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Interval Estimation Interval Estimation Point Estimate: A single number used to estimate a population value. Interval Estimate: A range within which a population value probably lies. Interval Estimator (Confidence Interval): A formula that tells us how to use sample data to calculate an interval that estimates a population parameter. We will calculate interval estimates for the population mean and the population proportion. Confidence Interval for the Population Mean, . Confidence Interval for the Population Mean The Confidence Interval (C.I.) for the population mean is given by: C.I. = x z 2 n Where z 2 is the z value with an area of 2 to its right. The parameter is the standard deviation of the sampled population. Confidence Interval for the Population Mean When is not known and n is large, the confidence interval is calculated by: s x z 2 n Where s is the sample standard deviation. The confidence coefficient is the probability that a randomly selected confidence interval will contain the true population parameter. The confidence level is the confidence coefficient expressed as a percentage. Interpreting a Confidence Interval for the Population Mean When we form a 100(1 )% confidence interval for the population mean, we usually express our confidence in the interval by saying “we can be 100(1 )% confident that lies between the lower and upper bounds of the confidence interval.” This means that in repeated sampling 100(1 )% of the resulting intervals will contain . Confidence Interval for the Population Mean Example 1: A random sample of size n = 100, from a population with variance 2 = 14400 has mean x = 800. Construct a 95% confidence interval for the population mean, . Confidence Interval for the Population Mean Example 2: A biochemist wishes to estimate the mean vitamin content of a fruit. From a random sample of size 36, he calculates the sample mean to be 12.5mg and the standard deviation to be 3.0mg. Construct a 99% confidence interval for the true vitamin content of the fruit. Confidence Interval for the Population Mean Example 3: Unoccupied seats on flights cause airlines to lose revenue. Suppose a large airline wants to estimate its average number of unoccupied seats per flight over the past year. To do this, the records of 225 flights are randomly selected and the number of unoccupied seats is noted for each of the sampled flights. The sample mean and standard deviation are 11.6 and 4.1 seats respectively. Estimate the mean number of unoccupied seats per flight during the past year using a 90% confidence interval. Confidence Interval for the Population Proportion Confidence Interval for the Population Proportion The C.I. for the population proportion is given by: pq p(1 p) C.I. = p z 2 ˆ = p z 2 ˆ n n When p is not known and the sample is large, the formula is given by: p(1 p) ˆ ˆ p z 2 ˆ n Confidence Interval for the Population Proportion Example 4: A firm is considering producing and promoting a new product. It is estimated that the new product will be profitable if at least 5 percent of the population buy it within the first year of its introduction. A survey of 200 people has shown that 6 people would buy the product. i. Give a point estimate for the proportion of buyers in the population. ii. Construct a 95% C.I. for the true population proportion. Confidence Interval for the Population Proportion Example 5: A food products company conducted a market study by randomly sampling and interviewing 1000 consumers to determine which brand of breakfast cereal they prefer. Suppose 313 consumers were found to prefer the company’s brand. How would you estimate the true fraction of all consumers who prefer the company’s cereal brand? Construct a 95% confidence interval for the proportion of all consumers who prefer the brand. Determining the Sample Size The appropriate sample size for making an inference about a population parameter depends on the desired reliability or the maximum allowable error. The maximum allowable error when forming a confidence interval for the population mean is given by: E z 2 n Transposing this formula and solving for n yields: z 2 n E Determining the Sample Size The maximum allowable error when forming a confidence interval for the population proportion is given by: p (1 p ) ˆ ˆ E z 2 n Transposing this formula and solving for n yields: 2 z n p(1 p) ˆ ˆ E Determining the Sample Size Example 6: Recall example 2, what sample size is required if the 95% C.I. is to be done so that the maximum error of the estimate is 0.5mg? Example 7: Recall example 5, suppose the company wished to estimate its market share to within 0.015 with a 95% C.I., what sample size is required?

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interval estimation, confidence interval, interval estimate, Point estimate, conﬁdence interval, sample size, population mean, point estimation, 95% confidence, sample mean

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posted: | 4/16/2010 |

language: | English |

pages: | 16 |

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