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AP Statistics ________________________ Activity 10 Estimating with CONFIDENCE Name Date_________________________ That Sure Was TACKY!!!! Materials: A thumb tack. When you flip a FAIR coin, it is equally likely to land “HEADS” or “TAILS.” The question for discussion. DO THUMBTACKS BEHAVE IN THE SAME WAY? Explain. In this activity you will toss a thumbtack several times and observe whether the tack comes to rest with the point (UP or the point down (D). The question we are trying to answer is: What Proportion of the Time does a Tossed Thumbtack Settle with Its Point UP (U)? 1. If you could toss your thumbtack over and over and over again, what proportion of all tosses do you think would settle with the point up (U)? What is your guess? _________ = _________ is a parameter. 2. Let’s conduct the simulation. Toss your thumbtack 50 times. Record the result of each toss ( U or D ) in the table below. The the third column, calculate the cumulative proportion of points up (U) tosses. Toss Outcome ( U or or D ) Cumulative Prop. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Toss Outcome ( U or or D ) Cumulative Prop. 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 22 23 24 25 47 48 49 50 3. Construct a SCATTER PLOT of Tosses and Cumulative Proportion. Number of Tosses on the horizontal axis and the cumulative proportion of ( U ) on the vertical axis. Connect the consecutive points with line segments. Does the overall proportion of ( U’s ) seem to be approaching a single value? If yes, please name the value.__________ If no, explain. 4. The n = 50 tosses is a SRS from the POPULATION of all possible tosses. The PARAMETER is the (UNKNOWN) POPULATION proportion of tosses that would land up ( U ). What is your BEST ESTIMATE for sample of 50 ?. We call the estimate value _________ , the proportion of ( U’s) in your thumbtack tosses. Record your _______ = _______, How does it compare with the conjecture you made in step number 1? Explain. 5. If you tossed your thumbtack 50 additional times, would you expect to get the same value of ________? Explain. If we repeat this process, the MEAN of the SAMPLING DISTRIBUTION of ________ is equal to the POPULATION PROPORTION . ie. p¦ = ___________ ___________ is unknown. A question we would like to answer is: HOW FAR WILL YOUR SAMPLE PROPORTION ______ BE FROM THE TRUE VALUE OF THE POPULATION PROPORTION ? If the sampling distribution is approximately normal, then the Empirical Rule, _______, _______, _______ tells us that approximately _________ of all sample proportion values, _______, will be within TWO (2) STANDARD DEVIATIONS OF . 6. The Sampling Distribution of _______ will be approximately NORMAL if the RULE OF THUMB #2 is satisfied. Rule of Thumb #2 id: _____________________________________________________________________ Verify the Conditions for your SAMPLE. 7. Estimate the STANDARD DEVIATION of the SAMPLING DISTRIBUTION by computing p¦ = =_________ = _________ 8. Construct the INTERVAL _________ sample. _______________ _________ * based on your __________*_____ _____ _____ The INTERVAL is ( _____________________ , _____________________ ) This INTERVAL is called a CONFIDENCE _________________ for the sampling distribution of________. The center of your confidence interval is _________. The length of your confidence interval is __________ 9. Draw the SKETCH OF A NORMAL DISTRIBUTION with your proportion mean and your proportion standard deviation of your sample of 50 thumbtack tosses. Label the horizontal axis and mark off a scale. Construct all of your classmates confidence intervals one after another below the horizontal axis accurately with a ruler. 10. We have ________ samples from the same population of THUMBTACKS gave these 95% CONFIDENCE INTERVAL. We state that in the long run, 95% of all samples give an interval that contains the population proportion . So, approximately 95% of the intervals, the interval _________ will contain the true population proportion . _________ *