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6.1 Confidence Intervals for the fd l f h Mean (Large Samples) Mean (Large Samples) Estimating Population Parameters Estimating Population Parameters • Point Point Estimate is a single value estimate for Estimate is a single value estimate for the population parameter. Most likely is the sample mean. • Interval estimate is an interval, or range of value, used to estimate a population parameter • Level of confidence is the probability that the interval estimate contains the population parameter. Level of Confidence Level of Confidence Level of Confidence Z‐score Level of Confidence Level of Confidence Level of Confidence Z‐score 90% 1 645 1.645 Level of Confidence Level of Confidence Level of Confidence Z‐score 90% 1 645 1.645 95% 1.96 Level of Confidence Level of Confidence Level of Confidence Z‐score 90% 1 645 1.645 95% 1.96 99% 2.575 Finding the z‐score Finding the z score Level of confidence is 88% Level of confidence is 88% Look on the chart and find .06 Look on the chart and find .06 and tell the corresponding z‐ score 6% About 1.555 88% 6% Finding Point Estimate Finding Point Estimate Market researchers use the number of sentences per advertisement as a measure of readability for magazine advertisements. The following represents a random sample of readability for magazine advertisements. The following represents a random sample of the number of sentences found in 54 advertisements. Find the point estimate of the population mean. 9 20 18 19 9 16 16 9 11 13 22 16 5 18 6 6 5 12 25 17 23 7 10 9 10 10 5 11 18 18 9 9 17 13 11 7 25 17 23 7 10 9 10 10 5 11 18 18 9 9 17 13 11 7 14 6 11 12 11 15 6 12 14 11 4 9 18 12 12 17 11 20 A good point estimate is the sample mean. d h l About 12 481 About 12.481 Margin of Error Margin of Error Margin of Error, E, is the greatest possible distance between the point estimate and the value of the parameter it is estimating given the level of confidence. When n is greater than 30, the sample standard deviation, s, can be used in place of sigma. Finding the Margin of Error Finding the Margin of Error Use the data from the first example and a 95% confidence level and a standard deviation of 5 to find the margin of error for the mean number of sentences in all magazine advertisements. You are 95% confident that the margin of error for the population mean is You are 95% confident that the margin of error for the population mean is about 1.333 sentences. Confidence Interval Confidence Interval Construct a Confidence Interval Construct a Confidence Interval Construct a 95% confidence interval for the mean number of sentences in all magazine advertisement. advertisement. From the previous examples With a 95% confidence, you can say that the population mean number of With a 95% confidence, you can say that the population mean number of sentences is between 11.148 and 13.814 Guidelines for Constructing a Confidence Interval fd l • Find Find the n, the standard deviation, and the the n the standard deviation and the sample mean (point of estimate) • Find the z‐score that corresponds to the given Find the z score that corresponds to the given level of confidence • Find the margin of error Fi d h i f • Find the left and right endpoints of the confidence interval Constructing a Confidence Interval Constructing a Confidence Interval A college admissions director wishes to estimate the mean age of all students currently enrolled In a random sample of 20 students the mean age is found to be currently enrolled. In a random sample of 20 students, the mean age is found to be 22.9 years. From past studies, standard deviation is known to be 1.5 years, and the population is normally distributed. Construct a 90% confidence interval of the population mean age. Find the z‐score from the chart that corresponds to 90% Find the margin of error Find the confidence interval With 90% confidence, you can say that the mean age of all the say that the mean age of all the students is between 22.348 and 23.452. Constructing a Confidence Interval Constructing a Confidence Interval Construct a 90% confidence interval of the population mean age for all the college students in the previous example if the sample size is increased to 30 students. students in the previous example if the sample size is increased to 30 students. With 90% confidence, you can say that the mean age of all the students is between 22.449 and 23.351 Determine the Minimum Sample Size Determine the Minimum Sample Size You want to estimate the mean number of sentences in a magazine advertisement. How many magazine advertisements must be included in the sample if you want to be 95% confident that the sample mean is within one sentence of the population mean. Assume the standard deviation is 5. So you should include at least 97 magazine advertisements in your sample