Stat 11

March 13, 2006

Homework #6 (due Friday, March 17)

 

This homework is due at the start of class Friday, March 17.  You may work in groups (across sections if you like), consult with others, or use any references or tools that seem useful, but you must write up your solutions yourself.

 

Critical values:

 

1.  Draw a picture showing  t*a/2, n-1  (as a point on the horizontal axis) and the probabilities  C  and  a/2  (as areas under a curve).

 

2.  If C = 0.80, what are  a,  z*a/2,  and  t*a/2, 15 ?

 

3.  What are  t*0.005,  100  and  z*0.005  ?   (You would use one of these critical values for a 99% CI when n = 101.)

 

4.  When  n  is large,  t*a/2, n-1  is close to  z*a/2.  Is the difference greater when C is large or when C is small?

 

(Nothing to turn in on the next two problems)

 

X.  Verify that  z*a/2  (in Table D) agrees with    =NORMSINV(a/2)   (in Excel) and that t*a/2, n-1  (in Table D) agrees with    =TINV (a, n-1 )    (in Excel).

 

Y.  If possible, learn how to find  z*a/2  and  t*a/2, n-1  on your calculator.  (The t-critical values you want may be labeled as “two-tailed” critical values.)

 

Confidence Intervals for Means:

 

5.   We weighed a random sample of 15 newly hatched turtles, and found an average mass of 8.0  grams.  We know from our extensive experience with turtles that the (population) standard deviation of their masses is  3.0  grams.  If  m  is the average mass of all newly hatched turtles, what is a  95%  confidence interval for  m ?

 

            Also, what is a  99%  confidence interval for  m ?

 

6.  We also weighed a random sample of  20  newly hatched salamanders, and found an average mass of  4.5  grams.  Lacking experience with salamanders, we also calculated the standard deviation of our sample, finding  s = 1.4  grams.  If  m  is the average mass of all newly hatched salamanders, what is a 95% confidence interval for  m ?

 

            Also, what is an  94%  confidence interval for  m ?   (You’ll need to go beyond the table for this one.)

7.  Construct a 95% confidence interval for  m,  if you can, from these measurements:

 

            80.0    82.0    82.5    83.0    83.0   84.0   86.5             ()

                       

 

8.  Construct a 95% confidence interval for  m,  if you can, from these measurements:

 

            4.00     4.00    4.01    4.03    4.10   4.20   4.57            ()

 

 

Confidence intervals for proportions

 

9.  Among your sample of 400 relays, exactly 16 were defective. 

a.  What is ?

b.  Give a 90% confidence interval for  p,  the fraction of all relays that are defective.

 

 

10.  In another factory, only 4 of 400 relays in your sample were defective.

a.  What is ?

b.  Give a 99% confidence interval for p at this factory.

c.  If you don’t like your answer, explain what went wrong.

 

 

11.  The following statement is wrong:

 

            (WRONG)  For a given confidence level, halving the margin of error requires a sample twice as large.

 

            Fix it.

 

 

12.  The marketing manager wants you to plan a survey to find out what fraction of adults in this county read the local sports page.  In nearby counties, the fraction varies from 15% to 40%.

 

            “I don’t want to waste money on this survey,” says the manager.  “On the other hand, I don’t want to run much risk of its being wrong by more than five percentage points.”

 

            What sample size will you recommend ?

 

 (end)