SAMPLING DISTRIBUTION MODELS TODAY YOU WILL NEED: PENCIL SCRATCH PAPER A PARTNER (YOUR CHOICE) ONE THUMBTACK PER GROUP Z-SCORE CHART
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1 SAMPLING DISTRIBUTION MODELS TODAY YOU WILL NEED: PENCIL SCRATCH PAPER A PARTNER (YOUR CHOICE) ONE THUMBTACK PER GROUP Z-SCORE CHART
2 FLIPPING THUMBTACKS PART 1 I want to know the probability that, when you flip a thumbtack, it lands point up. 1. Assign jobs - one person is the tosser and one person is the recorder. 2. Flip the thumb tack five times and record the sample proportion of ups. (i.e. 0.20, 0.80 etc.). 3. Record your data on the board (only one proportion per group) 4. Sketch a histogram (or dotplot) of the class results. 5. Describe the distribution (CUSS).
3 FLIPPING THUMBTACKS PART 2 I want to know the probability that, when you flip a thumbtack, it lands point up. 1. Flip the thumb tack twenty more times and record the sample proportion of ups (out of 25 total). 2. Record your data on the board (only one proportion per group) 3. Sketch a histogram (or dotplot) of the class results. 4. Describe the distribution (CUSS).
4 FLIPPING THUMBTACKS PART 3 I want to know the probability that, when you flip a thumbtack, it lands point up. 1. Flip the thumb tack 75 more times and record the sample proportion of ups (out of 100 total). 2. Record your data on the board (only one proportion per group) 3. Sketch a histogram (or dotplot) of the class results. 4. Describe the distribution (CUSS).
5 FLIPPING THUMBTACKS CONCLUSIONS On your paper, answer the following questions: 1. What changes occurred to your histogram as the number of tosses increased? 2. Are your thumbtack flips considered Bernoulli trials? 3. Does the standard deviation become larger or smaller as the sample sizes increase? What about the mean? 4. Could you use a binomial model to predict the number of ups we would get in proportions of 200 tosses? How? 5. Why could you not use a binomial model to predict the number of ups we would get in proportions of 10,000 tosses? What would be a better way to predict this?
6 SKETCH THE NORMAL MODEL OF ALL CARS ON THE INTERSTATE, 80% EXCEED THE SPEED LIMIT. WHAT PROPORTION OF SPEEDERS MIGHT WE SEE AMONG THE NEXT 50 CARS? Check the conditions to make sure we re allowed to use a Normal model: 10% condition Success / failure condition What s the mean? What s the standard deviation?
7 SKETCH THE NORMAL MODEL WE DON T KNOW IT, BUT 52% OF VOTERS PLAN TO VOTE YES ON THE UPCOMING SCHOOL BUDGET. WE POLL A RANDOM SAMPLE OF 300 VOTERS. WHAT MIGHT THE PERCENTAGE OF YES-VOTERS APPEAR TO BE IN OUR POLL? Check the conditions to make sure we re allowed to use a Normal model: What s the mean? 10% condition Success / failure condition What s the standard deviation?
8 CALCULATING PROBABILITIES GROOVY M&M S ARE SUPPOSED TO MAKE UP 30% OF THE CANDIES SOLD. IN A LARGE BAG OF 250 M&M S, WHAT IS THE PROBABILITY WE GET AT LEAST 25% GROOVY CANDIES? Check the conditions to make sure we re allowed to use a Normal model: What s the mean? 10% condition Success / failure condition What s the standard deviation?
9 CALCULATING PROBABILITIES In the previous problem, we used the mean and standard deviation of a sample to predict the mean and standard deviation of the population. What if we already know the mean and standard deviation of the population, and we want to find the mean and standard deviation of a sample?
10 SKETCH THE NORMAL MODEL SAT SCORES SHOULD HAVE A MEAN OF 500 AND A STANDARD DEVIATION OF 100. WHAT ABOUT THE MEAN OF RANDOM SAMPLES OF 20 STUDENTS? Check the conditions to make sure we re allowed to use a Normal model: What s the mean? What s the standard deviation? Random sampling condition. Independence assumption. 10% condition
11 SKETCH THE NORMAL MODEL SPEEDS OF CARS ON A HIGHWAY HAVE MEAN 52 MPH AND A STANDARD DEVIATION OF 6 MPH, AND ARE LIKELY TO BE SKEWED TO THE RIGHT (A FEW VERY FAST DRIVERS). DESCRIBE WHAT WE MIGHT SEE IN RANDOM SAMPLES OF 50 CARS? Check the conditions to make sure we re allowed to use a Normal model: What s the mean? What s the standard deviation? Random sampling condition. Independence assumption. 10% condition
12 CALCULATING PROBABILITIES AT BIRTH, BABIES AVERAGE 7.8 POUNDS, WITH A STANDARD DEVIATION OF 2.1 POUNDS. A RANDOM SAMPLE OF 34 BABIES BORN TO MOTHERS LIVING NEAR A LARGE FACTORY THAT MAY BE POLLUTING THE AIR AND WATER SHOWS A MEAN BIRTHWEIGHT OF ONLY 7.2 POUNDS. IS THAT UNUSUALLY LOW? Check the conditions to make sure we re allowed to use a Normal model: What s the mean? What s the standard deviation? Random sampling condition. Independence assumption. 10% condition
13 ACTIVITY-BASED STATISTICS Spinning Pennies
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