314 CHAPTER 6 Confidence Intervals The flowchart describes when to use the standard normal distribution and when to use the t@distribution to construct a confidence interval for a population mean. Is known? Use the t-distribution with and n− 1 degrees of freedom. E = tc n s Use the standard normal distribution with E = zc s s n Yes Is the population normally distributed or is n≥ 30? You cannot use the standard normal distribution or the t-distribution. Yes No Section 6.1 Section 6.2 . No Notice in the flowchart that when both n 6 30 and the population is not normally distributed, you cannot use the standard normal distribution or the t@distribution. Choosing the Standard Normal Distribution or the t@Distribution You randomly select 25 newly constructed houses. The sample mean construction cost is $299,000 and the population standard deviation is $46,000. Assuming construction costs are normally distributed, should you use the standard normal distribution, the t@distribution, or neither to construct a 95% confidence interval for the population mean construction cost? Explain your reasoning. SOLUTION Is the population normally distributed or is n Ú 30? Yes, the population is normally distributed. Note that even though n = 25 6 30 you can still use either the standard normal distribution or the t-distribution because the population is normally distributed. Is s known? Yes. Decision: Use the standard normal distribution. TRY IT YOURSELF 4 You randomly select 18 adult male athletes and measure the resting heart rate of each. The sample mean heart rate is 64 beats per minute, with a sample standard deviation of 2.5 beats per minute. Assuming the heart rates are normally distributed, should you use the standard normal distribution, the t@distribution, or neither to construct a 90% confidence interval for the population mean heart rate? Explain your reasoning. Answer: Page A40 EXAMPLE 4 Picturing the World Two footballs, one filled with air and the other filled with helium, were kicked on a windless day at Ohio State University. The footballs were alternated with each kick. After 10 practice kicks, each football was kicked 29 more times. The distances (in yards) are listed. (Source: The Columbus Dispatch) Air Filled 1 9 2 0 0 2 2 2 2 5 5 5 5 6 6 2 77788888999 3 1 1 1 2 3 3 4 Key: 10 9 = 19 Helium Filled 1 1 2 1 4 1 2 2 2 3 4 6 6 6 2 7 8 8 8 9 9 9 9 3 0 0 0 0 1 1 2 2 3 3 4 5 3 9 Key: 10 1 = 11 Assume that the distances are normally distributed for each football. Apply the flowchart at the right to each sample. Construct a 95% confidence interval for the population mean distance each football traveled. Do the confidence intervals overlap? What does this result tell you?
RkJQdWJsaXNoZXIy NjM5ODQ=