Download Understanding Sampling Distribution & Margin of Error in Proportions' Confidence Intervals and more Study notes Business Management and Analysis in PDF only on Docsity! Chapter 9 Confidence Intervals for Proportions © 2011 Pearson Education, Inc. 1 Business Statistics: A First Course 2 9.2 The Sampling Distribution for Proportions Because each sample is different, he observed proportion of successes, , varies from sample to sample. However, the ’s vary in a way that we can model and predict. The distribution of many ’s from a large number of samples is called the sampling distribution of . The sampling distribution of is a Normal distribution centered at the true proportion, p. The standard deviation of the sampling distribution, also called the standard error: © 2011 Pearson Education, Inc. p̂ p̂ p̂ p̂ p̂ n qp n pp pSDSE ˆˆˆ1ˆ ˆ 5 9.5 A Confidence Interval What Does 95% Confidence Really Mean? Below are confidence intervals produced by simulating 20 samples. The purple dots are the of the 20 samples; the orange lines show the confidence intervals. The green line is the true p. Not all intervals capture the true proportion. 5% of the time, the confidence interval misses the true p. © 2011 Pearson Education, Inc. p̂ 6 9.6 Margin of Error: Certainty vs. Precision The confidence interval is given by SEzp *ˆ The width of the interval on each side of is called the margin of error (ME). The margin of error is half the width of the confidence interval. The wider the interval, the larger the margin of error. ME gives the maximum error between p and , for the given confidence level. In general, the confidence interval can be written: where Note that for a given confidence interval, the Margin of Error is always half the width. For example (0.4, 0.6) has ME = 0.1. p̂ MEestimate © 2011 Pearson Education, Inc. SEzME * p̂ 7 9.6 Margin of Error: Certainty vs. Precision The more confident we want to be about including the true p, the larger the margin of error. We can be very confident p is inside a very wide interval, but we can’t be very confident p is inside a narrow interval. Every confidence interval is a balance between confidence and interval width (precision). The greater the confidence, the wider the interval must be to capture the true p. A narrow interval will have a low confidence of capturing the true p. Wide confidence intervals have a high chance of capturing p, but also a large margin of error. Narrow intervals have a small margin of error, but a smaller chance of capturing p. © 2011 Pearson Education, Inc. 10 9.8 Assumptions and Conditions 3) 10% Condition: The population must be at least 10 times larger than the sample. 4) Sample size: The number of successes np ≥ 10, and also the number of failures nq ≥ 10. © 2011 Pearson Education, Inc. 11 9.9 Sample Size To get a narrower interval for the same confidence, we must choose a larger sample. Example: Suppose a company wants to offer a new service and wants to estimate the proportion of customers to within 3% who are likely to purchase this new service, with 95% confidence. How large a sample do they need? 0.03 is the margin of error, and for 95% confidence z* = 1.96: qpqp ME z n n qp zME ˆˆ 03.0 96.1 ˆˆ ˆˆ * 22* What is the value of and ? © 2011 Pearson Education, Inc. p̂ q̂ 12 9.9 Sample Size The worst case scenario occurs when = 0.50. This results in the largest possible n, so it covers all the other cases. Better to err with too large a sample size, than too small. Sample size always rounds up, because rounding down leads to a smaller sample size and a bigger error. The company will need at least 1068 respondents to keep the margin of error to within 3%, with 95% confidence. p̂ © 2011 Pearson Education, Inc. 10681.10675.05.0 03.0 96.1 ˆˆ 03.0 96.1 22 qpn
