invest_3ed.pdf

# We previously used simulation and the binomial

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We previously used simulation and the binomial distribution to determine which values were plausible for the underlying probability that a kissing couple leans right. In particular, we found 0.5 and 0.74 were not plausible values but 0.6667 was. Now you will consider applying the normal model as another method for producing confidence intervals for this parameter. (a) With a sample size of 124 kissing couples, does the Central Limit Theorem predict the normal probability distribution will be a reasonable model for the distribution of the sample proportion? Note : When you do not have a particular value to be tested for the process probability, it’s reasonable to use the sample proportion in checking the sample size condition for the CLT. (Note: This is equivalent to making sure there are at least 10 successes and at least 10 failures in the sample.) (b) Do you have enough information to describe and sketch the distribution of the sample proportion as predicted by the Central Limit Theorem? Explain. (c) Suggest one method for estimating the standard deviation of this distribution of sample proportions based on the observed sample data. Definition: The standard error of the sample proportion, SE( p ˆ ), is an estimate for the standard deviation of p ˆ (i.e., SD( p ˆ )) based on the sample data, found by substituting the sample proportion for S : SE( p ˆ ) = n p p ) ˆ 1 ( ˆ ± . (d) Now consider calculating a 95% confidence interval for the process probability S based on the observed sample proportion p ˆ . Calculate the standard error of p ˆ . Then, how far do you expect to see a sample proportion fall from the underlying process probability? [ Hint : Assuming a normal distribution 95% ….. ] SE( p ˆ ) = Plausible distance =

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Chance/Rossman, 2015 ISCAM III Investigation 1.10 83 (e) Use the distance in (d) and the observed sample proportion of p ˆ = 0.645 to determine an interval of plausible values for S , the probability that a kissing couple leans to the right. An approximate 95% confidence interval for the process probability based on the normal distribution would be p ˆ r 2 n p p / ) ˆ 1 ( ˆ ± . That is, this interval extends two standard deviations on each side of the sample proportion. We know that for the Normal distribution roughly 95% of observations (here sample proportions) fall within 2 SDs of the mean (here the unknown population proportion), so this method will capture the process probability for roughly 95% of samples. However, we should admit that the multiplier of 2 is a bit of a simplification. So how do we find a more precise value of the multiplier to use, including for values other than 95%? We will use technology to do this. Keep in mind that the z -value corresponding to probability C in the middle of the distribution, also corresponds to having probability (1 ± C )/2 in each tail.
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