Question

Hey guys! Looking for some help with statistics homework. These online classes have made it a little hard to understand some of the specifc stuff like finding Z-score.

1) The average yearly Medicare Insurance benefit per person was $4064 in a recent year. The benefits are normally distributed with a standard deviation of $460 and 20 patients are randomly sampled:

a) Find the mean and standard deviation of the sampling distribution of the sample mean:

b) Find the probability that the mean benefit is less than $3800

c) Find the probability that the mean benefit is more than $4100

d) Find the probability that the mean benefit is between $3950 and $4150

Answer 1

1. 
   Answer: consider: xi yearly medicare insurance bepefit per person Xny Niel,8) eli mean = $ 4064 ri standard deviation = $ 460 Di sample size = 20 of To find mean and standard deviation Sampling distribution of sample mean X^ Nllll,0) : : * Nalor). - central limit theorem says z= X-4 ANCO,1) olañ is mean of sampling distribution of sample mean E $ 4064 Standard deviation of sampling distoia bution of sample mean is olva 460 =102.86 ($) A4
   
   bil. Here we want to compute. P(X < $ 3800) P ( X < 3800) pl X-ell < 3800 - Il time torr sin • central limit theorem says __20 X-l a NOU ... Poño Plz < 3800 – 4064 102.86 Pl < 3890,- 4054) = P( Z <= 2.5666) D o SP(Z > 2.5666) . due to symmetry = 0:0051 . from 2- table. ZNOU DES0.00577 -205666 Answer: probability that is less than $3800 is the mean benefit 0.0051
   
   Here we want to computer pc> $4100) Pl x > 4100) =P(x- & 4100-l. I slik sinn / > 4100 - 4064 102.86 .: central limit theorem says. z- -- X- MAN (0,1) siño Plz> 0.3499) 0.3632 from PIZ.0.35) z table 2>NOO, O 0.3632 0 0.3499 Answeré probability that the mean benefit is more than $4100 is 0.3632
   
   d here we want to compute, PC $ 3950 < x < $4150 ) =p3950-all cx-ul Toivo sivá < 4150-ell 1 orra no central limit theorem seys 2- x-all NCO, U Porn P ( 3950-4064 102.86 : 2 <4150-4064 _ 102.86 SPC-1:1083 2< 0.8361) = 1- P(Z <=101083) -P(Z > 0.8361) = 1-PC27151083) -PC2>0.8361). due to symmetry = 1- 0.1339 - 0.2015 = 006646 from z-table 2->NCOI). 0.6646 LE -1.1083 0.8361 Answer: probability that the meao benefit is beteween $3950 and $4150 is 0.6666