Question

pleaee help me out in this question (a.,b,c,d)

STA 2023-Worksheet 2 Be sure to show all your work for full credit. NO WORK, NO CREDIT. Due by Saturday, March 16th. Round all final answers to two decimal places unless otherwise indicated. Don't round until your final answer! 1. The distribution of heights of 6-year-old girls is approximately normally distributed with a mean of 46.0 inches and a standard deviation of 2.7 inches. Carina is 6 years old, and her height is 0.96 standard deviations above the mean. Her friend Aloni is also 6 years old and is at the 93rd percentile of the height distribution At what percentile is Carina's height? Draw and label the Normal model that corresponds to this situation. (3 pts) a. b. How does her height compare to Aloni's? (1 pt) c. What percent of 6-year-old giris wouid you expect to be taller than 49.4 inches? Draw and label the Normal model that corresponds to this situation. (3 pts)

Answer 1

$1.~a.~Let~X=heights~of~6-year~old~girls\sim Normal(46,~2.7^2).\\ Percentile~of~Carina's~height=P(X\leq 46+0.96*2.7)*100\\ =\Phi(0.96)*100=83.15$

83.15% 48.592 40 45 50

$b.~Height~of~Carina=46+0.96*2.7=48.592\approx 48.6~inches\\ Let,~Aloni's ~height=c~inches~then\\ P(X\leq c)=\Phi\left(\frac{c-46}{2.7} \right )=0.93\\=\Phi( 1.48)~or,~c=46+1.48*2.7=49.996 \approx~50~inches.\\ Aloni~is~taller~than~Carina.\\ c.~Required~percentage=P(X>49.4)*100\\ =(1-\Phi((49.4-46)/2.7))*100=10.40\%\\$

5 父 10.4% 49.4T 40 45 50

$d.~ P(X\leq c)=\Phi\left(\frac{c-46}{2.7} \right )=0.30=\Phi(-0.52)\\ or,~c=46-0.52*2.7=44.60~inches$

5 8 3 30% 44.6 40 45 50