Lesson 7 Confidence Intervals: The basics

Recall is the mean of the sample and s is the standard deviation of the sample. Where μ is the mean of the population and σ is the standard deviation of the population.

Proportion just means a percentage

While we use μ to represent they are not always exactly the same. The margin of error helps give a range of values that we would expect μ to fall between. The confidence level is the probability that the value of the parameter μ falls within the interval. If the sampling procedure was repeated many times we would expect the confidence percent of the intervals to capture the true parameter. Confidence Level

The point estimate is 63% and the margin of error is 4%. We are 95% confident that the interval 59% to 67% captures the true proportion of adults who support the death penalty.

Example: A large company is concerned that many of its employees are in poor physical condition, which can result in decreased productivity. To determine how many steps each employee takes per day, on average, the company provides a pedometer to 50 randomly selected employees to use for one 24-hour period. After collecting the data, the company statistician reports a 95% confidence interval of 4547 steps to 8473 steps. a)What is the point estimate that was used to create the interval? What is the margin of error? b) Interpret the confidence level(%)c) Interpret the confidence interval (endpts) To find the point estimate find the middle of the interval: 6510 The margin of error would be = ± 1963 If we took many samples, we could expect 95% of the intervals capture the true mean steps. We are 95% confident that the interval 4547 to 8473 captures the true mean steps of employees in a 24-hour period.

Calculating a 95% Confidence Interval: point estimate ± margin of error statistic ± critical value  standard deviation ± 2s For a critical value we use 2 because we want the estimate to fall within two standard deviations (95%). (Remember the empirical rule for a normal distribution.) For their semester project in Statistics, Ann and Tori wanted to estimate the average weight of an Oreo to determine if the average weight was less than advertised. They selected a random sample of 36 cookies and found the weight of each cookie (in grams). The mean weight was grams with a standard deviation of a)Construct and interpret a 95% confidence interval for the mean weight of an Oreo cookie. b) On the packaging, the stated serving size is 3 cookies (34 grams). Does the interval in part (a) provide convincing evidence that the average weight of an Oreo cookie is less than advertised? = and s = ± 2(0.0817) ± to