Math 203, Introduction to Statistics,
Central College, Spring 2001 Exam 3 Review Sheet.
  1. As part of a promotion for a new type of cracker, free trial samples are offered to shoppers in a local supermarket. The probability that a shopper will buy a package of crackers after tasting the free sample is 0.22. Different shoppers can be regarded as independent trials. If X is the number of the next 200 shoppers that do not buy the crackers after tasting the free sample, then X has a binomial distribution.
    1. In this setting, what is a success? What are the values of n and p?
    2. How likely is it that exactly 155 shoppers, of the next 200 that try a sample, do not buy a package of crackers?
    3. What are the mean and standard deviation of the random variable X?
    4. If you wanted to estimate the probability that 165 or more (of the next 200) shoppers do not buy a package of the crackers after tasting the free sample, what normalcdf command would you use?
  2. 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. Is the result significant at the 5% level?
    6. Based on the sample above, give a 95% confidence interval for the mean bursting pressure of this type of PVC pipe.
    7. In common English, describe what the interval from part (f) tells you.

      8.  
  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 matched pairs 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 due to spreading the potassium applications over the growing season.
  2. The time needed for college students to complete a certain paper and pencil maze follows a normal distribution with a mean of 30 seconds and a standard deviation of 3 seconds. You wish to see if the mean reaction time m decreases after students exercise vigorously. You have a group of 15 students exercise vigorously for 30 minutes and then complete the maze. You decide to run a significance test at the 1% level, based on the mean,  of the 15 times of the students.
    1. Define the null and alternative hypothesis and sketch a (well labeled) normal density curve that illustrates the acceptance interval (the values of  that will cause you to accept H0).
    2. Several possible results from the test above are stated below. In each case, decide if the null hypothesis is accepted or rejected.
      1. The p-value of the test is 0.09.
      2. The p-value of the test is 0.02.
      3. The test yields  = 31.80 seconds.
      4. The test yields  = 33.29 seconds.
      5. The test yields  = 30.71 seconds.
    3. Describe in common words what a type 1 error means in this case, and what a type 2 error means in this case.
    4. What is the probability of a type 1 error in this case?
    5. What is the probability of a type 2 error if m = 32.5 seconds?
    6. What is the power of this test against the alternative m = 33 seconds?
  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.
    1. If you want the margin of error to be $150, how large should your sample be?
    2. Assuming a sample of size n = 25 yields  = $2319, find the 90% confidence interval based on this sample.
    3. Describe in words what the interval from part (b) tells you.
  4. If you wish to use T-procedures on a data set of size n = 15, what characteristics (shape, skewness, outliers etc.) should the data set have, and what characteristics must the data set avoid?
      5.  
  5. 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. You may assume that the distribution of weight gains for both diets are approximately normal. 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? Fully describe and carry out the test you use to make this decision.
Diet 1 Diet 2
1 = 5.80 pounds 2 = 7.27 pounds
 n1 = 8  n2 = 9
 S1 = 1.613 pounds  S2 = 0.99 pounds