1. MATH&146 Final Review
Part 2

2. Example 1
A building contractor has a chance to buy an odd lot of used bricks at an auction. She is interested in determining the proportion of bricks in the lot that are cracked and therefore unusable for her current project, but she does not have enough time to inspect all bricks. Instead, she checks 100 bricks to determine whether each is cracked and finds 6 of those bricks were unusable.
What is the population and parameter of interest?

3. Example 1 continued
Give the sample statistic for the proportion of bricks that were unusable.
Based on this sample, the building contractor should expect how many bricks from the lot to be unusable?
Give the 95% confidence interval for the proportion of bricks that were unusable.

4. Example 2
As part of its twenty-fifth reunion celebration, the Class of 1990 of State University mailed a questionnaire to its members. One of the questions asked the respondent to give his or her total income the previous year. Of the 820 members of the class of 1990, the university alumni office had addresses for 583. Of these, 421 returned the questionnaire.

5. Example 2 continued
The reunion committee computed the mean income given in the responses and announced, "The members of the class of 1990 have enjoyed resounding success. The average income of class members is $120,000!" Identify how each type of bias (selection bias, nonresponse bias, response bias) may be present.

6. Example 3
A 2010 SurveyUSA poll that asked 500 Los Angeles residents, "What is the best hamburger place in Southern California? Five Guys Burgers? In-N-Out Burger? Fat Burger? Tommy's Hamburgers? Umani Burger? Or somewhere else?" The distribution of responses by gender is shown below.
Male
Female
Total
Five Guys Burgers 5 6 11
In-N-Out Burger
162
181
343
Fat Burger 10 12 22
Tommy's Hamburgers 27 54
Umani Burger 1
Other 26 20 46
Not Sure 13 18
248
252
500

7. Example 3 continued
What is the probability that a randomly chosen male likes Fat Burger the best?
What is the probability that a randomly chosen person likes In-N-Out best or that person is female?
Male
Female
Total
Five Guys Burgers 5 6 11
In-N-Out Burger
162
181
343
Fat Burger 10 12 22
Tommy's Hamburgers 27 54
Umani Burger 1
Other 26 20 46
Not Sure 13 18
248
252
500

8. Example 4
Compute the mean, median, and mode for the following set of numbers. Which measure of central tendency best describes the set of measurements?
Compute the standard deviation. What is the z-score for 188?

9. Example 5
A medical doctor uses a diagnostic test to determine if her patient has rheumatoid arthritis. The doctor will prescribe treatment only if she thinks the patient has arthritis. In a sense, the doctor is using a null and an alternative hypothesis to decide whether or not to administer treatment.

10. Example 5 continued The hypotheses might be stated as:
H0: The person does not have arthritis (no arthritis)
HA: The person has arthritis
What would be Type 1 and Type 2 errors for these hypotheses?
State the consequences of the doctor committing a Type 1 or Type 2 error.

11. Example 6
According to Harper's magazine, the time spent by kids in front of the television set per year can be modelled by a normal distribution with a mean equal to hours and a standard deviation equal to 250 hours.
What percent of kids watch television for less than hours per year?
What percent of kids watch more than 1600 hours?
What are the boundaries for the middle 50% of time spent by kinds in front of the television?

12. Example 7
A student wanted to assess whether her dog Muffin tends to chase one of her balls more often than the other. She rolled both a green ball and a red ball at the same time and observed which ball Muffin chose to chase. Repeating this process a total of 100 times, the student found that Muffin chased the green ball 57 times and the red ball 43 times.
Construct hypotheses regarding whether or not Muffin preferred one ball over the other.
What proportion of the time in the sample did Muffin choose the green ball?

13. Example 7 continued
Have we met the conditions to use the normal distribution for inference?
Compute the standard error, test statistic, and p-value.
Write a suitable conclusion based on your p-value.
Create a 95% confidence interval to estimate the proportion of times Muffin chooses the green ball over the red ball. Are the results from the interval consistent with your conclusion in part (e)?

14. Example 8
The manager of an assembly process wants to determine whether or not the number of defective articles manufactured depends on the day of the week the articles are produced the following information.
Identify the expected count for the number of defective articles on Wednesday.
Day of Week M Tu W Th F
Total
Nondefective 85 90 95
455
Defective 15 10 5 45
100
500

15. Example 8 continued What are the degrees of freedom for this test?
For the test statistic χ2 = 8.547, compute the p- value.
What is the conclusion of the test?
Day of Week M Tu W Th F
Total
Nondefective 85 90 95
455
Defective 15 10 5 45
100
500

16. Example 9
Waiters are expected to keep track of their income from tips and report it on their income tax forms. The Internal Revenue Service suspects that one waiter (we'll call him "Fred") has been under-reporting his income, so they are auditing his tax return. An IRS agent goes through the restaurant's files and obtains a random sample of 80 credit card receipts from people Fred served. The average tip size shown on these receipts was $9.68 with a standard deviation of $2.72.

17. Example 9 continued
Create a 90% confidence interval for the mean size of all of Fred's credit card tips. On his tax return Fred had claimed that his tips averaged $8.73. Based on their confidence interval, does the IRS have a case against him? Explain.

18. Example 10
At the end of their first day at training camp, 10 new recruits participated in a rifle- shooting competition. The same 10 competed again at the end of a full week of training and practice. Their results are shown in the following table.
Recruit
First day
One week later
Diff 1 72 75 3 2 29 43 14 62 63 4 60 5 68 61 –7 6 59 13 7 73 12 8 82 9 38 47 10 48 –5
diff = one week later – first day

19. Example 10 continued
Does this set of 10 pairs of data show that there is a significant amount of improvement in the recruits' shooting abilities during the week?
Recruit
First day
One week later
Diff 1 72 75 3 2 29 43 14 62 63 4 60 5 68 61 –7 6 59 13 7 73 12 8 82 9 38 47 10 48 –5
diff = one week later – first day

20. Example 11
An agronomist hopes that a new fertilizer she has developed will enable grape growers to increase the yield of each grapevine. To test this fertilizer she applied it to 44 vines and used the traditional growing strategies on 47 other vines. The fertilized vines produced a mean of 58.4 pounds of grapes with standard deviation 3.7 pounds, while the unfertilized vines yielded an average of 52.1 pounds with standard deviation 3.4 pounds of grapes. Do these experimental results confirm the agronomist’s expectations?