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Midterm Exam

Psychology 406

Fall 2007
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Name: \rule{5in}{.02in}

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Whenever possible, use fractions throughout.




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I. ELISA tests are used to screen donated blood for the presence of the AIDS (HIV) virus.  The test actually detects antibodies, substances that the body produces when the virus is present.  When antibodies are present, ELISA is positive with probability 0.997 and negative with probability 0.003.  When the blood tested is not contaminated with AIDS antibodies, ELISA gives a positive result with probability 0.015 and a negative result with probability 0.985.  Suppose that 1\% of a large population carries the AIDS antibody in their blood.

The following results may be useful in answering the questions below:

\[ P(A) = P(A|B)P(B) + P(A|\bar{B})P(\bar{B}) \]

\[ P(A|B) = \frac{P(B|A)P(A)}{P(B|A)P(A) + P(B|\bar{A})P(\bar{A})} \]

(1) What is the probability that the ELISA test for AIDS is positive for a randomly chosen person from this population?


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(2) What is the probability that a person has the antibody given that the ELISA test is positive (for a randomly chosen person from this population)?


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(3) What is the probability that a person doesn't have the antibody given that the ELISA test is positive (for a randomly chosen person from this population)?

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(4) What is the joint probability that a person doesn't have the antibody and the ELISA test is positive (for a randomly chosen person from this population)?


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II. Suppose I have a set of observations, $X_{1},\ldots,X_{N}$,  and
I turn them into $Z$-scores, $Z_{1},\ldots,Z_{N}$.  If I carry out a
further transformation to $T_{i} = 10 Z_{i} + 50$,  what is the
resulting mean and variance of  $T_{1},\ldots,T_{N}$?


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 Besides just giving the answers, demonstrate how you got them using our rules for  summations (and how I might have done this in class).


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III. Suppose $X$ is a random variable that takes on the following values with the probabilities listed:

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     P($X$ = -1) = 10/32; P($X$ = 0) = 20/32; P($X$ = 1) = 1/32; P($X$ = 2) = 1/32

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 a)   E($X$):

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 b)  Var($X$):

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 c)  Mode:

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  d) Median:

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  e) E($X^{2}$):

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f) Standard deviation of $X$:

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 g) P($X \ge 0$):

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 h)  E(-3$X$ - 7):

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 i) Var(-3$X$ - 7):

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j) $X$  represents a``fair'' game (yes or no):




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IV. Let $X$ be a continuous random variable with the cumulative
distribution function given above.

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a) Calibrate the pointer above to correspond to the cumulative
distribution function.

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b)  P($X$ = 2):

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c)  P($3 \le  X  \le 5)$:






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V. As part of an article that appeared in the \emph{NY Times}
(10/9/2003) (which is attached), and after Arnie won the California
recall, a section is given on ``How the Poll was Conducted''
beginning on the second page. Explain and show what the section
means, in terms of formulas you know, about the margin of error in
estimating population proportions. Does the plus or minus two points
require any rounding up to two decimal places?




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VI.  On a particular standardized reading test, the mean score for
     50 low-achieving students was 61.3 with a standard deviation
     of 10.2.  According to the publisher's manual, the mean score
     in the general population should be 70.  For now, assume that
     the population from which the low-achieving students were drawn
     can be represented by a random variable $X$ that is N($\mu$,$\sigma^2$).
     If necessary, use interpolation in the tables to obtain the
     appropriate $t$-value for use (and show how you did this).

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     (a) To assess whether the population mean for the low-achieving
          group might differ from that of the general population
          used to standardize the reading test, carry out a test of
          $H_{o}: \mu  = 70$ versus $H_{1}: \mu < 70$ at a fixed alpha level of .01.






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     (b) Construct a 99\% confidence interval for the unknown mean
          $\mu$ for the low-achieving population.



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     (c) Assuming that the standard deviation in the low-achieving population is a known value of 10.2, what sample size would be required to have          a 99\% confidence interval be 4 points in total length?  Show your work.



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VII. In a taste test of two versions of cola (e.g., pepsi versus
     coke), I asked 6 people independently which they prefer
     (blindfolded and randomly deciding on each trial whether pepsi or
     coke is given first to the person to taste).  Suppose the
     preference for pepsi is represented by a random variable $Y$ taking
     on a value of 1 and the preference for coke represented by the
     random variable $Y$ taking on a value of 0.

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     We let $\hat{p}$  denote the observed proportion of the 6 people who prefer pepsi to
     coke, and assume in the population from which these people are
     drawn that P($Y$ =1) = 1/2.

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     (a) Find the probability distribution for $\hat{p}$:

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          Values of $\hat{p}$    \hspace{10ex}          Probability



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     (b) E($\hat{p}$) =





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        (c) Var($\hat{p}$) =

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VIII. I have sampled 10 hyperactive children and have blood pressure
      scores before and after the administration of a tranquilizer.
      The data are as follows:

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\begin{tabular}{ccc}

                    Subject & Before    &   After \\
                        1   &     155   &    110 \\
                        2   &     160   &    120 \\
                        3   &     140   &    145 \\
                        4   &     145    &   125 \\
                        5   &     155   &    145 \\
                        6   &     160   &    115 \\
                        7   &     150   &    120 \\
                        8   &     175   &    155 \\
                        9   &     170   &    145 \\
                       10   &     165   &    140 \\

\end{tabular}

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     Suppose $\mu_{B}$ and $\mu_{A}$ refer to the population means for the before and
     after scores, respectively.

     (a) Using the usual t-test procedure for dependent samples, carry
          out a test of $H_{o}: \mu_{B} - \mu_{A} = 0$ versus $H_{1}: \mu_{B} - \mu_{A} > 0$ at a fixed
          alpha level of .01.





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     (b) Construct a 95\% confidence interval on $\mu_{B} - \mu_{A}$.







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     (c) Using the sign test, state the hypothesis you would now test
          that would be analogous to what as carried out in (a).
          Provide the exact $p$-value.

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IX. I am interested in the effects of a certain chemical that is
     typically sprayed on fruit to preserve it during transit, and
     decide to carry out an experiment on maze learning behavior in
     rats to see if the chemical might have any noticeable influence.
     Given the 36 rats I have available, 18 are randomly assigned to a
    ``chemical additive'' diet and 18 to a regular diet.  Some of the data follow, where the dependent variable is the number of trials
   it takes a rat to make five consecutive successful runs
 (no rat      was   given more than 20 trials).  Use the summary
 statistics to answer the two questions, (a) and (b).

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\begin{tabular}{cccc}

     Chemical Additive Diet & & Regular Diet & \\
           7    &   20  &            8 &     9 \\
           8    &    9  &            9 &    13 \\
          12    &   14  &           10  &   20  \\
          20    &   16  &           14  &   18  \\
                        $\vdots$     &      $\vdots$   &            $\vdots$  &  $
                        \vdots$  \\[2ex]

Sum: &             270      & Sum: &               222 \\
Sum of Squares: & 4430      &   Sum of Squares: &         2950 \\

\end{tabular}

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The assumptions are made that the data in the (independent) groups I and II came from a $N(\mu_{1},\sigma_{1}^{2})$ and  $N(\mu_{2},\sigma_{2}^{2})$, respectively.

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     (a) Give the 95\% confidence interval on $\mu_{1} - \mu_{2}$ using the
          approximate degrees-of-freedom calculated from the data.  If necessary, use          interpolation in the tables to obtain the    appropriate $t$-value          for use (and show how you did this).



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     (b) Carry out a test of $H_{o}: \mu_{1} - \mu_{2} = 0$ versus $H_{a}: \mu_{1} - \mu_{2} \ne 0$, using the
         ``pooled'' version of the $t$-test at a fixed alpha level of .05. Again, if          necessary find the appropriate $t$-value for use by
          interpolation (and show how you did this).




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X.  Completion

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 a) The sampling distribution for the number of
                        successes in $n$ independent observations on a
                        population characterized by a dichotomous outcome
                        of success/failure, is called the \rule{2.0in}{.02in}.

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 b) If I have 6 individuals and need to form a
                        committee of 3 from this group, how many different
                        ways could this be done? \rule{2.0in}{.02in}
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 c) The sample proportion $\hat{p}$, since E($\hat{p}$) = $p$, is said
                        to be a(n) \rule{2.0in}{.02in} estimator of $p$.

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 d) An estimator $\hat{\theta}$ is said to be \rule{2.0in}{.02in} if $\hat{\theta}$                        converges              to                    $\theta$ asympototically.

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 e) The robustness of several hypothesis testing
                        procedures is based on the fact that sample means
                        are approximately normal when simple random
                        samples are taken and the sample size gets large.                      This result is called the \rule{2.0in}{.02in}.

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 f) Generally, and other things being equal, power
                        decreases both as the sample size \rule{2.0in}{.02in}, and as
    the alternative hypothesis value being considered
                        gets \rule{2.0in}{.02in} the value specified under $H_{o}$.

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 g) One-tailed hypothesis tests are also referred to
                        as \rule{2.0in}{.02in} hypothesis tests.

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 h) In calculating power for a specific value
                        within the alternative hypothesis, 1.0 minus the power is called the \rule{2.0in}{.02in}.

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 i) The estimated degrees-of-freedom for a two-independent samples $t$-test with 10 observations in
             one group and 15 in the second must fall between
                        the values of \rule{2.0in}{.02in}and \rule{2.0in}{.02in}.

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 j) The assumption in the ``pooled'' two-independent
                        sample $t$-test that $\sigma_{1}^{2} = \sigma_{2}^{2}$ appears to not be very
                        important (i.e., we have robustness) as long as
                        \rule{2.0in}{.02in}.

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 k) 5 ``factorial'' is equal to \rule{2.0in}{.02in}.

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 l) If $X_{1} \sim N(0,5)$ and $Y_{1} \sim N(1,2)$ and $X_{1}$ and $Y_{1}$
                        are independent, then $3X_{1} - 4Y_{1}$ is \rule{2.0in}{.02in}.

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m) 95\% confidence intervals are ``naturally connected''
                        with fixed $\alpha$-level two-tailed tests, where $\alpha$ is
                        equal to \rule{2.0in}{.02in}.

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 n) If the \rule{2.0in}{.02in} is as small or smaller than $\alpha$,
                        we then typically say that the data are statistically significant
                        at level $\alpha$.

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 o) The standard error of a statistic is also called
                        the \rule{2.0in}{.02in} of the statistic.
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 p) The variance of a $t$-distribution with $k$ degrees-
                        of-freedom converges to \rule{2.0in}{.02in} as $k$ goes to
                        infinity.

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 q) A continuity correction is sometimes used when the
                        probabilities from a binomial distribution are
                        approximated by probabilities from a(n)
                        \rule{2.0in}{.02in}.

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XI. I have a test statistic $T$ that has the following distribution
under $H_{0}$ and $H_{1}$:

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\begin{tabular}{ccc}


$T$     &   Under $H_{0}$   &      Under $H_{1}$ \\
1    &     1/8  &           2/8 \\
2     &    2/8   &          2/8 \\
3      &   2/8    &         2/8 \\
4       &  2/8     &        1/8 \\
5        &  1/8     &        1/8 \\

\end{tabular}

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If you have a decision rule that says to reject $H_{0}$  if $T$ = 1,
2, or 3, find:

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a) Type I error:

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b) Type II error:

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c) Power:

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