Students Tutorial Answers Week7

  BES Tutorial Sample Solutions, S2 2010  It will be posted on BES website with one week delay WEEK 7 TUTORIAL EXERCISES (To be discussed in the week starting August 30) 1   From several years’ records, a fish market manager has determined that the weight of deep sea bream sold in the market (  X  ) is approximately normally distributed with a mean of 420 grams and a standard deviation of 80 grams Assuming this distribution will remain unchanged in the future, calculate the expected proportions of deep sea bream sold over the next year weighing (a)    between 300 and 400 grams 300    400  30042080    40042080     15    025    0    150    025    0433200987  03345 (b)    between 300 and 500 grams 300    500  30042080    50042080     15    1    0    150    1    0433203413  07745 (c)   more than 600 grams    600    600 42080       225    050    225    0504878  00122   2   In a certain large city, household annual incomes are considered approximately normally distributed with a mean of 40,000 and a standard deviation of 6,000 What proportion of households in the city have an annual income over 30,000? If a random sample of 60 households were selected, how many of these households would we expect to have annual incomes between 35,000 and 45,000?        ~40000,6000  )    30000    30000 400006000       167    05 0    167    05 04525  09525 So   9525%   of    households   in   the   city    have   annual    incomes    greater    than   30,000   35000    45000  35000400006000    45000 400006000     083    083    20    083    202967  05934 Therefore   we   expect    05934(60) ≈  36   households   in   the   sample   to   have   annual    incomes   between   35,000   and    45,000    3   In a certain city it is estimated that 40% of households have access to the internet A company wishing to sell services to internet users randomly chooses 150 households in the city and sends them advertising material For the households contacted: (a)   Calculate the probability that less than 60 households have internet access? Let     X    be   the   number    of    households   contacted    that    have   internet    access   Then   assume    X    is   a   binomial    random   variable   with   n=150   and     p=04   Because   n   is   large   we   can   use   the   normal    approximation   to   the   binomial    where:       15004  60       1  1500406  36   Thus   incorporating   the   continuity    correction   we   need    to    find:        60       59       595      595606       0083    050    0083    0500319  04681 (b)   Calculate the probability that between 50 and 100 (inclusive) households have internet access? 50      100  495      1005    495606   1005606     175    675    0    1750    675    0459905  09599 (c)   Calculate the probability that more than 50 households have internet access?      51       505      505 606       1583    05 04429  09429