Math 156 Sample Exam 3 Formulas Binomial mean np standard ...

Math 156 Sample Exam 3
Formulas Binomial mean np standard deviation np(1! p) probability
n n
C k
k p k (1! p) n! k
z-score z = x ! µ
"
z-statistic z = x ! µ
" / n
standard error ! n confidence interval !
"
x ± z * sample size n = z *! %
$ '
n
# m &
t = x ! µ
s / n standard error sn confidence interval x ± t * s n
2
matched pairs t =
x D
s D
/
n
1. A slot machine has three wheels. Each wheel has 10 pictures. The first wheel has one bell, the
second two bells, and the third three bells. What is the chance a spin shows exactly two bells
2. You forgot to study for your sociology quiz. It is a 8 question multiple choice quiz, with 4
possible answers for each question. You must score 6 or more correct to get a passing grade. If
you just randomly guess:
a) What is the mean and standard deviation of the number of questions you get correct
b) What is the probability of getting exactly 6 correct

3. Use the normal approximation to estimate the probability that in 800 tosses of a fair coin,
there would be more than 409 heads.
4. Produce a 95% confidence interval for the mean, if your sample consists of 200 items, whose
mean is 4.31 and the assumed population standard deviation is 8.
5. Suppose a 95% confidence interval for the mean of a population, computed from a sample of
size 500, is (160, 190).
a) What is the estimated mean from the data
b) What is the margin of error
c) Explain in plain language what the interval tells you.
d) Would you expect around 95% of the sampled values to fall in this interval Why or why not
e) What must be true in order for the interval to be valid

6. a) Consider two confidence intervals for the mean of a population:
413.1 ± 3.12 and 413.1 ± 2.62 . One is a 95% confidence interval and one is a 99% confidence
interval. Which is which Defend your answer.
b) Consider two confidence intervals for the mean of a population:
217.2 ± 4.12 and 217.2 ± 3.43 . One is from a sample of size 25 and one is from a sample of size
36. Which is which Defend your answer.
7. a) Suppose we wish a 99% confidence interval to have a margin of error of .25, and the
population standard deviation is assumed to be 1.5. Determine what sample size is required.
b) Suppose the standard error of the mean is .05 and the sample size is 900. About how big is the
standard deviation
c) In a recent study it was claimed that lengths of trout in Lake Pueblo have a mean different than
those upstream, which average 14 inches. State the null and alternative hypothesis in this setting.
d) Suppose a test results in a p-value of .16. What can you say about the null hypothesis and
alternative hypothesis

8. a) Is it possible for a result not to be significant at the 1% level but considered significant
at the 5% level Explain.
b) If a two-sided test with H 0
: µ = 25 vs. H a
: µ ! 25 fails to reject the null hypothesis at the
5% level, what can you say about the corresponding 95% confidence interval
c) Suppose the null hypothesis is true. What is the chance the hypothesis test produces a result
that is deemed statistically significant at the 5% level

9. a) Suppose H 0
:µ = 15 H a
:µ > 15 and z = -2.35. What are your conclusions What is the p-
value
b) Suppose H 0
: µ = 15 H a
: µ < 15 and z = -2.35. What are your conclusions What is the p-
value
10. Perform a test of H 0
: µ = 10 vs. H a
: µ ! 10 if n = 50 , ! = 1.5 , and x = 10.5 .
Quote an approximate p-value from the tables in your conclusions.

12. Nine overweight men agree to try a new diet pill for three weeks. Below are their weights
before and after the three week period.
before after
185 192
236 224
188 184
195 200
204 191
224 221
268 271
193 182
284 285
Determine an appropriate test procedure to see if the pills work.
Check any necessary conditions needed for the procedure to be valid.
Perform the test, even if you think the conditions are not met.
State conclusions, including a p-value.