Population Sample Statistic Parameter Proportion p Count Mean

Population Sample Statistic: Parameter: Proportion p Count Mean Median

Estimate population proportion, with a confidence interval, from data of a random sample.

p Sample Proportion = p Population

p There is 95% chance that will fall inside the interval Sample Proportion = p Population ND

p There is 95% chance that will fall inside the interval Sample Proportion = p Population

p There is 95% chance that will fall inside the interval Sample Proportion = p Population

Open the Fathom file ‘estimate 1. ftm’

The proportion of “yes” in the population is given by the slider value p. (In this example, p =0. 75)

Assume that the population proportion is an unknown, and we are going to estimate it by suggesting a 95% confidence interval based on the data of one random sample. Size of this sample is n = 20.

The proportion of “yes” in this sample is The estimated standard deviation of is Margin of error = 2(0. 080) = 0. 160 The 95% confidence interval is (0. 85 – 0. 16, 0. 85 + 0. 16) or (0. 69, 1. 01)

Based on the result of this sample, we are 95% confident that the true proportion p lies between 0. 69 and 1. 01. 0 0. 2 0. 4 0. 6 0. 8 1 95% conf. Int. Note that the interval will vary from sample to sample, but if we repeat the sampling process indefinitely with samples of the same size, we will expect 95% of these intervals to capture the true proportion. To shorten the interval, we have to increase the sample size.

Note that the interval will vary from sample to sample, but if we repeat the sampling process indefinitely with samples of the same size, we will expect 95% of these intervals to capture the true proportion. 0 0. 2 Confidence Intervals from different samples 0. 4 0. 6 0. 8 1

Use an experiment record sheet to record more confidence intervals from other samples of the same size.

Some intervals may not be able to capture the true proportion.

To estimate with a larger sample, double click on the ‘Sample of Data’ collection to open its inspector and adjust the sample size.

The proportion of “yes” in this sample is and the sample size is n = 80. The estimated standard deviation of is Margin of error = 2(0. 0467) = 0. 0934 The 95% confidence interval is (0. 775 – 0. 0934, 0. 775 + 0. 0934) or (0. 682, 0. 868)

We are now 95% confident that the true proportion lies between 0. 682 and 0. 868. The interval is shorter when the sample size is increased from 20 to 80. Sample size = 80 0 0. 2 0. 4 0. 6 0. 8 1 95% conf. Int. Sample size = 20 0 0. 2 0. 4 0. 6 0. 8 95% conf. Int. 1

Example: Halloween Practices and Beliefs An organization conducted a poll about Halloween practices and beliefs in 1999. A sample of 1005 adult Americans were asked whether someone in their family would give out Halloween treats from the door of their home, and 69% answered ‘yes’. Construct a 95% confidence interval for p, the proportion of all adult Americans who planned to give out Halloween treats from their home in 1999. Adapted from Rossman et al. (2001, p. 433)

Sample size = 1005 Sample proportion = 0. 69 Estimated standard deviation of sample proportions = Margin of error = 2(0. 0146) = 0. 0292 95% confidence interval is 0. 69 0. 0292 We are 95% confident that the population proportion lies between 0. 6608 and 0. 7192. 0 0. 2 0. 4 0. 6 0. 8 95% conf. Int. 1

Example: Personal Goal According to a survey in a university, 132 out of 200 first-year students in a random sample have identified “being well-off financially” as an important personal goal. Give a 95% confidence interval for the proportion of all firstyear students at the university who would identify being well-off as an important personal goal. Adapted from Moore & Mccabe (1999, p. 597)

Sample size = 200 Sample proportion = 132/200 = 0. 66 Estimated standard deviation of sample proportions = Margin of error = 2(0. 0335) = 0. 067 95% confidence interval is 0. 66 0. 067 We are 95% confident that the population proportion lies between 0. 593 and 0. 727. 0 0. 2 0. 4 0. 6 95% conf. Int. 0. 8 1

Estimate population mean, with a confidence interval, from data of a random sample.

Sample Mean = Population

There is 95% chance that will fall inside the interval Sample Mean = s. d. = Population ND

There is 95% chance that will fall inside the interval Sample Mean = s. d. = Population

There is 95% chance that will fall inside the interval Sample s. d. = s Mean = Population

Open the Fathom file ‘estimate 2. ftm’

This summary table record the true mean and standard deviation of the population, where are supposed to be unknowns.

Assume that the population mean is an unknown, and we are going to estimate it by suggesting a 95% confidence interval based on the data of one random sample. Size of this sample is n = 20.

The sample mean and standard deviation are The estimated standard deviation of is Margin of error = 2(4. 52) = 9. 046 The 95% confidence interval is (29. 85 – 9. 05, 29. 85 + 9. 05) or (20. 80, 38. 90)

Based on the result of this sample, we are 95% confident that the true mean lies between 20. 80 and 38. 90. 0 10 20 30 40 50 95% conf. Int. Note that the interval will vary from sample to sample, but if we repeat the sampling process indefinitely with samples of the same size, we will expect 95% of these intervals to capture the true mean. To shorten the interval, we have to increase the sample size.

Example: Protein Intake A nutritional study produced data on protein intake for women. In a sample of n = 264 women, the mean of protein intake is grams and the standard deviation is s = 30. 5 grams. Estimate the population mean and give a 95% confidence interval. Adapted from Bennett et al. (2001, p. 401)

Estimated standard deviation of the sample means = Margin of Error = 2(1. 9) = 3. 8 grams 95% confidence interval is 59. 6 3. 8 grams We can say with 95% confidence that the interval ranging from 55. 8 grams to 63. 4 grams contains the population mean.

Example: Body Temperature A study by University of Maryland researchers investigated the body temperatures of n = 106 subjects. The sample mean of the data set is and the standard deviation for the sample is . Estimate the population mean body temperature with a 95% confidence interval. Adapted from Bennett et al. (2001, p. 403)

Estimated standard deviation of the sample means = Margin of Error = 2(0. 06 F) = 0. 12 F 95% confidence interval is 98. 20 F 0. 12 F We can say with 95% confidence that the interval ranging from 98. 08 F to 98. 32 F contains the population mean.

Normal Distribution 99. 7% 95% 68% m – 3 s m – 2 s m–s m m+s m + 2 s m + 3 s