**Math 221 statistics for decision making**

**Final Exam**

You should work each of the following on your own, thenreview the solutions guide. DO NOT look at the solutions guidefirst.

1. Explain the difference between a population and a sample. Inwhich of these is it important to distinguish between the two in orderto use the correct formula? mean; median; mode; range; quartiles;variance; standard deviation.

2. The following numbers represent the weights in pounds of six 7-year old children in Mrs. Jones’ 2nd grade class.

{25, 60, 51, 47, 49, 45}

Find the mean; median; mode; range; quartiles; variance; standarddeviation.

3. If the variance is 846, what is the standard deviation?

4. If we have the following data

34, 38, 22, 21, 29, 37, 40, 41, 22, 20, 49, 47, 20, 31, 34, 66

Draw a stem and leaf. Discuss the shape of the distribution.

5. What type of relationship is shown by this scatter plot?

6. What values can r take in linear regression? Select 4 values in thisinterval and describe how they would be interpreted.

7. Does correlation imply causation?

8. What do we call the r value.

9. To predict the annual rice yield in pounds we use the equation

y ˆ = 859 + 5.76x1 + 3.82x2, where x1 represents the number of acresplanted (in thousands) and where x2 represents the number of acresharvested (in thousands) and where r2 = .94.

a) Predict the annual yield when 3200 acres are planted and 3000are harvested.

b) Interpret the results of this r2 value.

c) What do we call the r2 value?

10. The Student Services office did a survey of 500 students in whichthey asked if the student is part-time or full-time. Another questionasked whether the student was a transfer student. The results follow.

**Transfer Non-Transfer Row Totals**

**Part-Time ** 100 110 210**Full-Time** 170 120 290**Column Totals** 270 230 500

a) If a student is selected at random (from this group of 500students), find the probability that the student is a transfer student. P(Transfer)

b) If a student is selected at random (from this group of 500students), find the probability that the student is a part time student.P (Part Time)

c) If a student is selected at random (from this group of 500students), find the probability that the student is a transfer studentand a part time student. P(transfer ∩ part time).

d) If a student is selected at random (from this group of 500students), find the probability that the student is a transfer student ifwe know he is a part time student. P(transfer | part time).

e) If a student is selected at random (from this group of 500students), find the probability that the student is a part time given heis a transfer student. P(part time | transfer)

f) Are the events part time and transfer independent? Explainmathematically.

g) Are the events part time and transfer mutually exclusive. Explainmathematically.

11. A shipment of 40 television sets contains 3 defective units. Howmany ways can a vending company can buy five of these units andreceive no defective units?

12. How do you recognize a discrete distribution?

13. The random variable X represents the annual salaries in dollars ofa group of teachers. Find the expected value E(X).

X = {$35,000; $45,000; $55,000}.

P(35,000) = .4; P(45,000) = .3; P(55,000) = .3

14. How do you recognize a binomial experiment?

15. An advertising agency is hired to introduce a new product. Theagency claims that after its campaign 61% of all consumers arefamiliar with the product. We ask 7 randomly selected customerswhether or not they are familiar with the product.

a) Is this a binomial experiment? Explain how you know.

b) Use the correct formula to find the probability that, out of 7customers, exactly 4 are familiar with the product. Show yourcalculations.

16. How do you recognize a Poisson experiment?

17. The mean number of cars per minute going through theEisenhower turnpike automatic toll is about 7. Find the probability thatexactly 3 will go through in a given minute using the correct table,formula, or Excel function.

18. How do you recognize a normal distribution?

19. Label the following as continuous or discrete distributions.

a) The lengths of fish in a certain lake.

b) The number of fish in a certain lake.

c) The diameter of 15 trees in a forest.

d) How many trees are on a farmer’s acre.

20. Jack weighs 160 pounds and his sister weighs 110 pounds. If themean weight for men his age is 175 with a standard deviation of 14pounds and the mean weight for women is 145 with a standarddeviation of 10 pounds, determine whose weight is closer to“average.” Write your answer in terms of z-scores and areas underthe normal curve.

21. On a dry surface, the braking distance (in meters) of a certain caris a normal distribution with mu = µ = 45.1 m and sigma = σ = 0.5

(a) Find the braking distance that corresponds to z = 1.8

(b) Find the braking distance that represents the 91st percentile.

(c) Find the z-score for a braking distance of 46.1 m

(d) Find the probability that the braking distance is less than orequal to 45 m

(e) Find the probability that the braking distance is greater than46.8 m

(f) Find the probability that the braking distance is between 45 mand 46.8 m.

22. A drug manufacturer wants to estimate the mean heart rate forpatients with a certain heart condition. Because the condition is rare,the manufacturer can only find 14 people with the condition currentlyuntreated. From this small sample, the mean heart rate is 101 beatsper minute with a standard deviation of 8.

(a) Find a 99% confidence interval for the true mean heart rate of allpeople with this untreated condition. Show your calculations.

(b) Interpret this confidence interval and write a sentence thatexplains it.

23. Determine the minimum required sample size if you want to be80% confident that the sample mean is within 2 units of the population mean given sigma = 9.4. Assume the population is normallydistributed.

24. A social service worker wants to estimate the true proportion ofpregnant teenagers who miss at least one day of school per week onaverage. The social worker wants to be within 5% of the trueproportion when using a 90% confidence interval. A previous studyestimated the population proportion at 0.21.

(a) Using this previous study as an estimate for p, what sample sizeshould be used?

(b) If the previous study was not available, what estimate for pshould be used?

25. Suppose you are performing a hypothesis test on a claim about apopulation proportion. Using an alpha = .04 and n = 90, what twocritical values determine the rejection region if the null hypothesis is

Ho: p = 0.54?

(a) ± 1.96

(b) =± 2.05

(c) ± 2.33

(d) none of these

26. A restaurant claims that its speed of service time is less than 15minutes. A random selection of 49 service times was collected, andtheir mean was calculated to be 14.5 minutes. Their standarddeviation is 2.7 minutes. Is there enough evidence to support theclaim at alpha = .07. Perform an appropriate hypothesis test, showingeach important step. (Note: 1st Step: Write Ho and Ha; 2nd Step:Determine Rejection Region; etc.)

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