3. The Director of Student Loans wants to know the mean amount owed on student loans at the time of his/her graduation. The director selects a sample of 20 graduating students and contacts each to find the information. 298
Characteristics of the t -distribution 1. It is, like the z distribution, a continuous distribution . 2. It is, like the z distribution, bell-shaped and symmetrical . 3. There is not one t distribution , but rather a family of t distributions . All t distributions have a mean of 0, but their standard deviations differ according to the sample size, n. 4. The t distribution is more spread out and flatter at the center than the standard normal distribution As the sample size increases, however, the t distribution approaches the standard normal distribution 299 Comparing the z and t Distributions when n is small, 95% Confidence Level 300
Confidence Interval Estimates for the Mean Use Z -distribution If the population standard deviation is known OR the sample is greater than or equal to 30 . Use t -distribution If the population standard deviation is unknown AND the sample is less than 30 . 300 n z X V r n s t X r When to Use the z or t Distribution for Confidence Interval Computation 301 OR n >= 30
Confidence Interval for the Mean – Example using the t- distribution A tire manufacturer wishes to investigate the tread life of its tires. A sample of 10 tires driven 50,000 miles revealed a sample mean of 0.32 inch of tread remaining with a standard deviation of 0.09 inch. Construct a 95 percent confidence interval for the population mean. Would it be reasonable for the manufacturer to conclude that after 50,000 miles the population mean amount of tread remaining is 0.30 inches? n s t X s x n n 1 , 2 / unknown) is (since dist. - t the using C.I. the Compute 09 . 0 32 . 0 10 : problem in the Given 0 r ! ! ! D V 301 NOTE: Degrees of Freedom (df) = n - 1 Student’s t -distribution Table 302
The manager of the Inlet Square Mall, near Ft. Myers, Florida, wants to estimate the mean amount spent per shopping visit by customers. A sample of 20 customers reveals the following amounts spent. Confidence Interval Estimates for the Mean – Calculated Mean and Standard Deviation 302 Confidence Interval Estimates for the Mean – By Formula $60. be o unlikely t is mean population that the conclude we Hence, interval. confidence in the not is $60 of value The $50. be could mean population that the reasonable is It : Conclude $53.57 and $45.13 are interval confidence the of endpoints The 22 . 4 35 . 49 20 01 . 9 093 . 2 35 . 49 20 01 . 9 35 . 49 unknown) is (since dist. - t the using C.I. the Compute 19 , 025 . 1 20 , 2 / 05 . 1 , 2 / r ! r ! r ! r ! r 0 0 t n s t X n s t X n D V 303
Confidence Interval Estimates for the Mean – Using Excel 304 A Confidence Interval for a Proportion ( ʌ ) The examples below illustrate the nominal scale of measurement. 1. The career services director at Southern Technical Institute reports that 80 percent of its graduates enter the job market in a position related to their field of study.