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#### Business statistics        I       Practice Test 5                      (Chapter        7      &     8)        1

Practice Test 5                      (Chapter        7      &     8)

 1.                                                                                                        From a          population  containing   5         items (A,      B,        C,        D,            and    E),       all       possible        samples        of        size    2         (n=2)            are      to        be       drawn.                      How   many samples        are      possible?      List     all       the            samples.                   Using simple           random        sampling,     what  is         the            probability  of        drawing       each   sample          of        size            2?                   What is         the      probability  of        drawing            each   item   in        the      sample;        that    is,        what  is            P(A),  P(B),....,         P(E)? Suppose       item   A         corresponds           to        random            number        1,        B         corresponds           to        2,        C            corresponds           to        3         and    so        on.      From the            random        number        table, you    selected        the      digits            28652.         What is         a          simple           random            sample          of        size    2         that    can     be       formed            using these five     digits?

2.

Assume       a          finite  population  contains       95       items.            Use     a          random        number        table  to        draw a          random        sample          of        size    15.

 3.                                                                                                        The    following     sample          data   indicate        the      number        of            days   absent           for      6         employees  in        a          company:                            5,        8,        10,      7,        10,      14       Find   the      point estimate       of        the      population  mean.                        Find   the      point estimate       of        the      population            standard      deviation.

 4.                                                                                    Suppose       that    the      mean of        a          population  is            200    with   a          standard      deviation     of        50.      A         simple            random        sample          of        size    100    (n=100)       is            selected        from  this     population. The    sample          mean x            will     be       used  to        estimate       the      population  mean μ.         Find   the      expected      value of        x or     E         ( x        ).         Find   the      standard      deviation     of        x          or        the            standard      error of        the      mean.            What distribution            would           the      sample          mean x            follow?         Why? What does   the      sampling      distribution            of        x            show?

5.

Describe     the      central          limit   theorem.      What is         the      importance of          this     theorem       in        sampling      distribution?

6.

The    mean price  of        a          particular    brand            of        a          digital          camera         is         \$200, with   a          standard      deviation     of        \$50.          Suppose       these values           are      the      population  mean and    standard          deviation     for      this     brand            of        camera;        that    is,        μ          =          200    and    s =

50.    From this     population, a          random        sample          of        size    100          (n=100)       is         selected        to        estimate       the      population  mean          price, μ.

1. Find   the      probability  that    the      sample          mean will     be       within          ±         5         of        the      population  mean,            or        the      probability  that          the      sample          mean x          is         between       \$195 and    \$205?
2. Find   the      probability  that    the      sample          mean will     be       within          ±         10       of        the      population  mean,            or        the      probability  that          the      sample          mean x          is         between       \$190 and    \$210?

7.

Suppose     you    draw a          sample          of        size    50       from  a          population  with   a          standard      deviation     of        10       (σ        =          10).    Find   the      standard      error of        the      mean if         (a)      the          population  is         infinite          (b)      the      population  size    is          N=50,000,  (c)       N=      5000,            (d)      N=500.         Comment    on          your  results.

8.

The   mean annual          starting         salary            for      the      accounting  major            was          \$30,   393    with   a          standard      deviation     of        \$2000.          Suppose       this          value holds for      the      population  of        graduates    with   accounting  majors;          that    is,        ?          =         \$30,393       and    the      standard      deviation,    σ         =          \$2,000.         Find   the      probability  that    the      sample          mean salary            will          be       within           ±250 of        the      population  mean if         you    select a          random        sample          of        30,      50,      100,   200,   and    400?  What happens          to        the      probability  values           as        the      sample          size    was          increased?   Would          you    prefer           a          large  sample          to        estimate          the      population  mean?           Why?

9.

Suppose     the      population  proportion  (p)      of        a          population  is         0.4          (p=0.40).     A         simple           random        sample          of        100    is          selected        from  this     population. (a) What      is         the      expected      value          of        p ?

1. What         is         the      standard      deviation     of        p ?
2. What         is         the      sampling      distribution            of        p ,        or        what        distribution            would           the      sample                                  proportion  p        follow?

1. Draw         a          sketch           and    show the      sampling      distribution            of        sample          proportion, p .

10.

Suppose     that    a          prime?time show of        Fox     TV      network       is         watched          by       40%  of        the      audience      in        a          city.    You    select a          random          sample          of        200    viewers        so        that    the      sample          proportion p          can     be       used  to        estimate       the      population  proportion, p.

1. Find           the      probability  the      sample          proportion  will     be       within        3%     or        ±0.03            of        the      population  proportion?
2. Find           the      probability  the      sample          proportion  will     be       within        5%     or        ±0.05            of        the      population  proportion?

 11.                                                                                                      A         sample          of        size    50       was    drawn           from  a            population  with   a          known          ?.        The    sample          mean            was    found            to        be       32.      Suppose       that    the            population  standard      deviation,    ?=6.                           Find   a          90%  confidence  interval        for      the      population            mean.            Find   a          95%  confidence  interval        for      the      population            mean.            Find   a          99%  confidence  interval        for      the      population            mean.            Compare     the      three intervals.     What is         the      effect of            increasing   the      confidence  level   on       the      confidence            interval?

 12.                                                                                                      The    following     are      the      scores           on       IQ        tests   of        a            sample          of        20       students       at        a          university:  120                            118                            119                            132                                        135                122                                        118                                        139                                        140                128                            125                            115                            135                                        139                                        130                                        112                                        129                                        140                                        115                                        120    What      is         the      point estimate       of        the      population  mean IQ     score?           What      is         the      point estimate       of        the      population     standard      deviation?   Find       a          95%  confidence  interval        for      the      population     mean IQ        score?

 13.                                                                                                      A         simple           random        sample          of        20       machining   jobs            indicated     the      average        processing  time   of        17.25            minutes        with   a          sample          standard      deviation     of            3.3      minutes.       Find   a          90%  confidence  interval        for      the      population            mean processing  time.  Find   a          95%  confidence  interval        for      the      population            mean processing  time.               Find   a          99%  confidence  interval        for      the      population            mean processing  time.

 14.                                                                                                      An      analyst          wants            to        determine   the      mean salary            of            recent           graduates    in        Mechanical Engineering.           The            margin          of        error in        estimating   the      mean is         to        be            within           \$500 with   a          95%  level   of        confidence. The            analyst          found            a          report           by       the      Department            of        Labor that    estimated    the      standard      deviation     to        be            \$2000.                      What         is         the      required      sample          size?  If     the      error in        estimating   the      mean is         to        be        within           \$200 and    all       other data   being the      same,        then   what  is         the      sample          size?  If     the      error in        estimating   the      mean is         to        be        within           \$100 and    all       other data   being the      same,        then   what  is         the      sample          size?  Compare  the      results          of        (a),     (b),     and    (c).      Why  is        the      sample          size    much bigger            in        part    (c)?

 15.                                                                                                      Out     of        a          random        sample          of        814    individuals working            in        a          city,    562    indicated     that    they   were  satisfied            with   their  working       condition.                What is         the      point estimate       of        the      proportion  of            working       people          who   felt      they   are      satisfied            with   their  working       condition?   What is         the      margin          of        error at        a          90%            confidence?            Compute      a          90%  confidence  interval        for      the            proportion  of        working       population  who   were  satisfied            with   their  working       condition.

 16.                                                                                                      A         recent           survey          about the      state   of        economy      by       a            news media            indicated     that    approximately       35%  of            the      population  believed       that    the      economy      is            improving.  It         was    felt      that    the      sample          size    used  for            this     study was    too      small and    the      margin          of        error            was    too      large.             Determine  the      sample          size    that    will     be       required,     if           the      margin          of        error is         3.0% with   a          0.95           reliability,    or        with   a          confidence  interval        of           95%. Determine  the      sample          size    that    will     be       required,     if           the      margin          of        error is         2.0% with   a          0.95           reliability,    or        with   a          confidence  interval        of           95%.

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