Our exam will cover chapters 13 to 16 of Moore's text. Because you may need z* values, they will appear on the exam. If you can do most of the problems below in a timely fashion, with little or no aid from your notes or text, I am confident that you are well prepared for this exam.

1. For each setting below, decide if the random variable X has a binomial distribution. If X does have a binomial distribution, describe what a success is, what is n, what is p, and then calculate the mean and standard deviation of X. If it does not have a binomial distribution, explain why (which properties of the binomial setting are NOT satisfied?).
1. Grandma Lumpit makes wonderful chocolate chip cookies, but her strength just isn't what it used to be, and she has trouble mixing the chips evenly in the dough. Some cookies have no chips, and others have as many as 15 chips. I select one of Grandma Lumpit's cookies at random and let X = the number of chips in the cookie.
2. In fact, 7% of Grandma Lumpit's cookies have no chips. I select a dozen of her cookies at random (from a huge supply) and let X = the number of cookies with one or more chips.
3. A collection of 3 dozen of Grandma Lumpit's cookies contains 3 cookies with no chips. I select 10 of these 36 cookies at random and let X = the number I get with no chips.
2. The department of transportation for Des Moines has found that 25% of all Des Moines parking tickets issued are NOT paid off within one month of their issue date. The department selects an SRS of 100 tickets issued one month ago and calculates X = the number that have not been paid off. The random variable X has a binomial distribution.
1. For this setting, what is a success? What is n? What is p?
2. What is the probability that X = 24?
3. What is the probability that exactly 80 of the tickets HAVE been paid off?
4. What are the mean and standard deviation of X?
5. Is it safe to use the Normal approximation to the binomial for this random variable? Explain.
6. What normalcdf command would approximate the probability that X is 30 or more?
3. A manufacturer of PVC irrigation pipes claims that the mean bursting pressure of their pipes is 450 psi (pounds per square inch). Furthermore, it is known that the bursting pressures for this particular type of pipe have a distribution that is very close to Normal, and the standard deviation is 70 psi. 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.

1. The sample gives reason to suspect that z-procedures should not be used in this setting? Explain.
   We will examine the results of a significance test both by ignoring the trouble with the sample and by removing the trouble.
2. Define the null and alternative hypothesis for a significance test to help decide if the bursting pressure is lower than claimed by the manufacturer.
3. Carry out the test described in part (b) and report the p-value of this test.
4. Explain in common language what this p-value tells you.
5. Do you think that the watchdog group has a legitimate fear?
6. Now remove the outlier in the sample and repeat parts (c) to (e).
   
7. With the outlier removed, give a 95% confidence interval based on this sample.
8. In common terms, what does the phrase "95% confident" mean?
9. Would you expect roughly 95% of the bursting pressures for individual pipes of this variety to lie in your interval? Explain.
    
2. 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:
   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 (if one treatment has high yields in a given year, then it is more likely that another treatment will also have higher than normal yields that year, due to the favorable growing conditions). 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. A matched pairs test is appropriate, so calculate the differences "treatment 2 - treatment 1". You should assume that these differences have a standard deviation of 130 pounds of dry matter per acre.
      
   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 90% 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. Does it seem that the gain is practically significant?
   4. Explain why the value zero not being in your confidence interval from part (c) indicates that the results of your test from part (b) are significant at the 5% level?
       
3. After many years of working your way up the ranks in a local grocery chain, you've just landed a well deserved promotion to assistant manager at your chain's brand new store. One of your responsibilities is to decide where the major sale items for a given week are displayed. These items can be dislayed at the front-end (near the check-out registers) of an aisle, or on the back-end of an aisle. Your head manager has suggested that the premier sale item for next week, Miracle Whip, be displayed on the front-end of the middle aisle in your store (more or less the "best" location for visibility). It is known that weekly sales of Miracle Whip, when on special and displayed in this location, are Normally distributed with a mean of 22.7 cases and standard deviation of 4.8 cases (much better sales figures than when Miracle Whip is not part of the weekly advertisement). Having stocked shelves for what seems like centuries, you've noticed over the years that when Miracle Whip goes on sale, it seems to sell better when displayed on the back-end of an aisle. You'd like to tell your manager that this is the case, but you're a bit nervous about rocking the head manager's boat based on your gut feelings from late night work over several years. You decide to test your claim that Miracle Whip (when on sale) sells better off of a back display than it does when displayed up front at the 5% level. If the results of your test are significant at the 5% level, you will use your authority to over-ride the head manager's suggestion and assign Miracle Whip to a back display. If the test yields a p-value greater than 0.05, you will simply follow the suggestion of the head manager and put the Miracle Whip display up front. You base your test on the mean weekly sales of Miracle Whip (when on special and displayed at the back-end of an aisle), for an SRS of 36 weeks.
   1. State the null and alternative hypothesis for your test.
   2. Your alternative hypothesis from part (a) should involve the parameter m (the "mean" of the "population"), for this particular situation give a precise description of this parameter, i.e. what does m represent?
   3. Restate "the test rejects H₀" in terms of the statistic .
   4. Given below are several possible results of your test. For each such result, descide whether you display Miracle Whip at the front or back of an aisle.
      1. The p-value of the test is p = .48.
      2. The test statistic is z = 0.001.
      3. Your SRS has = 30.5953.
      4. Your SRS has = 24.0159.
   5. Give an example (there are gagillions of them) of a specific value of m and , where your test commits a type 2 error.
   6. If you place Miracle Whip on a back display, even though weekly sales of Miracle Whip (when part of the ad) do not sell better there, did the test commit a type 1 or type 2 error?
   7. If you place Miracle Whip on a front display, even though it actually sells better on a back display, did your test commit a type 1 or type 2 error?
   8. If the power of your test against the alternative m = 26.0 is 0.993, what is the probability of a type 2 error against the alternative m = 26.0?
   9. What is the power of your test against the alternative m = 25?
4. 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.
   1. How large should your sample be?
   2. If you desire a 95% confidence interval instead of a 90% interval (with the same margin of error), would your sample need to be bigger or smaller?

5. 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 repair cost is significant at the 5% level.  Give a 95% confidence interval for the difference in costs for these two types of  bumber guards to answer this question. Assume that the difference in costs has a population standard deviation of  $35.
   

6. 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?
7. A simple random sample of 12 graduates of a secretarial school typed on the average 78.2 words per minute and the standard deviation for all graduates is known to be 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?