Good morning, friends.

Transcription

1 Good morning, friends. Have your homework out. Get a yellow and pink highlighter; just one set per group. Have last time s notes out. Copy the dot plots on the post-its at the back of the room wall on your notes (make sure the dotplot you make is under the right heading). Await further instructions.

2 Sampling Distributions 7.2: Sample Proportions

3 The Sampling Distribution for proportions How good of a predictor of p is p-hat? Let s determine the shape, center, and spread of the sampling distribution of p-hat. Distribution of the population Machine full of candy Distribution of the sample The breakdown of each color count for a sample Distribution of the sample proportion The proportion of the sample which is orange Cool demo 1. Launch the applet and change p = 0.45

4 2. Draw Samples What was your sample proportion of orange candies? Was it close to the actual population proportion, p = 0.45?

5 3. Draw Samples 9 more times. Find the mean of your 10 sample proportions. Find the standard deviation. Add your statistics to the google doc shared to you on Friday, 1/6

6 more samples Turn animate off, and enter 390 in the number of samples, with a total, then, of 400 samples drawn. Fill out the shared doc with the indicated statistics. How would the shape, center, and spread of the sampling distribution of p- hat be described?

7 5. Change the sample size to 50 Reset the applet and take 400 samples of 50 candies. Fill out the shared doc with the indicated statistics. How would the shape, center, and spread of the sampling distribution of p- hat be described?

8 6. What if there was 15% orange candies instead of 45%? Reset the applet and take 400 samples of 25 candies when p = Reset the applet and take 400 samples of 50 candies when p = Find the statistics for each scenario and add them to the shared doc. How would the shape, center, and spread of the sampling distribution of p- hat be described for each?

9 Sampling Distribution of a sample proportion Choose an SRS of size n from a population of size N with proportion p of successes. Let p-hat be the sample proportion of successes. Center: The mean of the sampling distribution is Spread: As long as the 10% condition is satisfied**, the standard deviation of the sampling distribution of p-hat is** Shape: As n increases, the sampling distribution of p-hat becomes approximately normal. In order to perform Normal calculations, check that the Large Counts condition is satisfied**:

10 Example 1 About 75% of young adult Internet users (ages 18-29) watch online videos. Suppose that a sample survey contacts an SRS of 1000 young adult Internet users and calculates the proportion, p-hat in this sample who watch online videos. a. What is the mean of the sampling distribution of p-hat? Explain. b. Find the standard deviation of the sampling distribution of p-hat. Check that the 10% condition is met. c. Is the sampling distribution of p-hat approximately Normal? Check that the Large Counts condition is met. d. If the sample size were 9000 rather than 1000, how would this change the sampling distribution of p-hat?

11 Example 1 Answers About 75% of young adult Internet users (ages 18-29) watch online videos. Suppose that a sample survey contacts an SRS of 1000 young adult Internet users and calculates the proportion, p-hat in this sample who watch online videos. a. What is the mean of the sampling distribution of p-hat? 0.75 b. Find the standard deviation of the sampling distribution of p-hat Check that the 10% condition is met. Yep; There are more than 10,000 young adult Internet users. c. Is the sampling distribution of p-hat approximately Normal? Yes, both np = 750, and n(1-p) = 250. Check that the Large Counts condition is met. d. If the sample size were 9000 rather than 1000, how would this change the sampling distribution of p-hat? The sampling distribution would be approximately normal, with mean The standard deviation would be smaller, at

12 Using Normal Distribution As long as the sample size is large enough for np and n(1-p) to both be at least 10, we can assume the sampling distribution of p-hat is normally distributed. Under these conditions, we can then find the probability of an SRS with a p-hat in a specified interval.

13 Example 2: Normal Calculations involving p-hat A polling organization asks an SRS of 1500 first-year college students how far away their home is. Suppose that 35% of all firstyear students attend college within 50 miles of home. Find the probability that the random sample of 1500 students will give a result within 2 percentage points of this true value.

14 Example 2: process: Step 1 Step 1: Draw a picture of the distribution and the values of interest a. the mean of the sampling distribution of p- hat b. The standard deviation of the sampling distribution of p-hat Include checking the 10% condition c. Check the large counts condition

15 Example 2: Process: Step 2 Step 2: Perform calculations: Find the standardized scores for the boundary values, and the corresponding areas. Z = and z = 1.63 Areas: =

16 Example 2: Process: Step 3 Answer the question: About 90% of all simple random samples of size 1500 will give a result within 2 percentage points of the true proportion of students who attend college within 50 miles of home.

17 Example 3: Convincing Evidence? A newspaper report claims that 40% of all U.S. adults went to church last week. The Gallup Poll asked a random sample of 1785 adults whether they attended church during the past week. Of the poll respondents, 44% said they did attend church last week. a. Find the probability of obtaining a sample of 1785 adults in which 44% or more say they attended church last week if the newspaper report s claim is true. (Don t forget your check-steps 10% condition and Large Counts condition) b. Does this poll give convincing evidence against the claim? Explain.

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