MATH399 Final Exam Review Questions Solutions Guide Complete Solution
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1. Determine if each of the following represents nominal, ordinal, interval, or ratio data.
a. Movie ratings
b. Test scores
c. Colors of shirts
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; variance; standard deviation.
3. If the variance is 846, what is the standard deviation?
Solution: standard deviation = square root of variance = sqrt(846) =
29.086
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 sampling technique is (a) a wheat field is divided into sections and a random sample of stocks are taken from each section, and (b) a manufacturer measures every 80th bottle to test for quality?
6. If a data set with a normal distribution has a mean of 32 and a standard deviation of 4.9, what percent of the data would you expect to find between 27.1 and 36.9? What range encompasses 99.7% of these data?
7. Determine the regression equation for the following data:
X: 4 5 7 3 2 7 8 9 7 6 4 6 5 9 3 2 1
Y: 9 7 8 6 9 5 6 4 3 6 7 8 9 3 4 2 4
Solution: y^ = 6.442 – 0.108x
8. If the correlation coefficient is -.749, what would be the value of the coefficient of determination?
Solution: The coefficient of determination is (-0.749)2 = 0.561.
9. To predict the annual rice yield in pounds we use the equation
yˆ = 859 + 5.76x1 + 3.82x2 , where x1 represents the number of acres planted and where x2 represents the number of acres harvested and where r2 = .94.
a) Predict the annual yield when 3200 acres are planted and 3000 are harvested.
b) Interpret the results of this r2 value.
Solution:
(a) yˆ = 859 + 5.76*3200 + 3.82*3000
= 859 + 18432 + 11460
= 30751 which is 30,751 pounds of rice
(b) 94% of the variation in the annual rice yield can be explained by the number of acres planted and harvested. The remaining 6% is unexplained and is due to other factors or to chance.
10. The Student Services office did a survey of 500 students in which they asked if the student is part-time or full-time. Another question asked 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
Totals 270 230 500
a) If a student is selected at random (from this group of 500 students), find the probability that the student is a transfer student. P (Transfer)
b) If a student is selected at random (from this group of 500 students), 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 500 students), find the probability that the student is a transfer student and a part time student. P(transfer AND part time).
d) If a student is selected at random (from this group of 500 students), find the probability that the student is a transfer student if we know he is a part time student. P(transfer | part time).
e) If a student is selected at random (from this group of 500 students), find the probability that the student is a part time given he is a transfer student. P(part time | transfer)
f) Are the events part time and transfer independent? Explain mathematically.
g) Are the events part time and transfer mutually exclusive. Explain mathematically.
(b) The total number of part time students is 210. The total number of students in the survey is 500. P(Part Time) = 210/500 = .42
(c) From the table we see that there are 100 students which are both transfer and part time. This is out of 500 students in the sample. P(transfer AND part time) = 100/500 = .20
(d) This is conditional probability and so we must change the denominator to the total of what has already happened. There are
100 students which are both transfer and part time. There are 210 part time students. P(transfer | part time) = 100/210 ≈ .4762
(e) P(part time | transfer) = 100/270 ≈ .3704
(f) The definition of independent is P(A|B) = P(A). To test we ask if P(part time | transfer) = P(part time)? Is .3704 = .42? No, there for the events are not independent.
We could also test P(transfer | part time) = P(transfer). Is .4762 =
.54? Again, the answer is no.
(g) For events to be mutually exclusive their intersection must be 0. In part c we found that P(transfer ∩ part time) = 100/500 = .20. Therefore the events are not mutually exclusive.
11. A shipment of 40 television sets contains 3 defective units. How many ways can a vending company can buy five of these units and receive no defective units?
Solution: There are 37 sets which are not defective. There are
37C5 ways to get 5 sets with none defective. 37C5 = 435, 897. Thus, there are 435,897 ways to get 5 sets with non defective.
12. In the US, 28% of people consider snickerdoodle as their favorite cookie. You asked 5 people if their favorite cookie was snickerdoodle. Create the probability distribution for the possible outcomes.
13. The random variable X represents the annual salaries in dollars of a group of teachers. Find the expected value E(X).
X = {$35,000; $45,000; $55,000} where,
P(35,000) = .4; P(45,000) = .3; P(55,000) = .3.
Solution: E(X) = 35,000*.4 + 45,000*.3 + 55,000*.3
= $44,000
14. An advertising agency is hired to introduce a new product. The agency claims that after its campaign 61% of all consumers are familiar with the product. We ask 7 randomly selected customers whether or not they are familiar with the product. (a) find the probability that, out of 7 customers, exactly 4 are familiar with the product; (b) find the probability that at least 3 customers are familiar with the product; and (c) find the probability that at most 5 are familiar with the product.
15. The mean number of cars per minute going through the Eisenhower turnpike automatic toll is about 7. Find the probability that exactly 3 will go through in a given minute.
Solution: Poisson with average of 7, P(3) = .052
16. 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.
17. On a dry surface, the braking distance (in meters) of a certain car is a normal distribution with µ = 45.1 m and σ = 0.5
(a) Find the braking distance that represents the 91st percentile.
(b) Find the probability that the braking distance is less than or equal to 45 m
(c) Find the probability that the braking distance is greater than
46.8 m
(d) Find the probability that the braking distance is between 45 m and 46.8 m
18. A drug manufacturer wants to estimate the mean heart rate for patients with a certain heart condition. The manufacturer finds 62 people with the condition. From this sample, the mean heart rate is 101 beats per minute with a standard deviation of 8.
(a) Find a 99% confidence interval for the true mean heart rate of all people with this condition.
(b) Interpret this confidence interval and write a sentence that explains it.
(b) We are 99% confident that the true mean heart rate of all people with this heart condition is between 98.39 and 103.62.
19. Determine the minimum required sample size if you want to be
80% confident that the sample mean is within 2 units of the population mean given sigma = 9.4. Assume the population is normally distributed.
20. A social service worker wants to estimate the true proportion of pregnant teenagers who miss at least one day of school per week on average. The social worker wants to be within 5% of the true proportion when using a 90% confidence interval. A previous study estimated the population proportion at 0.21.
(a) Using this previous study as an estimate for p, what sample size should be used?
(b) If the previous study was not available, what estimate for p should be used?
(b) If no estimate of p is known, we must use p = 0.5 to have a large enough sample size to meet the desired maximum error.
21. A restaurant claims that its speed of service time is less than 15 minutes. A random selection of 49 service times was collected, and their mean was calculated to be 14.5 minutes. Their standard deviation is 2.7 minutes. Is there enough evidence to support the claim at alpha = .07. Perform an appropriate hypothesis test, showing each important step. (Note: 1st Step: Write Ho and Ha; 2nd Step: Determine Rejection Region; etc.) [Show Less]