Question

Section 6.2 ܐܝܝ A company that makes adjustable dog collars wants to ensure that they will fit dogs of all sizes. Assume that the collar length for dogs is normally distributed with a mean of 18 inches and a standard deviation of 3 inches. a) What is the maximum length the collars should adjust to that will allow 97.5% of dogs to use them? Round your answer to the nearest inch. Section 6.2 continued... b) What is the minimum length the collars should adjust to so that only 1% of dogs are too small to use them? Round your answer to the nearest inch.

Answer 1

$(a) \ 97.5th \ percentile \ calculation\\ \\ z \ value \ for \ 97.5th \ percentile\\ z_{97.5th} \ percentile \ =1.96 , \ using \ z \ percentile \ table \\ \\mean \ \mu =18 \ and \ standard \ deviation \ \sigma=3\\ \\ by \ z-score \ formula \\ z= \frac{x-\mu}{\sigma}\rightarrow x=\mu+(z*\sigma)\\ \\ \rightarrow \ x=18+(1.96*3)\\ \\\rightarrow 18+5.88\\ \\\rightarrow x=23.88\\ \\\rightarrow x= 24 (nearest \ inch)\\ \\(b) 1st \ percentile \ calculation \\z_{1st} \ percentile \ =-2.33, \ using \ z \ percentile \ table \\ \\ mean \ \mu =18 \ and \ standard \ deviation \ \sigma =3 \\ \\by \ z \ formula \\ \\z= \frac{x-\mu}{\sigma} \rightarrow x=\mu+(z*\sigma)\\ \\\rightarrow x=18+(-2.33*3)\\ \\\rightarrow 18-6.99\\\rightarrow x=11.01\\\rightarrow x = 11(nearest \ inch )$