Confidence Intervals

Confidence Intervals A plausible range of values for the population parameter is called a confidence interval. Using a sample statistic to estimate a parameter is like fishing in a murky lake with a spear. Using a confidence interval is like fishing with a net.

Confidence Interval For a single Proportion

Notation for Numerical Summaries Population Proportion = p Sample Proportion = Sample size = n

Example: Binge Drinking in College Binge drinking is alcohol consumption that results in a blood alcohol concentration (BAC) of 0.08 or higher. For typical adults, this corresponds to 5 or more drinks (males), or 4 or more drinks (females), in about 2 hours. College presidents have indicated that binge drinking is the most serious problem on campuses. How can we estimate the proportion of students in U.S. four-year colleges that are binge drinkers?

Obtaining a point estimate Select a simple random sample of students at U.S. four-year colleges. The sample size is this example is n = 1000. The proportion of students in this sample that reported binge drinking is = 0.4323. = 0.4323 is the point estimate.

The sampling distribution This is the sampling distribution for the population proportion. This sampling distribution is not very helpful since we do not know the population proportion p. However, we do know the sample proportion , To make use of this fact, we switch the roles of p and .

The sampling distribution using This is the sampling distribution using the sample statistic in place of the population parameter. This can be determined easily. All we need to know is the sample size and the sample proportion.

Computing a 95% confidence interval A random sample of 1000 college students were asked if they had been binge drinking in the last two weeks. The proportion that responded they had was 0.4323.

Conclusion In a report: We are 95% confident that between 40.15% and 46.31% of all students in U.S. four-year colleges binge drink.

Conclusion In a report: We are 95% confident that between 40.15% and 46.31% of all students in U.S. four-year colleges binge drink. In a news article: A recent study shows that 43% of all students in U.S. four-year colleges binge drink. In a news headline: Nearly half of all college students binge drink, study finds.

What does 95% confident mean Suppose we took many samples and built a confidence interval from each using the formula point estimate ± 1.96 × SE. About 95% of those intervals would contain the true population proportion p.

Changing the confidence level The general formula for a confidence interval is point estimate ± z* × SE Confidence level 80% 90% 95% 98% 99% z* 1.28 1.65 1.96 2.33 2.58

Using TI 83/84 Calculator STAT -> TESTS -> 1-PropZInt x = number of “successes” n = sample size C-Level = confidence level as a decimal Example: In a survey of 1000 students attending four-year colleges, 43.23% reported they had participated in binge drinking. Based on this survey, we are 95% confident that between 40.13% and 46.27% of all college students binge drink.

Assumptions for confidence interval estimating a population proportion We have a random sample from the population The sample is large enough so that we see at least 15 observations of both possible outcomes

Confidence Interval For a single mean

Notation for Numerical Summaries Population Mean = μ Standard Deviation = σ Sample Mean = Standard Deviation = s Sample size = n

Example: Mercury Content in Dolphins Dolphins are at the top of the oceanic food chain, which causes dangerous substances such as mercury to concentrate in their organs and muscles. This is an important problem for both dolphins and other animals, like humans, who occasionally eat them. For instance, this is particularly relevant in Japan where school meals have included dolphin at times. How can we estimate the mean mercury level in dolphins?

Obtaining a point estimate Select a simple random sample of dolphins. The sample size is this example is n = 19. Some sample statistics are provided below, including the mean and standard deviation. (Measurements are in micrograms of mercury per wet gram of muscle).

Confidence Intervals for Means Still has form: point estimate ± margin of error. The construction is based on the t-distribution rather than a normal distribution when calculating the margin of error. This is because both the population mean and the population standard deviation must be estimated using the sample mean and sample standard deviation.

t-Distribution Figure 8.7 The t Distribution Relative to the Standard Normal Distribution. The t distribution gets closer to the standard normal as the degrees of freedom increases. The two are practically identical when df > 30.

95% confidence interval A random sample of 19 dolphins were tested for mercury. The mean was 4.4 μg/wet g with a standard deviation of 2.3 μg/wet g.

Conclusion In a report: We are 95% confident that the mean mercury level in dolphins is between 3.29 and 5.51 micrograms of mercury per wet gram of muscle.

Conclusion In a report: We are 95% confident that the mean mercury level in dolphins is between 3.29 and 5.51 micrograms of mercury per wet gram of muscle. In a news article: A recent study shows that the average mercury level in dolphins is 4.4 micrograms of mercury per wet gram of muscle. In a news headline: Possibly unsafe levels of mercury found in dolphins, study finds.

Using TI 83/84 Calculator STAT -> TESTS -> Tinterval = sample mean Sx = sample standard deviation n = sample size C-level = confidence level as a decimal Example: In a random sample of 19 dolphins the mean mercury level was 4.4 with a standard deviation of 2.3. Based on this sample, we are 95% confident that the mean mercury level in dolphins is between 3.29 and 5.51 micrograms of mercury per wet gram of muscle.

Assumptions for confidence interval estimating a population mean We have a random sample from the population Either sample size is large (30 or larger) or the population is normally distributed