Statistics 7: The Distribution of Sample Means

Learning Check 7.2 The standard distance between a sample mean and the population mean is 6 points for samples of n = 16 scores selected from a population with a mean of μ = 50. What is the standard deviation for the population?

24

Learning Check 7.4 For samples selected from a population with µ = 40 and σ = 20, what sample size is necessary to make the standard distance between the sample mean and the population mean equal to 2 points?

n = 100

Learning Check 7.3 A random sample of n = 9 scores is obtained from a population with µ = 50 and σ = 9. If the sample mean is M = 53, what is the z-score corresponding to the sample mean?

z = 1.00

•Within the distribution of sample means, the location of each sample mean can be specified by a z-score:

z = M - μ/ standard error σM

•In general, the difficulty of working with samples is that a sample provides an incomplete picture of the population.

hence sampling error

Learning Check 7.1 If all the possible random samples of size n = 5 are selected from a population with μ = 50 and σ = 10 and the mean is computed for each sample, then what value will be obtained for the mean of all the sample means?

50

Learning Check 7.5 If a sample of n = 25 scores is selected from a normal population with µ = 80 and σ = 10, then what sample means form the boundaries that separate the middle 95% of all sample means from the extreme 5% in the tails?

76.08 and 83.92

Learning Checking 7.3 A normal population has µ = 50 and σ = 8. A random sample of n = 16 scores from this population has a mean of 54. What is the z-score for this sample mean?

+2.00

Consider a population with a mean of 80 and a standard deviation of 20. Take a sample from the population and examine how accurately the sample mean represents the population mean n = 1, n = 4, n = 100

- standard error=σM=σ/square root of n 20/sqaure root of 1 = 20 20/square root of 4 = 10 20/square root of 100 = 2

•For each individual sample, you can measure the error (or distance) between the sample mean and the population mean.

-For some samples, the error will be relatively small, but for other samples, the error will be relatively large. -The standard error provides a way to measure the "average", or standard, distance between a sample mean and the population mean.

•The general concept of sampling error is that a sample typically will not provide a perfectly accurate representation of its population.

-More specifically, there typically is some discrepancy (or error) between a statistic computed for a sample and the corresponding parameter for the population.

•The primary use of the distribution of sample means is to find the probability associated with any specific sample.

-Recall that probability is equivalent to proportion. -Because the distribution of sample means presents the entire set of all possible sample means, we can use proportions of this distribution to determine probabilities.

•The average value of all the sample means is exactly equal to the value of the population mean. The formal statement of this phenomenon is that the mean of the distribution of sample means always is identical to the population mean. This mean value is called the expected value of M

-The sample mean is an example of an unbiased statistic, which means that on average the sample statistic produces a value that is exactly equal to the corresponding population parameter. -In this case, the average value of all the sample means is exactly equal to μ.

•The magnitude of the standard error is determined by two factors:

-The size of the sample: The law of large numbers states that the larger the sample size (n), the more probable it is that the sample mean will be close to the population mean. -The standard deviation of the population from which the sample is selected - standard error=σM=σ/square root of n

•The standard deviation of the distribution of sample means, σM, is called the standard error of M. The standard error provides a measure of how much distance is expected on average between a sample mean (M) and the population mean (μ).

-The standard error describes the distribution of sample means. •Measures how well an individual sample mean represents the entire distribution, specifically, how much distance is reasonable to expect between a sample mean and the overall mean for the distribution of sample means

•The value 525 is located above the mean by 25 points, which is exactly 1 standard deviation (in this case, exactly 1 standard error).

-Thus, the z-score for M = 525 is z = +1.00. -The unit normal table indicates that 0.1587 of the distribution is located in the tail of the distribution beyond z = +1.00. -Our conclusion is that it is relatively unlikely, p = 0.1587 (15.87%), to obtain a random sample of n = 16 students with an average SAT score greater than 525.

distribution of sample means

-means is the collection of sample means for all the possible random samples of a particular size (n) that can be obtained from a population. -Ex: population with N=4 scores, 2,4,6,8 Construct a distribution of sample means for n = 2 Make a histogram of the sample means Population mean = (2+4+6+8)/4= 20/4=5

Learning Check 7.2 . If random samples, each with n = 4 scores are selected from a population with µ = 80 and σ = 12, then how much distance is expected on average between the sample means and the population mean?

. 12/square root of 4 = 6 points

Learning Check 7.5 . If a sample is selected from a normal population with µ = 50 and σ = 20, which of the following samples is extreme and very unlikely to be obtained?

. M = 45 for a sample of n = 100 scores.

Learning Check 7.1 If all the possible random samples of size n = 25 are selected from a population with μ = 80 and σ = 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 whether the population is normal or not.

Learning Check 7.4 For samples selected from a population with µ = 100 and σ = 15, which of the following has the smallest standard error?

. a sample of n = 25 scores

Learning Check 7.3 A random sample of n = 25 scores is selected from a normally distributed population with μ = 500 and σ = 100. What is the probability that the sample mean will be less than 490?

0.3085

•Because of the central limit theorem, we know:

1.The distribution is normal because the population of SAT scores is normal. 2.The distribution has a mean of 500 because the population mean is μ = 500. 3.For n = 16, the distribution has a standard error of σM = 25: = s/Ön = 100 / Ö16 = 100/4 = 25

Learning Check 7.4 For a particular population, the standard distance between a sample mean and the population mean is 5 points for samples of n = 4 scores. What would the standard distance be for samples of n = 16 scores?

2.5 points

Learning Check 7.2 . If samples are selected from a population with μ = 80 and σ = 12, then which of the following samples will have the largest expected value for M?

All of the samples will have the same expected value.

Learning Check 7.1 All the possible random samples of size n = 2 are selected from a population with μ = 40 and σ = 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.

Learning Check 7.5 A random sample is obtained from a population with µ = 80 and σ = 10 and a treatment is administered to the sample. Which of the following outcomes would be considered noticeably different from a typical sample that did not receive the treatment?

n = 100 with M = 83

The Central Limit Theorem

•For any population with mean μ and standard deviation σ, the distribution of sample means for sample size n will have a mean of μ and a standard deviation of s/Ön and will approach a normal distribution as n approaches infinity. -Distribution of sample means for any population, no matter what shape, mean, or standard deviation •"Approaches" a normal distribution very rapidly; by the time the sample size reaches n = 30, the distribution is almost perfectly normal.

•Whenever a score is selected from a population, you should be able to compute a z-score that describes exactly where the score is located in the distribution. - If the population is normal, you also should be able to determine the probability value for obtaining any individual score.

•However, the z-scores and probabilities that we have considered so far are limited to situations in which the sample consists of a single score.

•The population of scores on the SAT forms a normal distribution with μ = 500 and σ = 100. -If you take a random sample of n = 16 students, what is the probability that the sample mean will be greater than M = 525?

•Restate this probability question as a proportion question: Out of all the possible sample means, what proportion has values greater than 525? -"All the possible sample means" refers to the distribution of sample means. The problem is to find a specific portion of this distribution.

The Shape of the Distribution of Sample Means

•The distribution is almost perfectly normal if either of the following two conditions is satisfied: -The population from which the samples are selected is a normal distribution. -The number of scores (n) in each sample is relatively large, around 30 or more.

•The sign tells whether the location is above (+) or below (-) the mean.

•The number tells the distance between the location and the mean in terms of the number of standard deviations.

Characteristics of the Distribution of Sample Means

•The sample means should pile up around the population mean. •The pile of sample means should tend to form a normal-shaped distribution. •In general, the larger the sample size, the closer the sample means should be to the population mean, μ.

sampling distribution

•is a distribution of statistics obtained by selecting all the possible samples of a specific size from a population.

sampling error

•is the natural discrepancy, or amount of error, between a sample statistic and its corresponding population parameter.