# Sampling Distributions

## Presentation on theme: "Sampling Distributions"— Presentation transcript:

Sampling Distributions
BPS chapter 11 © 2010 W.H. Freeman and Company

Parameters and statistics
The average fuel tank capacity of all cars made by Ford is 14.7 gallons. This value represents a Parameter because it is an average from all possible cars. Parameter because it is an average from all Ford cars. Statistic because it is an average from a sample of all cars. Statistic because it is an average from a sample of American cars.

The average fuel tank capacity of all cars made by Ford is 14.7 gallons. This value represents a Parameter because it is an average from all possible cars. Parameter because it is an average from all Ford cars. Statistic because it is an average from a sample of all cars. Statistic because it is an average from a sample of American cars.

Parameters and statistics
The fraction of all American adults who received at least one speeding ticket last year can be represented by .

The fraction of all American adults who received at least one speeding ticket last year can be represented by .

Parameters and statistics
The mean distance traveled in a year by a sample of truck drivers can be represented by .

The mean distance traveled in a year by a sample of truck drivers can be represented by .

Sample size We wish to estimate the mean price, , of all hotel rooms in Las Vegas. The Convention Bureau of Las Vegas did this in 1999 and used a sample of n = 112 rooms. In order to get a better estimate of  than the 1999 survey, we should Take a larger sample because the sample mean will be closer to . Take a smaller sample since we will be less likely to get outliers. Take a different sample of the same size since it does not matter what n is.

Sample size (answer) We wish to estimate the mean price, , of all hotel rooms in Las Vegas. The Convention Bureau of Las Vegas did this in 1999 and used a sample of n = 112 rooms. In order to get a better estimate of  than the 1999 survey, we should Take a larger sample because the sample mean will be closer to . Take a smaller sample since we will be less likely to get outliers. Take a different sample of the same size since it does not matter what n is.

Law of Large Numbers The Law of Large Numbers says that
If the population is large, the estimate for  would be better than if the population was small. is a good estimate for  as long as the size of the population was large. If we increase our sample size, will be a better estimate for . As long as the values for x are large, our estimate for  will be good.

The Law of Large Numbers says that If the population is large, the estimate for  would be better than if the population was small. is a good estimate for  as long as the size of the population was large. If we increase our sample size, will be a better estimate for . As long as the values for x are large, our estimate for  will be good.

Sampling distribution
If the first graph shows the population, which plot could be the sampling distribution of if all samples of size n = 50 are drawn? Plot A Plot B Plot C

If the first graph shows the population, which plot could be the sampling distribution of if all samples of size n = 50 are drawn? Plot A Plot B Plot C

Sampling distribution
Which of the following is true? The shape of the sampling distribution of is always bell-shaped. The shape of the sampling distribution of gets closer to the shape of the population distribution as n gets large. The shape of the sampling distribution of gets approximately normal as n gets large. The mean of the sampling distribution of gets closer to µ as n gets large.

Which of the following is true? The shape of the sampling distribution of is always bell-shaped. The shape of the sampling distribution of gets closer to the shape of the population distribution as n gets large. The shape of the sampling distribution of gets approximately normal as n gets large. The mean of the sampling distribution of gets closer to µ as n gets large.

Sampling distribution
True or false: The shape of the sampling distribution of becomes more normal the larger your sample size is. True False

True or false: The shape of the sampling distribution of becomes more normal the larger your samples size is. True False

Sampling distribution
True or false: The standard deviation of the sampling distribution of is always less than the standard deviation of the population when the sample size is at least 2. True False

True or false: The standard deviation of the sampling distribution of is always less than the standard deviation of the population when the sample size is at least 2. True False

Sampling distribution
The theoretical sampling distribution of Gives the values of from all possible samples of size n from the same population. Provides information about the shape, center, and spread of the values in a single sample. Can only be constructed from the results of a single random sample of size n. Is another name for the histogram of values in a random sample of size n.

The theoretical sampling distribution of Gives the values of from all possible samples of size n from the same population. Provides information about the shape, center, and spread of the values in a single sample. Can only be constructed from the results of a single random sample of size n. Is another name for the histogram of values in a random sample of size n.

Standard error What does measure? The spread of the population.
The spread of the ‘s. Different values could be.

What does measure? The spread of the population. The spread of the ‘s. Different values could be.

Sample size True or false: If we want the mean of all possible ‘s to equal , then we must use a large sample size. True False

Sample size (answer) True or false: If we want the mean of all possible ‘s to equal , then we must use a large sample size. True False

Sampling distributions
The following density curve represents waiting times at a customer service counter at a national department store. The mean waiting time is 5 minutes with standard deviation 5 minutes. If we took all possible samples of size n = 100, how would you describe the sampling distribution of the ‘s? Shape = right skewed, center = 5, spread = 5 Shape = less right skewed, center = 5, spread = 0.5 Shape = approx. normal, center = 5, spread = 5 Shape = approx. normal, center = 5, spread = 0.5

The following density curve represents waiting times at a customer service counter at a national department store. The mean waiting time is 5 minutes with standard deviation 5 minutes. If we took all possible samples of size n = 100, how would you describe the sampling distribution of the ‘s? Shape = right skewed, center = 5, spread = 5 Shape = less right skewed, center = 5, spread = 0.5 Shape = approx. normal, center = 5, spread = 5 Shape = approx. normal, center = 5, spread = 0.5

Central Limit Theorem Which is a true statement about the Central Limit Theorem? We need to take repeated samples in order to estimate . It only applies to populations that are Normally distributed. It says that the distribution of ‘s will have the same shape as the population. It requires the condition that the sample size, n, is large and that the samples were drawn randomly.

Which is a true statement about the Central Limit Theorem? We need to take repeated samples in order to estimate . It only applies to populations that are Normally distributed. It says that the distribution of ‘s will have the same shape as the population. It requires the condition that the sample size, n, is large and that the samples were drawn randomly.

Central Limit Theorem True or false: The Central Limit Theorem allows us to compute probabilities on when the conditions are met. True False

True or false: The Central Limit Theorem allows us to compute probabilities on when the conditions are met. True False

Sampling distributions
The following density curve shows a sampling distribution of created by taking all possible samples of size n = 6 from a population that was very left-skewed. Which of the following would result in a decrease of the area to the left of 0.5 (denoted by the vertical line)? Increasing the number of samples taken. Taking a different sample. Decreasing n. Increasing the number of observations in each sample.

The following density curve shows a sampling distribution of created by taking all possible samples of size n = 6 from a population that was very left-skewed. Which of the following would result in a decrease of the area to the left of 0.5 (denoted by the vertical line)? Increasing the number of samples taken. Taking a different sample. Decreasing n. Increasing the number of observations in each sample.

Sampling distributions
What effect does increasing the sample size, n, have on the center of the sampling distribution of ? The mean of the sampling distribution gets closer to the mean of the population. The mean of the sampling distribution gets closer to 0. The variability of the population mean is decreased. It has no effect. The mean of the sampling distribution always equals the mean of the population.

What effect does increasing the sample size, n, have on the center of the sampling distribution of ? The mean of the sampling distribution gets closer to the mean of the population. The mean of the sampling distribution gets closer to 0. The variability of the population mean is decreased. It has no effect. The mean of the sampling distribution always equals the mean of the population.

Sampling distributions
What effect does increasing the sample size, n, have on the spread of the sampling distribution of ? The spread of the sampling distribution gets closer to the spread of the population. The spread of the sampling distribution gets larger. The spread of the sampling distribution gets smaller. It has no effect. The spread of the sampling distribution always equals the spread of the population.

What effect does increasing the sample size, n, have on the spread of the sampling distribution of ? The spread of the sampling distribution gets closer to the spread of the population. The spread of the sampling distribution gets larger. The spread of the sampling distribution gets smaller. It has no effect. The spread of the sampling distribution always equals the spread of the population.

Sampling distributions
What effect does increasing the sample size, n, have on the shape of the sampling distribution of ? The shape of the sampling distribution gets closer to the shape of the population. The shape of the sampling distribution gets more bell-shaped. It has no effect. The shape of the sampling distribution always equals the shape of the population.

What effect does increasing the sample size, n, have on the shape of the sampling distribution of ? The shape of the sampling distribution gets closer to the shape of the population. The shape of the sampling distribution gets more bell-shaped. It has no effect. The shape of the sampling distribution always equals the shape of the population.

Sampling distributions
Which of the following would result in a decrease in the spread of the approximate sampling distribution of ? Increasing the sample size. Increasing the number of samples taken. Increasing the population standard deviation Decreasing the value of the population mean.

Which of the following would result in a decrease in the spread of the approximate sampling distribution of ? Increasing the sample size. Increasing the number of samples taken. Increasing the population standard deviation Decreasing the value of the population mean.

Sampling distributions
Time spent working out at a local gym is normally distributed with mean  = 43 minutes and standard deviation  = 6 minutes. The gym took a sample of size n = 24 from its patrons. What is the distribution of ? Normal with mean  = 43 minutes and standard deviation  = 6 minutes. Normal with mean  = 43 minutes and standard deviation  = minutes. Cannot be determined because the sample size is too small.

Time spent working out at a local gym is normally distributed with mean  = 43 minutes and standard deviation  = 6 minutes. The gym took a sample of size n = 24 from its patrons. What is the distribution of ? Normal with mean  = 43 minutes and standard deviation  = 6 minutes. Normal with mean  = 43 minutes and standard deviation  = minutes. Cannot be determined because the sample size is too small.

Control charts On a control chart, what are on the x and y axes?
Time is on the x-axis and the upper control limit is on the y-axis. The lower control limit is on the x-axis and the upper control limit is on the y-axis. Time is on the x-axis and values of the means are on the y-axis.

On a control chart, what are on the x and y axes? Time is on the x-axis and the upper control limit is on the y-axis. The lower control limit is on the x-axis and the upper control limit is on the y-axis. Time is on the x-axis and values of the means are on the y-axis.

Control charts Look at the following plots. Which show(s) a process that is out-of-control? Plot A Plot B Both plots Neither plot