Math 156 Sample Exam 3 Formulas Binomial mean np standard ...
Math 156 Sample Exam 3 Formulas Binomial mean np standard ...
Math 156 Sample Exam 3 Formulas Binomial mean np standard ...
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<strong>Math</strong> <strong>156</strong> <strong>Sample</strong> <strong>Exam</strong> 3<br />
<strong>Formulas</strong> <strong>Binomial</strong> <strong>mean</strong> <strong>np</strong> <strong>standard</strong> deviation <strong>np</strong>(1! p) probability<br />
n n<br />
C k<br />
k p k (1! p) n! k<br />
z-score z = x ! µ<br />
"<br />
z-statistic z = x ! µ<br />
" / n<br />
<strong>standard</strong> error ! n confidence interval !<br />
"<br />
x ± z * sample size n = z *! %<br />
$ '<br />
n<br />
# m &<br />
t = x ! µ<br />
s / n <strong>standard</strong> error sn confidence interval x ± t * s n<br />
2<br />
matched pairs t =<br />
x D<br />
s D<br />
/<br />
n<br />
1. A slot machine has three wheels. Each wheel has 10 pictures. The first wheel has one bell, the<br />
second two bells, and the third three bells. What is the chance a spin shows exactly two bells<br />
2. You forgot to study for your sociology quiz. It is a 8 question multiple choice quiz, with 4<br />
possible answers for each question. You must score 6 or more correct to get a passing grade. If<br />
you just randomly guess:<br />
a) What is the <strong>mean</strong> and <strong>standard</strong> deviation of the number of questions you get correct<br />
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,<br />
there would be more than 409 heads.<br />
4. Produce a 95% confidence interval for the <strong>mean</strong>, if your sample consists of 200 items, whose<br />
<strong>mean</strong> is 4.31 and the assumed population <strong>standard</strong> deviation is 8.<br />
5. Suppose a 95% confidence interval for the <strong>mean</strong> of a population, computed from a sample of<br />
size 500, is (160, 190).<br />
a) What is the estimated <strong>mean</strong> from the data<br />
b) What is the margin of error<br />
c) Explain in plain language what the interval tells you.<br />
d) Would you expect around 95% of the sampled values to fall in this interval Why or why not<br />
e) What must be true in order for the interval to be valid
6. a) Consider two confidence intervals for the <strong>mean</strong> of a population:<br />
413.1 ± 3.12 and 413.1 ± 2.62 . One is a 95% confidence interval and one is a 99% confidence<br />
interval. Which is which Defend your answer.<br />
b) Consider two confidence intervals for the <strong>mean</strong> of a population:<br />
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<br />
36. Which is which Defend your answer.<br />
7. a) Suppose we wish a 99% confidence interval to have a margin of error of .25, and the<br />
population <strong>standard</strong> deviation is assumed to be 1.5. Determine what sample size is required.<br />
b) Suppose the <strong>standard</strong> error of the <strong>mean</strong> is .05 and the sample size is 900. About how big is the<br />
<strong>standard</strong> deviation<br />
c) In a recent study it was claimed that lengths of trout in Lake Pueblo have a <strong>mean</strong> different than<br />
those upstream, which average 14 inches. State the null and alternative hypothesis in this setting.<br />
d) Suppose a test results in a p-value of .16. What can you say about the null hypothesis and<br />
alternative hypothesis
8. a) Is it possible for a result not to be significant at the 1% level but considered significant<br />
at the 5% level Explain.<br />
b) If a two-sided test with H 0<br />
: µ = 25 vs. H a<br />
: µ ! 25 fails to reject the null hypothesis at the<br />
5% level, what can you say about the corresponding 95% confidence interval<br />
c) Suppose the null hypothesis is true. What is the chance the hypothesis test produces a result<br />
that is deemed statistically significant at the 5% level
9. a) Suppose H 0<br />
:µ = 15 H a<br />
:µ > 15 and z = -2.35. What are your conclusions What is the p-<br />
value<br />
b) Suppose H 0<br />
: µ = 15 H a<br />
: µ < 15 and z = -2.35. What are your conclusions What is the p-<br />
value<br />
10. Perform a test of H 0<br />
: µ = 10 vs. H a<br />
: µ ! 10 if n = 50 , ! = 1.5 , and x = 10.5 .<br />
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<br />
before and after the three week period.<br />
before after<br />
185 192<br />
236 224<br />
188 184<br />
195 200<br />
204 191<br />
224 221<br />
268 271<br />
193 182<br />
284 285<br />
Determine an appropriate test procedure to see if the pills work.<br />
Check any necessary conditions needed for the procedure to be valid.<br />
Perform the test, even if you think the conditions are not met.<br />
State conclusions, including a p-value.