Handout THQ #5 at end of class. Day 49 Agenda: Handout THQ #5 at end of class.

Advanced Placement Statistics Section 9.3: Sample Means EQ: What does the Central Limit Theorem tell us about sampling distributions?

Use a cup to scoop out a sample of pennies from the sampling frame. Form groups of 2 to complete the Penny Activity Use a cup to scoop out a sample of pennies from the sampling frame. Randomly select 25 for your group then return the extra pennies to container. KEEP YOUR CUP. Follow the directions on your worksheet.

Creating dot plots to show shape of our sampling distributions: Go to the board and place a dot at the age for each of your pennies. Use the correct color marker to plot your five means for penny samples of size 1, size 5, and size 10 and your one mean for sample size 25. After everyone has done this, sketch the shape of each histogram below.  

Skewed Right Sample size Bell-shaped Normal Approximation CONCLUSION: Our original population distribution was not described as Normal nor was it bell-shaped. In fact it was __________________________.   However, as we increased the __________________, the distribution got closer and closer to a _________________ curve and could be approximated using a ___________. This property is called the _________. Skewed Right Sample size Bell-shaped Normal Approximation Central Limit Theorem

Sample Means --- averages of observations

Sample Means are less variable than single observations. Sample Means have a more normal distribution than single observations. A visual comparison of the distribution of sample means as the sample size increases.

RECALL: Sampling Proportions

What about Sampling Means? μ

True no matter what the shape of the population distribution. Central Limit Theorem ---SRS of size n taken from population with mean µ and standard deviation σ:   Sample Size LARGE ENOUGH

Central Limit Theorem  Law of Large Numbers Law of Large Numbers --- draw observations at random from any population with finite mean µ: Central Limit Theorem  Law of Large Numbers Sample Size Large Enough to use Normal Approximation Sample Mean approaches True Population Mean as observations increase.

SPARK NOTES FOR THIS SECTION:

SPARK NOTES FOR THIS SECTION:

SPARK NOTES FOR THIS SECTION:

In Class Assignment: We will do Worksheet: Sample Means together In Class Assignment: We will do Worksheet: Sample Means together. Finish it for HW. Recall: State, Plan, Do

PLAN: parameter of interest Worksheet: Sample Means   Follow the template we used in class for sampling distributions for proportions. 1. A survey found that the American family generates an average of 17.2 pounds of glass garbage each year. Assume the distribution is normal with a standard deviation of 2.5 pounds. STATE: What is the probability that a randomly selected American family will generate more than 18 pounds of garbage? PLAN: parameter of interest μ = the true mean pounds of glass garbage produced by American families annually

The problem states a family will be selected randomly. 1. A survey found that the American family generates an average of 17.2 pounds of glass garbage each year. Assume the distribution is normal with a standard deviation of 2.5 pounds.   What is the probability that a randomly selected family will generate more than 18 pounds of garbage? PLAN: randomness The problem states a family will be selected randomly. independence Standard Error not needed. Only selecting a single family.

1. A survey found that the American family generates an average of 17 1. A survey found that the American family generates an average of 17.2 pounds of glass garbage each year. Assume the distribution is normal with a standard deviation of 2.5 pounds.   What is the probability that a randomly selected family will generate more than 18 pounds of garbage? PLAN: large counts Problem states that distribution of garbage consumption for American families is Normally distributed.

1. A survey found that the American family generates an average of 17 1. A survey found that the American family generates an average of 17.2 pounds of glass garbage each year. Assume the distribution is normal with a standard deviation of 2.5 pounds.   What is the probability that a randomly selected family will generate more than 18 pounds of garbage? Do: μ = 17.2 σ = 2.5 n = 1 The probability that a randomly selected American family will generate more than 18 pounds of garbage each year is 37.45%.

μ = 17.2 σ = 2.5 n = 1 The probability that a randomly selected family will generate more than 18 pounds of garbage each year is 37.42%

The problem states the families will be selected randomly. b. What is the probability that the mean sample of 55 families selected randomly will be between 17 and 18 pounds? PLAN: randomness The problem states the families will be selected randomly. independence all families > 10(55) all families > 550 Condition met for Independence

PLAN: large counts n = 55 55 > 30 CLT states that sample size is large enough to use Normal Approx. n = 55 55 > 30

Do: The probability that the mean weight of yearly garbage of 55 randomly selected American families will be between 17 and 18 pounds is 71.5%.

c. If the distribution of glass garbage produced by the population were not normal, describe the distributions for a) and b). a) The sample size is only n = 1, therefore this distribution will not be normal. b) The sample size is n = 55. The CLT says that sample sizes of at least 30 will create a distribution that is approximately normal, even if the population distribution was not normal.

μ = the true mean annual cost per household for owning a dog 2. The average yearly cost per household of owning a dog is $186.80. Assume the standard deviation of the distribution is $32. Suppose we randomly select 50 households that own a dog. What is the probability that the sample mean for these 50 households is less than $175? What is the probability that the sample mean for these 50 households is less than $175? PLAN: parameter of interest μ = the true mean annual cost per household for owning a dog randomness The problem states the families will be selected randomly.

PLAN: independence large counts All households > 10(50) 2. The average yearly cost per household of owning a dog is $186.80. Assume the standard deviation of the distribution is $32. Suppose we randomly select 50 households that own a dog. What is the probability that the sample mean for these 50 households is less than $175? PLAN: independence All households > 10(50) All households > 500 Condition met for Independence large counts 50 > 30 CLT says normal distribution appropriate. n = 50

2. The average yearly cost per household of owning a dog is $186. 80 2. The average yearly cost per household of owning a dog is $186.80. Assume the standard deviation of the distribution is $32. Suppose we randomly select 50 households that own a dog. What is the probability that the sample mean for these 50 households is less than $175? Do: n = 50 The probability that a random sample of 50 households will have an average yearly cost for owning a dog less than $175 is 0.5%.

μ = the true mean salary of teachers in New Jersey 3. The average teacher’s salary in New Jersey (ranked first among states) is $52,174. Assume the distribution is normal with a standard deviation of $700. a. What is the probability that a randomly selected teacher makes less than $50,000 a year? PLAN: parameter of interest μ = the true mean salary of teachers in New Jersey randomness The problem states a teacher will be selected randomly.

Standard Error not needed. Only selecting a single teacher. 3. The average teacher’s salary in New Jersey (ranked first among states) is $52,174. Assume the distribution is normal with a standard deviation of $700. PLAN: independence Standard Error not needed. Only selecting a single teacher. large counts Problem states that distribution of teacher’s salaries in NJ is Normally distributed.

The probability that a randomly 3. The average teacher’s salary in New Jersey (ranked first among states) is $52,174. Assume the distribution is normal with a standard deviation of $700. Do: μ = 52,175 σ = 700 n = 1 The probability that a randomly selected teacher in NJ makes less than $50,000 is 0.09%.

μ = the true mean salary of teachers in New Jersey b. If we randomly sample 100 teachers’ salaries, what is the probability that the sample mean is less than $50,000? PLAN: parameter of interest μ = the true mean salary of teachers in New Jersey randomness The problem states a random sample of 100 teachers’ salaries will be obtained independence All teachers in NJ > 10(100) All teachers in NJ > 1000 Condition met for Independence

Do we need to reference the CLT? b. If we randomly sample 100 teachers’ salaries, what is the probability that the sample mean is less than $50,000? large counts PLAN: 100 > 30 CLT says normal distribution appropriate. n = 100 Do we need to reference the CLT? Do: n = 100

b. If we randomly sample 100 teachers’ salaries, what is the probability that the sample mean is less than $50,000? Do: n = 100 The probability that a random sample of 100 teachers in NJ have a mean salary less than $50,000 is 0%.

Assignment: p. 595 - 596 #31 – 34 p. 601 - 602 #35 – 40

THQ#5 Due MONDAY Day 54 Workday #1 Sampling Distribution Worksheet Review Ch 9 Day 55 (½) Workday #2 DG23 Worksheet Putting It all Together Begin Ch 10 THQ#5 Due MONDAY