Case Problem $3$ Metropolitan Research, Inc.

Metropolitan Research, Inc., a consumer research organization, conducts surveys designed to evaluate a wide variety of products and services available to consumers. In one particular study, Metropolitan looked at consumer satisfaction with the performance of automobiles produced by a major Detroit manufacturer. A questionnaire sent to owners of one of the manufacturer's full$-$sized cars revealed several complaints about early transmission problems. To learn more about the transmission failures, Metropolitan used a sample of actual transmission repairs provided by a transmission repair firm in the Detroit area. The following data show the actual number of miles driven for $50$ vehicles at the time of transmission failure.

$\backslash$table$[[$DATA $\sqrt[\phantom{2}]{f}$ile $,85,092,32,609,59,465,77,437,32,534,64,090,32,464,59,902],[$Auto$,39,323,89,641,94,219,116,803,92,857,63,436,65,605,85,861],[,64,342,61,978,67,998,59,817,101,769,95,774,121,352,69,568],[,74,276,66,998,40,001,72,069,25,066,77,098,69,922,35,662],[,74,425,67,202,118,444,53,500,79,294,64,544,86,813,116,269],[,37,831,89,341,73,341,85,288,138,114,53,402,85,586,82,256],[,77,539,88,798,,,,,,]]$

Managerial Report

Use appropriate descriptive statistics to summarize the transmission failure data.

Develop a $95\%$ confidence interval for the mean number of miles driven until transmission failure for the population of automobiles with transmission failure. Provide a managerial interpretation of the interval estimate.

Discuss the implication of your statistical findings in terms of the belief that some owners of the automobiles experienced early transmission failures.

How many repair records should be sampled if the research firm wants the population mean number of miles driven until transmission failure to be estimated with a margin of error of $5000$ miles? Use $95\%$ confidence.

What other information would you like to gather to evaluate the transmission failure problem more fully?