample means is exactly equal to the value of the ____ This mean value is called the ____
population mean expected value of M
Characteristics of the Distribution of Sample Means 1) The sample means should pile up around the _____ 2) The pile of sample means should tend to form a ____ distribution 3)In general, the larger the sample size, the ___ the sample means should be to the population mean, m
population mean normal-shaped closer
Once we have specified the complete set of all possible sample means (i.e., the distribution of sample means), we will be able to find the of selecting any of the specific sample means
probability
Z-score describes a sample mean's position in the distribution of sample means When the distribution of sample means is normal, it is possible to use z-scores and the unit normal table to find the ____ associated with any specific sample mean
probability
When n = 1, the standard error is identical to the ____
standard deviation When n = 1, sigmaM = sigma Can think of the s as the "starting point" for standard error As sample size increases beyond n =1, the sample becomes a more accurate representative of the population, and the standard error decreases
Further, samples are _____ (i.e., they are not all the same)
variable
How do you construct the distribution of sample means?
-Select a random sample of a specific size (n) from the population -Calculate the sample mean -Place the sample mean in a frequency distribution -Select another random sample with the same number of scores -Again, calculate the sample mean and add it to your distribution -Continue selecting samples and calculating means, over and over, until you have the complete set of all the possible random samples -At this point, your frequency distribution will show the distribution of sample means
Value of the central limit theorem?
1) It describes the distribution of sample means for any population 2) The distribution of sample means "approaches" a normal distribution very rapidly
The Shape of the Distribution of Sample Means Tends to be almost perfectly normal if either of the following two conditions is satisfied:
1) The population from which the samples are selected is a normal distribution 2) The number of scores (n) in each sample is relatively large, around 30 or more
We can use the distribution of sample means to answer probability questions about sample means E.g., if you take a sample of n = 2 scores from the original population, what is the probability of obtaining a sample with a mean greater than 7?
1/16
A sample of n = 4 scores has a standard error of 12. What is the standard deviation of the population from which the sample was obtained?
12 = stdev/2 24
If random samples, each with n = 9 scores, are selected from a normal population with mu = 80 and sigma = 18, and the mean is calculated for each sample, then how much distance is expected on average between M and mu?
18/3 = 6
____: provides a precise description of the distribution that would be obtained if you selected every possible sample, calculated every sample mean, and constructed the distribution of the sample mean
Central limit theorem Cornerstone of inferential statistics
Use this population to construct the distribution of sample means for n = 2
List all possible samples (4 x 4 = 16) There are 16 different samples 2, 2 2, 4 2, 6 2, 8 4, 2 4, 4 4, 6 4, 8 6, 2 6,4 6, 6 6, 8 8, 2 8, 4 8,6 8, 8 Then, compute the M for each of the 16 samples E.g., 2, 2 M = 2 The 16 sample means are then placed in a frequency distribution histogram
A sample of n = 16 scores is obtained from a population with mu = 70 and sigma = 20. If the sample mean is M = 75, then what is the z-score corresponding to the same mean?
SE = 20/4 = 5 z = 75-70/5 = 1 z = +1.00
If all the possible random samples of size n = 25 are selected from a population with mu = 80 and standard deviation = 10 and the mean is computed for each sample, then what shape is expected for the distribution of sample means?
The sample means tend to form a normal-shaped distribution
All the possible random samples of size n = 2 are selected from a population with mu = 40 and standard deviation 10 and the mean is computed for each sample. Then, all the possible samples of size n = 25 are selected from the same population and the mean is computed for each sample. How will the distribution of sample means for n = 2 compare with the distribution for n = 25?
The variance for n = 25 will be smaller than the variance for n = 2 but the two distributions will have the same mean
muM is used to represent the mean of the distribution of sample means However, muM =M
Unbiased statistic Exactly equal o the corresponding population parameter
Example: the population of scores on the SAT forms a normal distribution with m= 500 and s= 100. If you take a random sample of n = 16 students, what's the probability that the sample mean will be greater than M = 525?
Use the central limit theorem to see what the distribution looks like -The distribution is normal because the population of SAT scores is normal -The distribution has a mean of 500 because the population mean is 500 -For n = 16, the distribution has a standard error of sigmaM = 25 (100/4) We are interested in sample means greater than 525 Use a z-score to locate the exact position of M = 525 Above the mean by 25 points = +1 standard deviation Z-score for M = 525 is z = +1.00 Use unit normal table to find probability associated with z = +1.00 0.1587 of distribution is in the tail beyond +1.00 p = 0.1587
Example: the population of scores on the SAT forms a normal distribution with m= 500 and s= 100. n = 25 students. What is the exact range of values expected for the sample mean 80% of the time?
Use the central limit theorem to see what the distribution looks like -The distribution is normal because the population of SAT scores is normal -The distribution has a mean of 500 because the population mean is 500 -For n = 25, the distribution has a standard error of sM = 20 Our goal is to find the range of values that make up the middle 80% of the distribution 80% split is 40% Look up .4000 in column D z = 1.28 What score is 1.28? SE is 20 points 1.28*20=25.6 A distance of 25.6 in both directions produces a range of values from 474.4 to 525.6 *We are 80% confident that the value of the sample mean will be between 474.4 to 525.6
For a sample selected from a normal population with mu = 100 and sigma = 15, which of the following would be the most extreme and unrepresentative? a. M = 90 for a sample of n = 9 scores b. M = 90 for a sample of n =25 scores c. M = 95 for a sample of n = 9 scores d. M = 95 for a sample of n = 25 scores
b. M = 90 for a sample of n =25 scores 10/3= 3.33
Which of the following would cause the SEM to get smaller? a. Increasing both the sample size and standard deviation b. Decreasing both the sample size and standard deviation c. Increasing the sample size and decreasing the standard deviation d. Decreasing the sample size and increasing the standard deviation
c. Increasing the sample size and decreasing the standard deviation
For a normal population with mu = 80 and sigma = 20, which of the following samples is least likely to be obtained? a. M = 80 for a sample of n = 4 b. M = 84 for a sample of n = 4 c. M = 88 for a sample of n = 25 d. M = 84 for a sample of n = 25
c. M = 88 for a sample of n = 25 8/4 = 2
