Math 203, Introduction to Statistics,
Central College, Fall 2000 Exam 3 Review Sheet


  1. A manufacturer of PVC irrigation pipes claims that the mean bursting pressure of their pipes is 425 psi (pounds per square inch). A consumer watchdog group fears that the bursting pressure may be lower and tests a random sample of 10 such pipes. The bursting pressures are listed below.
401 359 383 427 414 415 389 463 394 428
    1. Define the null and alternative hypothesis for a significance test to help decide if the bursting pressure is lower than claimed by the manufacturer.
    2. Carry out the test described in part (a) and report the p-value of this test.
    3. Explain in common language what this p-value tells you.
    4. Do you think that the watchdog group has a legitimate fear?
    5. Based on the sample above, give a 95% confidence interval for the mean bursting pressure of this pipe.
    6. Explain in everyday language what your confidence interval from part (e) tells you.

      7.  
  1. An agricultural researcher reasons as follows: A heavy application of potassium fertilizer to grasslands in the spring seems to cause lush early growth but depletes the potassium before the growing season ends. Spreading the same amount of potassium over the growing season might increase yields. He therefore compares two treatments:
      2. Treatment 1: 100 lbs per acre in the spring;
      Treatment 2: 50, 25 and 25 pounds per acre applied in spring, early summer and late summer.
    The experiment is continued over several years because grass yields vary greatly from year to year. The yields (in pounds of dry matter per acre) are known to vary roughly with a normal distribution (over all years). The data observed in the experiment are given below.
     
    Treatment year 1 year 2 year 3 year 4 year 5
    1 3902 4281 5135 5350 5746
    2 3970 4271 5440 5490 6028
      1. Why is a one sample test appropriate here?
      2. Do the data give good evidence that treatment 2 leads to higher average yields? Fully describe the test you use to answer this question.
      3. Give a 98% confidence interval for the mean increase in yield (the difference between treatment 2 and treatment 1) due to spreading the potassium applications over the growing season.

        4.  
  2. The store manager of the local grocery store claims that customer complaints vary normally from week to week with a mean of 13 complaints per week and a standard deviation of 3 complaints per week. After hiring several new checkout clerks, the manager is worried that the average number of complaints per week has increased. The manager decides to test this hypothesis at the 5% level by selecting a random sample of 10 weeks and calculating  = the average number of complaints for these 10 weeks. If the average number of complaints is too high (significant at the 5% level), the manager will fire the new checkers.
    1. Define the null and alternative hypothesis and sketch a (well labeled) normal density curve that illustrates the rejection region (the values of  that will cause the manager to fire the checkers) and the acceptance region (the values of  that will cause the manager to NOT fire the new checkers).
    2. Several possible results from the test above are stated below. In each case, decide if the checkers are fired or not fired.
      1. The p-value of the test is 0.09.
      2. The p-value of the test is 0.02.
      3. The test yields  = 14.56.
      4. The test yields  = 13.29.
      5. The test yields  = 15.71.
    3. Describe in common words what a type I error means in this case, and what a type II error means in this case.
    4. What is the probability of a type II error if m = 15.5? What is the power of this test against the alternative m = 15.5?

      5.  
  3. Monthly salaries (in dollars) of recent graduates with a statistics course are known to vary normally with a standard deviation of $323 per month. You would like to estimate the mean monthly salary of all such graduates with a 90% confidence interval and a margin of error of $150. How large should your sample be?

    4.  
  4. To be considered lean, a 200 gram package of ground beef should contain 35 or fewer grams of fat (on average). Based on several complaints from customers of the Piggly Wiggly grocery store about ground beef that seems overly greasy (more fat than 35 grams per package), a consumer group decides to run a significance test at the 5% level. It is safe to assume that the grams of fat in the population of all 200 gram packages of ground beef are normally distributed with a standard deviation of 2.5 grams of fat (per package). The null hypothesis states that the mean fat content of all 200 gram ground beef packages (at this Piggly Wiggly store) is 35 grams, and the consumer group selects eighteen 200 gram packages at random and calculates = the average number of grams of fat per package.
    1. What is the alternative hypothesis for this test?
    2. In words (related to this specific example), what is a type 1 error?
    3. Give values (there are several correct answers) for m and  that would correspond to a type 1 error being made.
    4. In words (related to this specific example), what is a type 2 error?
    5. Give values (there are several correct answers) for m and  that would correspond to a type 2 error being made.
    6. What is the power of the consumer group's test against the alternative that m = 37 grams of fat per package?

      7.  
  5. Which of the following data sets are safe for using T-procedures? For the ones that are not safe, explain why they are not safe.
    1. A random sample of size 142 is selected from a population that is significantly skewed to the right.
    2. The data are: 80,22,17,131,-19,3,23,-1,20,-51,-3
    3. The data are: 11,18,1,4,6,19,6,3,13,1,-1

      4.  

       
       
       
       
       
       
       
       
       
       

For each of the problems that follow, fully describe and carry out the test that best fits the situation.
  1. Two experimental diets (diet 1 and diet 2) designed to add weight to malnourished 3rd world children are fed to independent and random samples of such children. The results in weight gains are summarized in the table below. Do the data support the claim that diet 2 adds significantly more weight than diet 1?  Give a 90% confidence interval for the mean increase in weight gain due to diet 2 over diet 1.  Explain in everyday language what this 90% confidence interval means.
Diet 1 Diet 2
1 = 5.80 pounds 2 = 7.27 pounds
 n1 = 8  n2 = 9
 S1 = 1.613 pounds  S2 = 0.99 pounds
  1. To compare two kinds of bumper guards, ten of each kind were mounted on a certain midsize car.  Then, each car was run into a concrete wall at 5 mph, and the following costs (in dollars) of the repairs were reported.  The manufacturer is trying to determine if the difference in cost is significant at the 5% level.  Give a 98% confidence interval for each kind of  bumber guard.
Bumper Guard A
545
495
506
447
530
510
487
539
559
531
Bumper Guard B
536
475
513
558
546
514
517
473
562
529
  1. An oceanographer wants to test, on the basis of the mean of a random sample of size 35 and at the 0.05 level of significance, whether the average depth of the ocean in a certain area is deeper than 72.4 fathoms, as has been recorded.  What will she decide if she gets a sample mean of 73.2 fathoms when the population standard deviation is 2.1 fathoms?
  1. A simple random sample of 12 graduates of a secretarial school typed on the average 78.2 words per minute with a sample standard deviation of 7.9 words per minute.  The admissions office of the school claims that their graduates type an average of 80 words per minute.  Do you believe the claim of the admissions office?
  1. The wing span of two varieties of sparrows is given in the tables below (in millimeters).  Is the difference between these two varieties significant at the 5% level?  What is the exact p-value?  Explain in everyday language what this p-value means.
Variety 1
162
159
176
165
164
145
157
128
154
Variety 2
147
180
153
135
157
153
141