AP Statistics 2017

calculator, the second method above (referred to as the “computed by software” method in the chart on the
previous page) is acceptable. Just be sure to report the degrees of freedom as given by the calculator so
that the method you say you are using matches your computation.

example: An airline is  interested  in  determining the average number  of  unoccupied  seats   for all of

its flights.    It  selects an  SRS of  81  flights and determines  that    the average number  of  unoccupied

seats   for the sample  is  12.5    seats   with    a   sample  standard    deviation   of  3.9 seats.  Construct   a

95% confidence  interval    for the true    number  of  unoccupied  seats   for all flights.

solution: The   problem states  that    the sample  is  an  SRS.    The large   sample  size    justifies   the

construction    of  a   one-sample  confidence  interval    for the population  mean.   For a   95%

confidence  interval    with    df  =   81  –   1   =   80, we  have,   from    Table   B,  t   *   =   1.990.  We  have    

    .

Note:   If  the problem had stated  that    n = 80  instead of  81, we  would   have    had df  =   80  –   1   =   79.

There   is  no  entry   in  Table   B   for 79  degrees of  freedom.    In  this    case    we  would   have    had to

round   down    and use df  =   60, resulting   in  t   *   =   2.000   and an  interval    of  

.   The difference  isn’t   large,  but the interval    is  slightly

wider.  (For    the record, we  note    that    the value   of  t   *   for df  =   79  is  given   by  the TI-84   as

invT(0.975,79)=1.99045. )

You can use the STAT    TESTS   TInterval function  on  the TI-83/84    calculator  to  find    a

confidence  interval    for a   population  mean    (a  confidence  interval    for a   population  mean    is  often

called  a   “one-sample”    t interval).    It’s    required    to  identify    your    procedure,  either  by  name    or  by

formula,    as  well    as  reporting   the calculator  answer. And don’t   forget  to  show    that    you have

checked the conditions  needed  to  construct   the interval.

example: Interpret  the confidence  interval    from    the previous    example in  the context of  the

problem.

solution: We    are 95% confident   that    the mean    number  of  unoccupied  seats   for all the airline’s

flights is  between 11.6    and 13.4    seats.

For large   sample  confidence  intervals   utilizing   z   -procedures,    it  is  probably    worth   memorizing  the

critical    values  of  z for   the most    common  C levels    of  0.90,   0.95,   and 0.99.   They    are:

example: Brittany   thinks  she has a   bad penny   because,    after   150 flips,  she counted 88  heads.  Find

a   99% confidence  interval    for the true    proportion  of  heads   for all possible    tosses  of  this    coin.

Do  you think   the coin    is  bad?

solution: First we  need    to  check   to  see if  using   a   z interval  is  justified.