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Statistics Final Exam Review Packet: Chapters 7-12

Show your work clearly with solutions or calculator commands. ex: normalcdf (Z, e99)

Chapter 7 : Random Variables

Mean of a Discrete Random Variable: [pic][pic]

Concept:

Differences between discrete and continuous random variables. Law of large numbers.

1. A small store keeps track of the number X of customers that make a purchase during the first hour that the store is open each day. Based on the records, X has the following probability distribution.

(a) Construct a probability histogram of X using the space below.

(b) What is the probability that no customers make a purchase during the first hour that the store is open?

(c) What is the probability that the number of customers that make a purchase during the first hour that the store is open is more than 1 but no greater than 3?

(d) Find the mean number of customers that make a purchase during the first hour that the store is open. Show products used to arrive at your answer.

(e) Find [pic]

(f) Find [pic]

(g) Circle the correct answer: This distribution is an example of a ( discrete ) ( continuous ) random

variable.

2. I toss a fair coin a large number of times. Assuming tosses are independent, which of the following

is true?

A) Once the number of flips is large enough (usually about 10,000) the number of heads will

always be exactly half the total of the number of tosses. For example, after 10,000 tosses I

B) The proportion of heads will be about 1/2, and this proportion will tend to get closer and closer

to 1/2 as the number of tosses increases.

C) As the number of tosses increases, any long run of heads will be balanced by a corresponding

run of tails so that the overall proportion of heads is 1/2.

D) All of the above.

3. A lottery offers one \$1,000 prize, one \$500 prize, and five \$100 prizes. One thousand tickets are sold at \$3 each.

a) Make a probability model of the player’s winning amount and probabilities.

b) What is player’s expected winning amount per bet (hint: find mean)?

c) In the long run, how much does the lottery company make from each single bet?

Chapter 8 : Binomial and Geometric Distributions

Binomial mean and standard deviation: Geometric mean and standard deviation:

[pic] [pic] [pic] [pic]

binompdf or binomcdf (n, p, x) geometpdf or geometcdf (p, x)

Concept:

Differences between binomial and geometric distributions. Normal approximation.

4. Identify whether the following setting is binomial, geometric, or neither. Explain clearly with the four requirements.

(a) Draw a card from a standard deck of 52 playing cards, observe the card, and replace the card within the deck. Count the number of times you draw a card in this manner until you observe a jack.

(b) UPS Company advertises that it ships 90% of its order within three working days. You select a random sample of 100 mail orders received in the past week and examine the probability that 90 of them were shipped on time.

5. In a large population of college students, 20% of the students have experienced feelings of math anxiety. If you take a random sample of 10 students from this population, the probability that exactly 2 students have experienced math anxiety is

A) 0.3020 C) 0.2013 E) 1

B) 0.2634 D) 0.5 F) None of the above

6. Refer to the previous problem # 5. Determine the standard deviation of the number of students in the sample who have experienced math anxiety.

7. A manufacturer produces a large number of toasters. From past experience, the manufacturer knows that approximately 2% are defective. In a quality control procedure, we randomly select 20 toasters for testing. We want to determine the probability that the number of these toasters is defective.

(a) Is this a binomial or geometric distribution? Explain clearly with four requirements.

(b) Determine the probability that exactly one of the toasters is defective.

(c) Find the probability that at most two of the toasters are defective.

(d) Find the mean and standard deviation for the random variable X in the toaster problem.

8. What rules of thumb must be satisfied in order to use the normal approximation to the binomial

distribution?

A) [pic] B) [pic] C) [pic] D) none of the above

9. When a computerized generator is used to generate random digits, the probability that any particular digit in the set {0, 1, 2, . . . , 9} is generated on any individual trial is 1/10 = 0.1. Suppose that we are generating digits one at a time and are interested in tracking the first occurrence of the digit 0.

(a) Is this a binomial or geometric distribution? Explain clearly with four requirements.

(b) Determine the probability that the first 0 occurs as the fifth random digit generated.

(c) How many random digits would you expect to have to generate in order to observe the first 0?

Chapter 9: Sampling Distributions

Sampling Distribution of [pic]: Sampling Distribution of [pic]:

1. [pic] 1. [pic]

2. [pic] 2. [pic]

Concepts:

1. Differences between parameter and statistics.

2. Differences between sample distributions for sample means and sample proportions.

3. Central Limit Theorem

10. Identify each of the bolded figures in the problem below as statistics or parameters. Use the appropriate notation to indicate each statistic and parameter.

A 1993 survey conducted by the local paper in Columbus, Ohio, one week before election day asked voters who they would vote for in the City Attorney’s race. Thirty-seven percent said they would vote for the Democratic candidate. On election day, 41 % actually voted for the Democratic candidate.

11. The distribution of the values taken on by a statistic in all possible samples from the same

population is called

A) the parameter. C) the sampling distribution.

B) bias. D) a table of random digits.

12. Below are histograms of the values taken by three sample statistics in several hundred samples from the same population. The true value of the population parameter is marked with an arrow.

[pic]

(a) Which statistic has the largest bias among these three? Explain.

(b) Which statistic has the lowest variability among these three? Explain.

(c) Which statistic has the largest sample size? Explain.

(d) Based on the performance of the three statistics in many samples, which is preferred as an estimate of the parameter? Why?

13. Power companies kill trees growing near their lines to avoid power failures due to falling limbs in storms. Applying a chemical to slow the growth of the trees is cheaper than trimming, but the chemical kills some of the trees. Suppose that one such chemical would kill 20% of sycamore trees. The power company tests the chemical on 250 sycamores.

(a) What are the unbiased mean and standard deviation of the sampling proportion of trees that are killed?

(b) Can you use the normal approximation for this study? Explain clearly with the two rules.

(c) What is the probability that at least 60 trees (24% of the sample) are killed?

14. The weights of newborn children in the United States vary according to the normal distribution with mean 7.5 pounds and standard deviation 1.25 pounds. The government classifies a newborn as having low birth weight if the weight is less than 5.5 pounds.

(a) What is the probability that a baby chosen at random weighs less than 5.5 pounds at birth?

(b) What is the probability that three randomly chosen babies have average birth weight less than 5.5 pounds?

Chapter 10: Statistical Inference

Margin of Error: [pic]

Concepts:

1. Stating null ([pic]) and alternate ([pic]) hypotheses.

2. Confidence Interval

3. Z test and interpret your results with p-value and α level.

4. Type I error (α error): falsely rejecting [pic]

5. Type II error (β error): falsely accepting [pic]

15. Suppose you administer a certain aptitude test to a random sample of 9 students in your school, and that the average score is 105. We want to determine the mean of the population of all students in the school. Suppose the population standard deviation is 15.

(a) What is the critical value, z*, for a 98% confidence interval (2 answers)?

(b) Calculate a 98% confidence interval for the mean score for the whole school. Sketch a graph to display the center, confidence interval and 3 probabilities. Circle the margin of error.

(c) Calculate a 90% confidence interval for the mean score for the whole school. Sketch a graph to display the center, confidence interval and 3 probabilities. Circle the margin of error.

(d) Compare your answers in (b) and (c), what is the relationship between margins of error and confidence levels?

16. It is believed that the average amount of money spent per U.S. household per week on food is about \$110, with standard deviation \$25. A random sample of 200 households in a certain affluent community yields a mean weekly food budget of \$105. We want to test the hypothesis that the mean weekly food budget for all households in this community is lower than the national average.

(a) State two hypotheses in notations and perform a significance test at the [pic] significance level. Interpret your results and state your conclusion clearly.

(b) Describe a Type I error in the context of this problem. What is the probability of making a Type I error?

(c) Describe a Type II error in the context of this problem.

17. At the bakery where you work, loaves of bread are supposed to weigh 1 pound, with standard deviation σ = 0.11 pounds. You believe that new personnel are producing loaves that differ from 1 pound. As supervisor of Quality Control, you want to test your hypotheses at the 0.05 significance level. You weigh 20 loaves and obtain the following weights, in pounds:

0.98 0.99 1.08 1.12 1.04 1.06 1.05 1.08 1.03 1.04

1.07 1.07 1.06 1.03 0.99 1.05 1.11 1.12 1.00 1.03

a) State the hypotheses both in words and symbols.

b) Find the sample mean.

c) Calculate the P-Value.

d) Is the result significant at the [pic] level? What is your conclusion?

e) Sketch a normal curve for your significance test. Mark the suggested population mean, sample mean, sampling standard deviation, P-value, and [pic]. Shade the area under the curve to show the P-value.

[pic]

18. A machine is designed to fill 16-ounce bottles of shampoo with a standard deviation of 0.1 ounce if the machine is working properly. Four bottles are randomly selected each hour and the number of ounces in each bottle is measured. The sample mean amount poured into the bottles is 16.05 ounces. What is 95% of the observations should occur in which interval?

(A) 16.05 and 16.15 ounces

(B) –.30 and +.30 ounces

(C) 15.95 and 16.15 ounces

(D) 15.90 and 16.20 ounces

(E) None of the above

Chapter 11 and 12: More about Statistical Inference

19. Data is collected from a certain medical sample. If the sample mean from a simple random sample

of 28 observations is11.4 and the sample standard deviation s = 1.7, what is the standard error?

20. Hallux abducto valgus (call it HAV) is a deformation of the big toe that is uncommon in youth and often requires surgery. Doctors used X-rays to measure the angle (in degrees) of deformity in 38 patients under the age of 21 who came to a medical center for surgery to correct HAV.

The angle is a measure of the seriousness of the deformity. Here are the data:

28 32 25 34 38 26 25 18 30 26 28 13 20 21 17 16 21 23 14 32 25 21 22 20 18 26 16 30 30 20 50 25 26 28 31 38 32 21

(a) We are willing to consider these patients as a random sample of young patients who require HAV

surgery. Give a 95 % confidence interval for the mean HAV angle in the population of all such

patients. Sketch a graph to display the center, confidence interval and 3 probabilities. Circle the

margin of error.

(b) Is there convincing evidence that the mean reading of the 38 angles of deformity collected in the

previous exercise is less than the expected population mean of 28?

State your hypotheses clearly in words and notations. Carry out a t-test, find the p-value, interpret your result at 0.05 significance level and state your conclusion clearly.

(c) Based on your result in (b), sketch a normal curve for your significance test. Mark the suggested population mean, sample mean, sampling standard deviation, P-value, and [pic]. Shade the area under the curve to show the P-value.

[pic]

21. Which of the following is an example of a matched-pairs design?

A) A track coach compares the long jump measurements of her upperclassmen

long jumpers with those of her underclassmen long jumpers.

B) A track coach compares the long jump measurements of her long jumpers

with the long jump measurements of a near-by high school’s track team.

C) A track coach compares the long jump measurements of her long jumpers

before and after a workshop on long jumping techniques.

D) A track coach compares the long jump measurements of her long jumpers

with nationally ranked long jumpers.

22. Mutual fund performance. Many mutual funds compare their performance with that of a benchmark, an index of the returns on all securities of the kind the fund buys. The Vanguard International Growth Fund, for example, takes as its benchmark the Morgan Stanley EAFE (Europe, Australasia, Far East) index of overseas stock market performance. Here are the percent returns for the fund and for the EAFE from 1982 (the first full year of the fund’s existence) to 2000. Does the fund significantly outperform its benchmark (EAFE)?

[pic]

a) Explain these hypotheses in words:

Ho: µ1 = µ2

H1: µ1 > µ2

(b) Complete the following table by entering the data into your graphing calculator.

| | |# of years ( ) |Sample Mean of Returns ( ) |Sample Standard Deviation ( ) |

|1 |Funds | | | |

|2 |EAFE | | | |

(c) Find the combined standard error.

d) T-test: find t score and P-value.

e) Interpret your results at significance level 0.05. State your conclusion clearly.

23. Do students tend to improve their Math SAT score the second time they take the test? A random sample of four students who took the test twice received the following scores:

Student 1 2 3 4__

First score 450 520 720 600

Second score 440 600 720 630

Assume that the change in SAT math scores (second score - first score) for the

population of all students taking the test twice is normally distributed with mean [pic].

You wish to test the hypotheses: [pic]. The p-value for the test is

A) 0.304 B) 0.386 C) 0.152. D) 0.2328.

24. In a matched-pairs experiment, 50 subjects tasted two unmarked cups of coffee (brand A and B) and said which he/she prefers. 31 of them prefer brand A coffee.

(a) Determine a 95 % confidence interval for this sample proportion. Sketch a graph to display the center, confidence interval and 3 probabilities. Circle the margin of error.

(b) Is there evidence to conclude that more than 50% of people prefer brand A coffee? Make two hypotheses, perform a Z-test, find P-value, interpret your result at significance level 0.05 and state your conclusion clearly.

(c) Based on your result in (b), sketch a normal curve for your significance test. Mark the suggested population proportion, sample proportion, sampling standard deviation, P-value, and [pic]. Shade the area under the curve to show the P-value.

[pic]

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