Section 3.2: Understanding and Interpreting Confidence Intervals

Section 3.2: Understanding and Interpreting Confidence Intervals Objective: estimating population parameters with sample statistics ABC News/Washington Post Poll. Jan. 12-15, 2015. N=1,003 adults nationwide. Margin of error ± 3.5. "Obama has proposed providing free tuition to attend community colleges at a cost to the federal government of sixty billion dollars over 10 years. Do you support or oppose this proposal?" 1/12-15/15 Support % 53 Oppose % 44 Unsure % 3 60 Minutes/Vanity Fair Poll. March 30-April 3, 2016. N=1,010 adults nationwide. Margin of error ± 4. "As far as you know, what do scientific studies conclude about global warming? Most scientific studies conclude that global warming is real. Most scientific studies conclude that global warming is not real. OR, Most scientific studies are divided, and there is no real consensus either way." ALL Republicans Democrats Independents Real % 61 39 78 60 Not real % 10 18 6 10 No real consensus % 26 42 13 28 Unsure/ No answer % 2 1 3 3 Confidence Interval Estimate Based on a Margin of Error 1) A confidence interval is 2) A common form of an interval estimate 3) The margin of error 1 Constructing confidence intervals using the margin of error 4) Example: Consider the results of a survey from Pew Research center. Based on a sample of 1500 U.S. adults, we can say that 6.9% of U.S. adults have a multicultural background. a) What are the cases? b) What is the variable? c) Is it quantitative or categorical? d) What is the relevant sample statistic? Give appropriate notation and the value of the statistic. e) What population parameter are we estimating with this sample statistic? f) The survey goes on to say that “The margin of sampling error is +/- 3 percentage points with a 95% level of confidence.” Use the margin of error to give a 95% confidence interval for the estimate. g) Interpret the interval in context h) Is 0.04 a plausible value of the population proportion? Is 0.12 a plausible value? Explain. 2 Constructing confidence intervals using the margin of error 5) Example: a random sample of 755 US cell phone users age 18 and older in May 2011 found that the average number of text messages sent or received per day is 41.5 messages with a margin of error of 12.2 with 95% confidence. Identify: a. The cases b. The variable c. Classify as quantitative or categorical d. The sample size with correct notation e. The sample statistic with correct notation f. The parameter that we are estimating with correct notation g. The 95% confidence interval estimate h. Interpret the interval in context i. Based on this interval, is the value given a plausible value of mu? Explain. i. Mu = 32 ii. Mu = 58 3 Constructing confidence intervals using the standard error 6) Remember the 95% rule? (Empirical rule) • For normal (bell-shaped) sampling distributions we expect 95% of sample statistics to be within _______ standard errors from the population parameter. • We can then assume that the sample statistic will be within two standard errors from the parameter 95% of the time. • We then expect that 95% of the intervals constructed around each sample statistic will contain the parameter. • We can then use the standard error to compute a 95% confidence interval – use the formula: • The success rate of a confidence interval is known as the ___________________________ 7) Read example 3.14 – page 181, section 3.2 and then try the following example. Example: In a random sample of 1480 adults in the US, 1258 of them consider a car a necessity. If the standard error is 1.5%, give and interpret the 95% confidence interval estimate for the proportion of all US adults who consider a car a necessity. 8) Is the Economy a Top Priority? A survey of 1,502 Americans in January 2012 found that 86% consider the economy a “top priority” for the president and congress. The standard error for this statistic is 0.01. Find and interpret a 95% confidence interval for the true proportion of all Americans that considered the economy a “top priority” at that time? 4 Standard deviation, standard error, margin of error 9) Explain the difference between: a. Standard deviation of a population b. Standard deviation of a sample c. The standard error d. The margin of error 10) Interpreting a Confidence Interval Using a sample of 24 deliveries described in “Diary of a Pizza Girl” on the Slice website, we find a 95% confidence interval for the mean tip given for a pizza delivery to be $2.18 to $3.90. Which of the following is a correct interpretation of this interval? Indicate all that are correct interpretations. a). I am 95% sure that all pizza delivery tips will be between $2.18 and $3.90. b). 95% of all pizza delivery tips will be between $2.18 and $3.90. c). I am 95% sure that the mean pizza delivery tip for this sample will be between $2.18 and $3.90. d). I am 95% sure that the mean tip for all pizza deliveries in this area will be between $2.18 and $3.90. e). I am 95% sure that the confidence interval for the mean pizza delivery tip will be between $2.18 and $3.90. 11) Biomass in Tropical Forests Using a sample of 4079 inventory plots, scientists give a 95% confidence interval of 9,600 to 13,600 tons for the mean amount of carbon per square kilometer in tropical forests. Clearly interpret the meaning of this confidence interval. 5 To do at home: Read section 3.2, examples 3.12 and 3.13 starting on page 179; then, answer the questions. 12) Read example 3.12 – Is a television set a necessity? a. What are the cases? b. What is the variable? Is it categorical or quantitative? c. What parameter are we estimating? d. What is the sample size? e. What is the observed sample proportion? Use proper notation. f. What is the margin of error used in the example? g. Use the margin of error to give a 95% confidence interval for the estimate. h. Based on the interval estimate from part (g); is 0.35 a plausible value of the population proportion? Is 0.44 a plausible value? 13) Adopting a Child in the US A survey of 1,000 American adults conducted in January 2013 stated that “44% say it’s too hard to adopt a child in the US.” The survey goes on to say that “The margin of sampling error is +/- 3 percentage points with a 95% level of confidence.” a). Give the notation and value of the relevant sample statistic. b). What population parameter are we estimating with this sample statistic? c). Use the margin of error to give a 95% confidence interval for the estimate. d). Is 0.42 a plausible value of the population proportion? Is 0.50 a plausible value? 6