|Except for the multiple choice questions below (1-3) and the last problem, all work and answers must be on separate paper.|
1. Circle the correct answer: To find confidence
intervals for the mean of a normal distribution, the t distribution is usually
used instead of the standard normal distribution because:
- a) the mean of the population is not known
b) the t distribution is more efficient
c) the variance of the population is usually not known
d) the standard error of the estimate is s/sqrt(n)
e) the sample mean is known.
3. Circle the correct answer: Suppose that
a significance test is such that the P-value does not change very much when
the assumptions of the procedures are violated. What is such a test called?
a. a t-test
b. a test with high power
c. a highly significant test
d. a robust test
In order to promote blood donation, a local chapter of The Red Cross quotes this study and suggests people lower their blood pressure by becoming regular blood donors. Explain why this type of study cannot be used to support such a claim.
Since it is an observational study, rather than an experiment with specific treatment, no causation can be attributed. Thus the Red Cross cannot say that donating blood causes your pressure to decrease.
5. In an experiment, the mean of the monthly salaries of a sample of men was $4000 and the salaries had standard deviation $270. The standard error of the mean of these salaries was $90. What was the sample size? Show your work.
6. An SRS of 100 of a certain popular model car
in 1993 found that 11 had a certain minor defect in the brakes. An SRS of
120 of this model car in 1994 found that 30 had the minor defect in the
brakes. We are interested in comparing the proportion of all cars of this
model in 1993 and 1994, respectively, that actually contain the defect.
a. Using the above data, and showing your work, compute the pooled sample proportion of defects.
b. In step c below, you will be asked to compute a 95% confidence interval for this problem. Clearly explain why the pooled sample proportion you computed in step a will or will not be used in this computation.
Since the two samples come from independent populations and, in contrast to a Test of Significance, a confidence interval makes no assumption that the populations are the same or have similar parameters (variances in this case), each sample must have freedom to influence the computations.
c. Compute by hand showing your steps so it is clear that you are working without the test feature of your calculator a 95% confidence interval for this problem.
d. Interpret this interval in terms of the problem (do not consider duality).
We have 95% confidence that those certain popular model cars in 1993 which had defects is between 4.12% and 23.9% less than the proportion of those certain popular model cars which had defects in 1994 [Jamie Holmes '98].
The difference in the population of the proportion of defects in model year 1993 cars and those in 1994 cars lies between approximately 0.04 and 0.24.
In other words, over the long run 95 out of every 100 similar samples will produce confidence intervals which will contain the difference of the population proportion of defects for the 1993 and 1994 models of this problem.
e. What would be the name of an appropriate test of significance to apply to this problem?
Two Sample Z-Test of Proportions
f. State an appropriate hull hypothesis for a test of significance for this problem and determine the results of that test of significance solely from the confidence interval you computed in step c.
Ho: The proportion of defects in the certain 1993 model is the same as the proportion of defects in the certain 1994 model .
Since the confidence interval in step c do not contain zero and this is a two sided test, duality implies that an appropriate test of significance would reject the null hypothesis at the 0.05 level.
g. Are the appropriate assumptions satisfied in order to do the appropriate test of significance for the given problem? Support your answer.
Yes they are. Since both samples are a) independent, b) SRSs and c) each has size >= 30.
7. Since most people are right-handed, a company
that designs machinery has traditionally placed the controls that demand
the most hand strength so that they will be used by the right hand.company
decides that it should test its assumption that the right hand of right-handed
adults tends to be stronger than the left hand.right-handed adults are selected
from employees of this company for the test.strength is measured by using
a calibrated hand gripper. The people test their left hand first and then
their right hand.hand strengths, in kilograms, for each person are given
DO NOT TYPE IN THIS DATA. IT IS AVAILABLE FROM
a. Briefly discuss a concern that is particular to the design of this study (not assumptions for a test of significance.
By always testing the left hand first the study
may be biased. Perhaps people generally try harder on a second try or perhaps
they tire after the first try.
b. Are the samples independent or dependent? Explain why this decision is important?
The samples are dependent since both the right
and left hand come from the same 9-people. This is important so that a matched
pairs test may be conducted [Jaime Bard, '98]. In contrast, if the samples
were independent a two-sample test would have to be used.
c. Demonstrate you can do the following BY HAND and with tables. You must show formulas and the substitution of values into the formulas. However, standard deviation(s) or stand error(s) can be determined from the calculator. See your teacher if you are not sure what you can do.
Do a fully annotated test of significance to determine if these
data support the conclusion that the right hand of right-handed adults tends
to be stronger than the left hand.
Since the samples are dependent, a paired t-test of differences is appropriate. Therefore the differences are computed below:
Ho: mu=0. The mean of the differences in hand
strength for each person is zero.
Ha: mu > 0. The mean of the the right hand strength for each person greater than that persons left hand strength.
Assumptions: It is unclear if the sample is an SRS and it is small. Nor do we know that the population is normal, so we proceed with great concern.
Apply a One Sample T-Test for a Population Mean.
Conclusion. If we are willing to disregard the assumptions of this test, this sample provides statistically significant evidence at the 0.05 level (p=.02, n=9, df = 8) that in fact the right hand strength for each person in a population similar to the one tested is greater that person's left hand strength.
8. In assessing the weather prior to leaving our residences on a spring morning, we make an informal test of the hypothesis "The weather will be fair today."the "best" information available to us, we complete the test and dress accordingly.would be the consequences of a Type I and Type II error?
From the choices below select and clearly explain your choice of the
correct answer. You may answer on this sheet.
a) Type I error: inconvenience in carrying needless rain equipment
Type II error: clothes get soaked
Type 1 Error: Rejecting Ho when Ho is true. So the weather will be fair but you "reject" that an bring an umbrella.
Type 2: Rejecting Ha when Ha is true. So it will rain but you "reject" that it will rain and get soaked.
b) Type I error: clothes get soaked
Type II error: inconvenience in carrying needless rain equipment
c) Type I error: clothes get soaked
Type II error: no consequence since Type II error cannot be made
d) Type I error: no consequence since Type I error cannot be made
Type II error: inconvenience in carrying needless rain equipment