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1 Intro to Confidence Intervals: A estimate is a single statistic based on sample data to estimate a population parameter Simplest approach But not always very precise due to variation in the sampling distribution Imagine taking many SRSs of 50 of the population of KSU freshman and asking the number of hours the student spent studying in the last 24 hours. The first sample has a mean of 2.3. The second has a mean of 2.5. The third has a mean of 3.1, etc. If collect all these means and show their distribution, we get a -shaped distribution with mean equal to the unknown μ and standard deviation. POPULATION: SRS n = μ =??? σ = 0.5 The figure below demonstrates the true meaning of a confidence interval. To say that is a confidence interval for the population mean μ is to say that, in repeated samples, % of these intervals capture the true population mean. The language of statistical inference uses this fact about what would happen in many samples to epress our confidence in the results of any one sample. SRS n = CONCLUSION: Let s say our sample gave us We see the resulting interval is, which can be written as. We say that we are 95% that the unknown mean number of hours studying in last 24 hours for all KSU freshmen is between and.

2 Make sure that you understand the basis for our confidence. There are only two possibilities: 1. The interval contains. 2. Our SRS was one of the few samples for which is not within points of the true μ. (Only % of all samples give such inaccurate results.) We cannot know whether our sample is one of the 95% for which the interval catches μ or whether it is one of the unfortunate 5%. The statement that we are 95% confident that the unknown μ lies in the interval is another way of saying, we got these numbers by a that gives correct results 95% of the time. NOTE: The interval of numbers is called the 95% for μ. The confidence is 95%. What does it mean to be 95% confident? Select the correct statement. 95% chance that μ is contained in the confidence interval The probability that the interval contains μ is 95% The method used to construct the interval will produce intervals that contain μ 95% of the time. STATEMENTS to memorize: Interpreting a confidence interval: We are % confident that the true mean (proportion) contet lies within the interval and. EXAMPLE- We are 95% confident that the true mean potassium level in the blood lies within the interval 2.97 and Interpreting a confidence level: The method used to construct the interval will produce intervals that contain the true mean (proportion) % of the time.

3 CONDITIONS (ASSUMPTIONS): One-sample z Confidence Interval for μ FORMULA: Standard error is: The estimated standard of the statistic When n is large (n 30), substitute for σ because the distribution is approimately. Confidence Level 80% 90% 95% 99% z critical value Margin of error gets smaller when z* gets smaller ( confidence level) σ gets smaller (less in the population) n gets larger (to cut the margin of error in half, n must be times as big) EXAMPLE 1: A certain filling machine has a true population standard deviation σ = ounces when used to fill ketchup or bottles. A random sample of 36 6 ounce bottles of ketchup was selected from the output from this machine and the sample mean was ounces. Find and interpret a 90% confidence interval estimate for the true mean ounces of ketchup from this machine.

4 EXAMPLE 2: A random sample of 50 MHS students was taken and their mean SAT score was 1250 with a standard deviation of 105. Find and interpret a 95% confidence interval for the mean SAT score of MHS students. EXAMPLE 3: The heights of MHS male students is normally distributed with σ = 2.5 inches. How large a random sample is necessary to be accurate within 0.75 inches with 95% confidence? One-sample t Confidence Interval for μ Just as normal distributions are distinguished from one another by their mean m and standard deviation s, t distributions are distinguished by a positive whole number called the number of degrees of freedom (df). There is a t distribution with 1 df, another with 2 df, etc. Properties of t-distributions: The t curve corresponding to any fied number of degrees of freedom is bell shaped and is centered at 0 (just like the standard normal curve.) The t curve is more out than the z curve. As the number of degrees of freedom increases, the spread of the corresponding t curve. As the number of degrees of freedom increases, the more closely the t curve resembles the curve.

5 The major difference between the confidence intervals when σ is known and not known is: when σ is known, we use a z critical value when σ is not known, we use a t critical value. EXAMPLE 1: Find the t critical value for: 80% confidence, n = 15 95% confidence, n = 24 95% confidence, n = 200 ASSUMPTIONS: FORMULA: EXAMPLE 2: Ten randomly selected Woodrow Wilson elementary students were each asked to list how many hours of television they watched per week. The results are Find and interpret a 90% confidence interval estimate for the true mean number of hours of television watched per week by Woodrow Wilson elementary students.

6 ASSUMPTIONS: One Sample z-interval for Proportions FORMULA: Since ρ is unknown and n is large, we estimate ρ with. Identify the standard error of the statistic: Identify the Margin of Error: EXAMPLE 1: For a project, a student randomly sampled 182 other students at a large university to determine if the majority of students were in favor of a proposal to build a field house. He found that 75 were in favor of the proposal. Let = the true proportion of students that favor the proposal. Find and interpret the 95% confidence interval.

7 EXAMPLE 2: The Gallup Youth Survey asked a random sample of 439 U.S. teens aged 13 to 17 whether they thought young people should wait to have se until marriage. Of the sample, 246 said Yes. Construct and interpret a 90% confidence interval for the proportion of all teens who would say Yes if asked this question. SAMPLE SIZE: Note: is a guessed value for the sample proportion. If no previous estimate is given, use the conservative estimate of. IMPORTANT: Always round the result to the nearest integer. EXAMPLE 3: If a TV eecutive would like to find a 95% confidence interval estimate within 0.03 for the proportion of all households that watch NYPD Blue regularly. How large a sample is needed if a prior estimate for p was 0.15? Suppose a TV eecutive would like to find a 95% confidence interval estimate within 0.03 for the proportion of all households that watch NYPD Blue regularly. How large a sample is needed if we have no reasonable prior estimate for p?