Math 203, Introduction to Statistics,
Central College, Spring 2000 Exam 2 Review Sheet Answers.
  1. What is the difference between a parameter and a statistic?

    2.  

     

    A parameter is associated with the population, a statistic with a sample.

  2. Give an example where using stratified random samples is appropriate.

    3.  

     

    A company has 35 male and 50 female employees. They need an SRS of size 10 to poll their employees on a newly proposed work project and would like the sample to match their employment ratios (of male to female) since the proposal may be viewed differently by the different sexes.

  3. Name two types of sampling methods that can give unreliable results.

    4.  

     

    Volunteer or self-response sampling and convenience sampling (as well as others).

  4. Researchers want to compare how well Miracle Whip Salad Dressing sells in two different locations in grocery stores: on a display in the front of the store (nearest the cash registers) and on a display at the back of the store (away from the cash registers). The 30 Super Valu stores in Iowa will be randomly assigned to two groups (15 stores each); those which sell Miracle Whip on a front display and those which sell Miracle Whip on a back display. No advertisements are used for any of the stores, but Miracle Whip is sold at the same slightly reduced price in each Super Valu store for a week. The total number of jars sold from front displays and back displays will be compared.
    1. Is this a study or an experiment? Explain what the difference is.

      2.  

       

      This is an experiment, actual treatments are being used (putting the items on displays). In a study, you simply observe, in an experiment, you subject participants to treatments and measure their responses.

    2. What are the response and explanatory variables?

      3.  

       

      Response = number of jars sold. Explanatory = location (front or back display).

    3. Assume that the 30 stores have been labeled with the numbers from 1 to 30. Use your randInt command to select the two groups (front display and back display). Explain how you used your randInt commands to determine these groups.

      4.  

       

      Use randInt(1,30,15) to select the 15 stores for front displays. Eliminate any duplicates with a single call to randInt(1,30).

    4. In comparing the total sales (number of jars sold) from the front-display group and the back-display group, a potential problem arises. The Miracle Whip jars sold may not have come from the displays, but perhaps were picked up from the normal shelf locations of Miracle Whip in each store. If a given store sells an above average number of jars of Miracle Whip (from the normal shelf location), then this store will have a false positive influence on the group to which it belongs (making either front or back displays look better). Explain how our design handles this potential problem.

      5.  

       

      Randomization should divide the "above average" stores evenly into the 2 groups.

    5. The total number of jars sold from front and back-display stores for this week are given in the table below. Calculate the mean number of jars sold for each group and the standard deviation (of the number of jars sold) for each group.

      6.  

       

      Front: mean = 85.67 std dev = 8.37
      Back: mean = 77.87  std dev = 10.14

    Number of Jars Sold
    Front 95 87 72 90 105 84 78 78 77 85 92 90 84 79 89
    Back 70 81 66 93 75 73 87 76 61 80 69 96 79 72 90
    1. Past market research indicates that each one of these stores would sell X jars of Miracle Whip in a week (at this price, from the normal shelf locations), where X is normally distributed with a mean of 75.2 jars and a standard deviation of 9.0 jars. Both groups did better than average (most likely due to the placing of Miracle Whip on a display), but the front-display stores out-performed both the back-display stores and the usual shelf sales, by quite a bit. Assuming that the front-display group is a sample of size 15 from a normal distribution with mean 75.2 and standard deviation of 9.0 jars (for a single store), calculate the probability that a sample of size 15 has an x-bar value greater than or equal to that of the 15 stores that sold Miracle Whip on a front display.

      2.  

       

      normalcdf(85.67, 9999, 75.2, 9/sqrt(15) ) = 0.000003313

    2. Based on your answer to the last part, do you think selling Miracle Whip on a front display increases sales or not?

      3.  

       

      The probability is tiny. I would say that front displays definitely improve sales over regular shelf locations.

    3. Now assume that weekly sales of Miracle Whip (in a single store) have a mean equal to the mean from the back-display group, and a standard deviation equal to the standard deviation of the back-display sample. How likely is it that a sample of size 15 from this distribution has an x-bar value greater than or equal to that of the front-display group? Do you think front-displays sell Miracle Whip better than back-displays, or is the difference just due to chance?

      4.  

       

      normalcdf(85.67, 9999, 77.87, 10.14/sqrt(15) ) = 0.00144. Again, the probability that the front display sales are just an "above average" group from the same distribution as the back display sales is tiny. It appears as if front displays definitiely out-sell back displays.

  5. The Minnesota Twins committed 0 to 4 errors (bumbles or bad throws) in each of their games in the 1999 season. The probability distribution for X = number of errors per game is given in the table below.
Minnesota Twins 1999 Season
X = Errors per Game 0 1  2  3 4
Probability of X 0.48 0.29  ?  0.05 0.02
    1. What is the probability that the 1999 Minnesota Twins committed 2 errors in a game?

      2.  

       

      0.16 (all probabilities must sum to 1).

    2. The average number of errors per game is defined as
    0*P(X = 0) + 1*P(X = 1) + 2*P(X = 2) + 3*P(X = 3) + 4 *P(X = 4)
      Calculate the average number of errors per game for the 1999 Twins.

      0*.48 + 1 *.29 + 2*.16 + 3*.05 + 4*.02 = 0.84 errors per game on average.

    1. What is the probability that in a single game, the Twins committed more than their average number of errors?

      2.  

       

      This is P(X = 1, 2, 3 or 4) = 0.52.

    2. How likely is it that the 1999 Minnesota Twins did not commit exactly 1 error in a game (i.e., they committed 0, 2,3 or 4 errors)?

      3.  

       

      1 - 0.29 = 0.71.

    3. What is the probability that the Twins committed an even number of errors in a game?

      4.  

       

      0.48 + 0.16 + 0.02 = 0.66.

    4. Given that errors are bad (in regards to whether or not you win a game), would you guess that the number of errors committed in a game, and whether or not you won the game are independent? Why or why not?

      5.  

       

      I'd guess they are NOT independent. If you make errors you played bad and are more likely to have lost, so numbers of errors and whether you win or lose, seem to have an influence on one another.

  1. Grandma Lumpit makes wonderful chocolate chip cookies, however, her eye sight is fading and not all of her cookies end up with chocolate chips in them (some have zero chocolate chips). In fact, each cookie, independent of all other cookies, has a 0.14 chance of having no chips. You buy a dozen of Grandma Lumpit's chocolate chip cookies (chosen at random from a large supply of cookies). You are interested in probabilities associated with
X =  the number of cookies you get which have no chips.
    The above setting describes a binomial random variable X.
    1. What is a success?

      2.  

       

      A cookie with no chips.

    2. What are the values of n and p?

      3.  

       

      n = 12, p = 0.14.

    3. What is the probability that you get 2 cookies with no chips?

      4.  

       

      binompdf(12,.14,2) = 0.2863.

    4. What is the expected number (or mean number) of cookies you receive without chips?

      5.  

       

      n*p = 12*.14 = 1.68.

    5. What is the standard deviation of the number of cookies in a dozen that have no chips?

      6.  

       

      sqrt(n*p*(1-p) ) = 1.202.

    6. What is the probability that all 12 of your cookies have chips?

      7.  

       

      This is P( X = 0) = binompdf(12,.14,0) = 0.164.

  1. Twenty percent of all M&Ms are red. If you select a random sample of n M&Ms and let X = the number that are red, then X is a binomial random variable with parameters n and p = 0.2.
    1. In a sample of size 25, what is the exact probability that X = 4?

      2.  

       

      binompdf(25, .2, 4) = 0.1867.

    2. In a sample of size 25, what is the exact probability that X < 3?

      3.  

       

      binomcdf(25, .2, 2) = 0.0982.

    3. Using the normal approximation to the binomial, what is the approximate probability that a random sample of 10,000 M&Ms has fewer than 1900 red M&Ms?

      4.  

       

      normalcdf(-9999, 1900, 10000*.2, sqrt(10000*.2*.8)) = 0.0062.
      normalcdf(-9999, 1899, 10000*.2, sqrt(10000*.2*.8)) = 0.0058 (either answer is fine).

  2. Assume that the height of six-year old girls is normally distributed with a mean of 46 inches and a standard deviation of 2.17 inches.
    1. How likely is it that a randomly selected six-year old girl is more than 4 feet tall?

      2.  

       

      normalcdf(48,9999,46,2.17) = 0.178.

    2. How likely is it that the average height of 6 randomly selected six-year old girls is more than 4 feet?

      3.  

       

      normalcdf(48,9999,46, 2.17 / sqrt(6) ) = 0.012