# RES/341 Week 5 Answer Guide - Statistics

*1. In an article in the Journal of Management, Morris, Avila and Allen studied innovation by surveying firms to find the number of new products introduced by the firms. A random sample of 100 California based firms are selected. Each firm is asked to report the number of new products introduced last year. The survey found that on average these firms introduced 5.68 products with a standard deviation of 8.70. Compute a 98% confidence interval for the new products introduced last year.*

Mean = 5.68, Deviation = 8.70, Z = 2.3263

8.70 / √100

[5.68 - 2.3263(8.70 / √100), 5.68 + 2.3263(8.70 / √100)]

5.68 - 2.3263(8.70 / √100) = 3.656119

5.68 + 2.3263(8.70 / √100) = 7.703881

**Lower Limit of 98% confidence interval = 3.656
Upper Limit of 98% confidence interval = 7.703**

*2. The score on the entrance test for a well known law school has a mean score of 200 points and a standard deviation of 50 points. At value should the lowest passing score be set if the school wishes only 2.5% of those taking the entrance test to pass?*** **

Mean = 200, Deviation = 50, Z = 1.960

200 + (1.960)(50) = 298

**The lowest passing score should be 298.00 points.**

*3. A tire manufacturer wishes to investigate the thread life of its tires. A sample of 10 tires driven 50,000 miles revealed a sample mean of 0.32 inches of thread remaining with a standard deviation of 0.09 inches. Construct a 95% confidence interval for the population mean. *

Mean = 0.32, Deviation = 0.09

-t(.025, df=9) = -2.262 and, t(.025, df=9) = 2.262

[0.32 – (2.262)(0.09 / √10), 0.32 + (2.262)(0.09 / √10)]

0.32 – (2.262)(0.09 / √10) = 0.25562

0.32 + (2.262)(0.09 / √10) = 0.38438

**Lower Limit of 95% Confidence Interval = 0.256
Upper Limit of 95% Confidence Interval = 0.384**

*Would it be reasonable for the manufacturer to conclude that after 50,000 miles the population mean amount of thread remaining is 0.30 inches?*

**Yes, this would be a reasonable conclusion because 0.30 is inside the upper and lower limits of the 95% confidence interval. This is also close to the sample mean of 0.32. **

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