Sampling Distribution of the Sample Mean Section 7.2 Sampling Distribution of the Sample Mean

For a finite population, the sampling distribution of the sample mean is the

For a finite population, the sampling distribution of the sample mean is the set of means from all possible samples of a specific size, n.

Simulated Sampling Distribution 4 steps to construct:

Simulated Sampling Distribution 4 steps to construct: 1) Take random sample from population

Simulated Sampling Distribution 4 steps to construct: 1) Take random sample from population 2) Compute summary statistic for sample

Simulated Sampling Distribution 4 steps to construct: 1) Take random sample from population 2) Compute summary statistic for sample 3) Repeat steps 1 and 2 many times

Simulated Sampling Distribution 4 steps to construct: 1) Take random sample from population 2) Compute summary statistic for sample 3) Repeat steps 1 and 2 many times 4) Display distribution of summary statistic

Find mean and standard deviation using Freq Find mean and standard deviation using Freq. Table of Population (not a sample distribution) Page 428

Find mean and standard deviation using Freq Find mean and standard deviation using Freq. Table of Population (not a sample distribution) L2 L1

Find mean and standard deviation using Freq Find mean and standard deviation using Freq. Table of Population (not a sample distribution) L1 L2 STAT CALC 1-Var Stats L1, L2

Find mean and standard deviation using Freq Find mean and standard deviation using Freq. Table of Population (not a sample distribution)

Find mean and standard deviation using a sample distribution (not of population) Suppose we make 5 sampling distributions of the sample mean for samples of size: n = 1 n = 4 n = 10 n = 20 n = 40

Find mean and standard deviation using a sample distribution, n=1 (from a population) Note: this column is a Sample Distribution using a sample size n=1 (looks just like the Population Frequency Graph) Note: this column is of a Population

Note: this graph is of a Population Using Population Freq. Table, generate Sampling Distributions of different sample sizes (next slides) n = 4 n = 10 n = 20 n = 40 4 Steps not shown…see Page 428 & 429 in textbook for calculations when n = 4 Note: this graph is of a Population

To clarify Step 1 in textbook, page 428, we can use a random table of digits OR a calculator to assign digits to possible outcomes Using random table of digits: 001- 524 525 - 725 726 - 904 905 - 974 975 – 999, 1000 Or using Calculator: MATH PRB randInt (1, 1000)

Results After repeating Steps 1-4 many times (i.e. Fathom) using different size samples, the results are:

As n increases, what happens to: Shape Center Spread

As n increases, what happens to: Shape: more normal Center Spread

As n increases, what happens to: Shape: more normal Center: stays same Spread

As n increases, what happens to: Shape: more normal Center: stays same Spread: decreases

Common Symbols Note: a calculator cannot tell if a list (L1 or L2) is from a population or sample distribution…you have to know which is which. Page 430

Properties of Sampling Distribution of the Sample Mean These properties depend on using ________ samples.

Properties of Sampling Distribution of the Sample Mean These properties depend on using random samples.

Properties of Sampling Distribution of the Sample Mean These properties depend on using random samples. These properties will cover center, spread, and shape.

Properties of Sampling Distribution of the Sample Mean If a random sample of size n is selected from a population with mean and standard deviation , then:

Properties of Sampling Distribution of the Sample Mean Center The mean, x, of the sampling distribution of x equals the mean of the population, : x =

Properties of Sampling Distribution of the Sample Mean Center The mean, x, of the sampling distribution of x equals the mean of the population, : x = In other words, the means of random samples are centered at the population mean.

Properties of Sampling Distribution of the Sample Mean Spread The standard deviation, x, of the sampling distribution, sometimes called the standard error of the mean, equals the standard deviation of the population, , divided by the square root of the sample size n. x =

Properties of Sampling Distribution of the Sample Mean Spread The standard deviation, x, of the sampling distribution, sometimes called the standard error of the mean, equals the standard deviation of the population, , divided by the square root of the sample size n. X = When sample size increases, spread ________.

Properties of Sampling Distribution of the Sample Mean Spread The standard deviation, x, of the sampling distribution, sometimes called the standard error of the mean, equals the standard deviation of the population, , divided by the square root of the sample size n. X = When sample size increases, spread decreases

Properties of Sampling Distribution of the Sample Mean Shape The shape of the sampling distribution will be approximately normal if the population is approximately normal.

Properties of Sampling Distribution of the Sample Mean Shape The shape of the sampling distribution will be approximately normal if the population is approximately normal. For other populations, the sampling distribution becomes more normal as n increases (Central Limit Theorem).

Using These Properties When can you use property that mean of sampling distribution of the mean is equal to the mean of the population, x = ?

Using These Properties When can you use property that mean of sampling distribution of the mean is equal to the mean of the population, x = ? Anytime you use random sampling. Shape of pop., size of sample, how large pop. is, or whether sample with or without replacement do not matter.

Using These Properties When can we use property that standard error of sampling distribution of the mean, x, is equal to ?

Using These Properties With a population of any shape and with any sample size as long as you randomly sample with replacement or

Using These Properties With a population of any shape and with any sample size as long as you randomly sample with replacement or you randomly sample without replacement and the sample size is less than 10% of population size.

Using These Properties 3. When can we assume the sampling distribution is approximately normal?

Using These Properties 3. When can we assume the sampling distribution is approximately normal? If we are told the population is approximately normally distributed (or bell-shaped or mound-shaped), you can assume sampling distribution is approximately normal regardless of sample size or _______.

Using These Properties 3. When can we assume the sampling distribution is approximately normal? If we are told the population is approximately normally distributed (or bell-shaped or mound-shaped), you can assume sampling distribution is approximately normal regardless of sample size or if we are told sample size is very large.

Using These Properties 3. When can we assume the sampling distribution is approximately normal? If we are told the population is approximately normally distributed (or bell-shaped or mound-shaped), you can assume sampling distribution is approximately normal regardless of sample size or if we are told sample size is very large (think Central Limit Theorem).

Using These Properties 4. Isn’t the size of the population really important?

Using These Properties 4. Isn’t the size of the population really important? As long as the sample was randomly selected and as long as the population is much larger than the sample, it does not matter how large the population is.

Example: Average Number of Children What is the probability that a random sample of 20 families in the United States will have an average of 1.5 children or fewer?

Example: Average Number of Children What is the probability that a random sample of 20 families in the United States will have an average of 1.5 children or fewer? Could add up heights of all bars ≤ 1.5

Example: Average Number of Children What is the probability that a random sample of 20 families in the United States will have an average of 1.5 children or fewer? Could add up heights of all bars ≤ 1.5 What shape is this?

Example: Average Number of Children What is the probability that a random sample of 20 families in the United States will have an average of 1.5 children or fewer? How do we find the area under the normal curve?

Average of 1.5 children or fewer? normalcdf(lower bound, upper bound, mean, standard deviation)

Example: Average Number of Children What is the probability that a random sample of 20 families in the United States will have an average of 1.5 children or fewer? Recall, for the population μ = 0.9 and σ = 1.1 What are the sampling distribution mean and standard deviation (standard error)?

Example: Average Number of Children What is the probability that a random sample of 20 families in the United States will have an average of 1.5 children or fewer? Recall, = 0.9 and = 1.1

What is the probability that a random sample of 20 families in the United States will have an average of 1.5 children or fewer? normalcdf (-1E99, 1.5, .9, .25) = 0.9918024711

Reasonably Likely Averages What average numbers of children are reasonably likely in a random sample of 20 families?

Reasonably Likely Averages What average numbers of children are reasonably likely in a random sample of 20 families? Reasonably likely values are those in the middle 95% of the sampling distribution.

Reasonably Likely Averages What average numbers of children are reasonably likely in a random sample of 20 families?

Reasonably Likely Averages What average numbers of children are reasonably likely in a random sample of 20 families? For approx. normal distributions, reasonably likely outcomes are those within 1.96 standard errors of the mean.

Reasonably Likely Averages What average numbers of children are reasonably likely in a random sample of 20 families? 0.9 1.96(0.25) which gives an average number of children between 0.41 and 1.39

Page 437, P7

Page 437, P7

Page 437, P7 population n = 4 n = 10

Page 437, P7 population n = 4 n = 10 mean ≈ 1.7 for each

Page 437, P7 population n = 4 n = 10 mean ≈ 1.7 for each SD ≈ 1

Page 437, P7 population n = 4 n = 10 mean ≈ 1.7 for each SE ≈ 0.3 SD ≈ 1

Page 437, P7 population n = 4 n = 10 mean ≈ 1.7 for each SE ≈ 0.3 SD ≈ 1 SE ≈ 0.5

Page 437, P7 a. Plot I is the population, starting out with only five different values and a slight skew. Plot III is for a sample size of 4, having more possible values, a smaller spread, and a more nearly normal shape. Plot II is for a sample size of 10, which has even more possible values, an even smaller spread, and a shape that is closest to normal of all the distributions.

Page 437, P7 b. Theoretically, the mean will be the same 1.7 for each distribution, which is consistent with the estimates.

Page 437, P7

Page 437, P7 d. The population distribution is roughly mound-shaped with a slight skew. The two sampling distributions; however, are approximately normal with the distributions becoming more nearly normal as the sample size increases. This is consistent with the Central Limit Theorem.

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Page 438, P10 Decrease in price means < 0

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Page 443, E21 normalcdf (510, 1E99, 500, 50)

Page 443, E21 normalcdf (510, 1E99, 500, 20)