Sec. 6.2 – Confidence Intervals for the Mean (Small Samples) If the distribution of a random variable x is approximately normal, then 𝑡= follows a _____________________. Characteristics of the t-distribution 1. The t-distribution is bell-shaped and symmetric about the mean. 2. The t-distribution is a family of curves, each determined by a parameter called the ____________ of ____________. 𝑑. 𝑓. = 3. The total are under the curve is 1 or 100%. 4. The mean, median and mode of the t-distribution are equal to zero. 5. As the degrees of freedom increase, the t-distribution approaches the ____________ _______________. After _______ d.f., the t-=distribution is very close to the standard normal distribution. The flowchart should describe when to use the normal distribution to construct a confidence interval for the population mean and when to use a t-distribution. Is 𝑛 ≥ 30? YES NO Is the population normally, or approximately normally, distributed? NO YES Is 𝜎 known? NO YES Should you use the normal distribution, t-distribution, or neither for the following? You randomly select 25 newly constructed houses. The sample mean construction cost is $181,000 and the population standard deviation is $28,000. Assume construction costs are normally distributed. In a random sample of 70 bolts, the mean length was 1.25 inches and the standard deviation was 0.05 inch. In a random sample of 19 patients at a hospital’s minor emergency department, the mean waiting time (in minutes) before seeing a medical professional was 23 minutes and the standard deviation was 11 minutes. Assume the waiting times are not normally distributed. You took a random sample of 12 two-slide toasters and found the mean price was $57.79 and the standard deviation was $19.05. Assume the prices are normally distributed. Find the critical value 𝑡𝑐 for a 95% confidence when the sample size is 15. You randomly select 16 coffee shops and measure the temperature of the coffee sold at each. The sample mean temperature is 162.0℉ with a sample standard deviation of 10.0℉. Find the 95% confidence interval for the mean temperature. Assume the temperatures are approximately normally distributed. You randomly select 20 mortgage institutions and determine the current mortgage interest rate at each. The sample mean rate is 6.22% with a sample standard deviation 0.42%. Find the 99% confidence interval for the population mean mortgage interest rate. Assume the interest rates are approximately normally distributed.