Sampling Distributions Chapter 7

First, a word from our textbook A statistic is a numerical value computed from a sample. EX. Mean, median, std. dev., etc. A parameter is a numerical value determined by the entire population and is assumed that the value is fixed, unchanging, and unknown.

Introduction to Statistics and Sampling Variability Consider a small population consisting of the board of directors of a day care center. Board member and number of children: Jay Carol Allison Teresa Anselmo Bob Roxy Vishal 5 2 1 0 2 2 1 3 Find the average number of children for the entire group of eight: m = 2 children parameter

Discovery question ONE: How is the parameter of the population related to a sampling distribution based on the population?

Introduction to Statistics and Sampling Variability Board member and number of children: Jay Carol Allison Teresa Anselmo Bob Roxy Vishal 5 2 1 0 2 2 1 3 List all possible samples of size 2. Calculate the average number of children represented by the group. Samples: Jay Carol 5 2 Jay Allison 5 1

Answer question ONE: the average of all possible values for a sampling distribution will equal the population parameter

Variability of a statistic What is the relationship between the population parameter and each sample statistic? The observed value of a statistic will vary from sample to sample. This fact is called sampling variability.

Sampling distributions If we calculated using only the first 3 columns of values, would we get the same results? Explain. How did the spread change from the population to the sampling distribution? Explain. If we created a distribution based on a sample size of 4 comment on the mean, spread and shape of the sampling distribution.

Definition In summary, a sampling distribution is the distribution of all possible values for a given sample size for a fixed population. Sampling distribution applet

Discovery question TWO: For a normal population, how will the shape and spread of a sampling distribution change as we increase the sample size? Population distribution: m = 16 s = 5

Discovery question TWO:

Answer question TWO: For a normal population, the shape of the sampling distribution remains mound shaped and symmetrical (taller/thinner)for all sample sizes. We can conclude the sampling distribution remains approximately normal. The standard deviation for the sampling distribution is equal to the population standard deviation divided by the square root of the sample size.

Sample means Formulas: sampling distribution parameter statistic mean standard deviation Formulas:

Example ONE The average sales price of a single-family house in the United States is $243,756. Assume that the sales prices are normally distributed with a standard deviation of $44,000.

Draw the normal distribution Draw the normal distribution. Within what range would the middle 68% of the houses fall? $199,756 $243,756 $287,756

Draw the sampling distribution for a sample size of 4 houses Draw the sampling distribution for a sample size of 4 houses. Within what range would the middle 68% of the samples of size 4 houses fall? $221,756 $243,756 $265,756

Draw the sampling distribution for a sample size of 16 houses Draw the sampling distribution for a sample size of 16 houses. Within what range would the middle 68% of the samples of size 16 houses fall? $232,756 $243,756 $254,756

Draw the sampling distribution for a sample size of 25 houses Draw the sampling distribution for a sample size of 25 houses. Within what range would the middle 68% of the samples of size 25 houses fall? $234,956 $243,756 $252,556

Example TWO Suppose the mean room and board expense per year at a certain four-year college is $7,850. You randomly select 9 dorms offering room and board near the college. Assume that the room and board expenses are normally distributed with a standard deviation of $1125.

Draw the population distribution. $4,475 $5,600 $6,725 $7,850 $8,975 $10,100 $11,225

What is the probability that a randomly selected dorm has room and board of less than $8,180? $4,475 $5,600 $6,725 $7,850 $8,975 $10,100 $11,225

What is the probability that a randomly selected dorm has room and board of less than $8,180? Given normal distribution

Draw the sampling distribution for a sample size of 9 dorms. $6,725 $7,100 $7,475 $7,850 $8,225 $8,600 $8,975

What is the probability that the mean room and board of the nine dorms is less than $8,180? $6,725 $7,100 $7,475 $7,850 $8,225 $8,600 $8,975

What is the probability that the mean room and board of the nine dorms is less than $8,180? Given normal distribution

What is the probability that the mean cost of a sample of four dorms is more than $7,250? Given normal distribution