MATH274 Grand Canyon Week 3 Statistics Binomial Distribution Questions Few questions I need help with, attached is my assignment that does need to be turned in today before midnight. 8.

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The College Board finds that the distribution of students’ SAT scores depends on the level of education their parents have. Children of

parents who did not finish high school have SAT math scores X with mean 449 and standard deviation 103. Scores Y of children of

parents with graduate degrees have mean 564 and standard deviation 105. Perhaps we should standardize to a common scale for equity.

Find numbers a, b, c, and d such that a + bX and c + dY both have mean 500 and standard deviation 100. (Round your answers to two

decimal places.)

a = 64.07

b = 0.97

c = 37.14

d = 0.95

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Slot machines are now video games, with winning determined by electronic random number generators. In the old days, slot machines

were like this: you pull the lever to spin three wheels; each wheel has 25 symbols, all equally likely to show when the wheel stops

spinning; the three wheels are independent of each other. Suppose that the middle wheel has 12 bells among its 25 symbols, and the left

and right wheels have 1 bell each.

(a) You win the jackpot if all three wheels show bells. What is the probability of winning the jackpot? (Round your answer to four

decimal places.)

(b) What is the probability that the wheels stop with exactly 2 bells showing? (Round your answer to four decimal places.)

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It is difficult to conduct sample surveys on sensitive issues because many people will not answer questions if the answers might

embarrass them. Randomized response is an effective way to guarantee anonymity while collecting information on topics such as

student cheating or sexual behavior. Here is the idea. To ask a sample of students whether they have plagiarized a term paper while in

college, have each student toss a coin in private. If the coin lands heads and they have not plagiarized, they are to answer “No.”

Otherwise they are to give “Yes” as their answer. Only the student knows whether the answer reflects the truth or just the coin toss, but

the researchers can use a proper random sample with follow-up for nonresponse and other good sampling practices. Suppose that in fact

the probability is 0.3 that a randomly chosen student has plagiarized a paper. Draw a tree diagram in which the first stage is tossing the

coin and the second is the truth about plagiarism. The outcome at the end of each branch is the answer given to the randomizedresponse question.

0.5

0.7

0.35

0.15

0.5

What is the probability of a “No” answer in the randomized-response poll?

0.35

If the probability of plagiarism were 0.16, what would be the probability of a “No” response on the poll?

0.42

Now suppose that you get 37% “No” answers in a randomized-response poll of a large sample of students at your college. What do you

estimate to be the percent of the population who have plagiarized a paper?

%

74

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Sheila’s doctor is concerned that she may suffer from gestational diabetes (high blood glucose levels during pregnancy). There is

variation both in the actual glucose level and in the blood test that measures the level. A patient is classified as having gestational

diabetes if the glucose level is above 140 milligrams per deciliter (mg/dl) one hour after a sugary drink is ingested. Sheila’s measured

glucose level one hour after ingesting the sugary drink varies according to the Normal distribution with μ = 128 mg/dl and σ = 10 mg/dl.

(a) If a single glucose measurement is made, what is the probability that Sheila is diagnosed as having gestational diabetes?

(Round your answer to four decimal places.)

(b) If measurements are made instead on 3 separate days and the mean result is compared with the criterion 140 mg/dl, what is

the probability that Sheila is diagnosed as having gestational diabetes? (Round your answer to four decimal places.)

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Here is a simple probability model for multiple-choice tests. Suppose that each student has probability p of correctly answering a question

chosen at random from a universe of possible questions. (A strong student has a higher p than a weak student.) The correctness of

answers to different questions are independent. Jodi is a good student for whom p = 0.8.

(a) Use the Normal approximation to find the probability that Jodi scores 74% or lower on a 100-question test. (Round your

answer to four decimal places.)

(b) If the test contains 250 questions, what is the probability that Jodi will score 74% or lower? (Use the normal approximation.

Round your answer to four decimal places.)

(c) How many questions must the test contain in order to reduce the standard deviation of Jodi’s proportion of correct answers to

half its value for a 100-item test?

questions

(d) Laura is a weaker student for whom p = 0.75. Does the answer you gave in (c) for standard deviation of Jodi’s score apply to

Laura’s standard deviation also?

Yes, the smaller p for Laura has no effect on the relationship between the number of questions and the standard deviation.

No, the smaller p for Laura alters the relationship between the number of questions and the standard deviation.

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Does delaying oral practice hinder learning a foreign language? Researchers randomly assigned 23 beginning students of Russian to begin

speaking practice immediately and another 23 to delay speaking for 4 weeks. At the end of the semester both groups took a standard

test of comprehension of spoken Russian. Suppose that in the population of all beginning students, the test scores for early speaking vary

according to the N(35, 7) distribution and scores for delayed speaking have the N(31, 6) distribution.

(a) What is the sampling distribution of the mean score x in the early speaking group in many repetitions of the experiment?

(Round your answers for s to two decimal places.)

Mean =

s =

What is the sampling distribution of the mean score y in the delayed speaking group?

Mean =

s =

(b) If the experiment were repeated many times, what would be the sampling distribution of the difference y – x between the

mean scores in the two groups? (Round your answer for s to two decimal places.)

Mean =

s =

(c) What is the probability that the experiment will find (misleadingly) that the mean score for delayed speaking is at least as

large as that for early speaking? (Round your answer to four decimal places.)

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In each of the following situations, is it reasonable to use a binomial distribution for the random variable X? Give reasons for your answer in

each case. If a binomial distribution applies, give the values of n and p.

(a) A poll of 200 college students asks whether or not you usually feel irritable in the morning. X is the number who reply that they do

usually feel irritable in the morning.

Yes, a binomial distribution is reasonable; B(200, 1/2).

Yes, a binomial distribution is reasonable; B(200, p).

Yes, a binomial distribution is reasonable; B(p, 200).

Yes, a binomial distribution is reasonable; B(1/2, 200).

No, a binomial distribution is not reasonable.

(b) You toss a fair coin until a head appears. X is the count of the number of tosses that you make.

Yes, a binomial distribution is reasonable; B(1/2, n).

Yes, a binomial distribution is reasonable; B(n, p).

Yes, a binomial distribution is reasonable; B(X, p).

Yes, a binomial distribution is reasonable; B(n, 1/2).

No, a binomial distribution is not reasonable.

(c)

Most calls made at random by sample surveys don’t succeed in talking with a person. Of calls to New York City, only one-twelfth

succeed. A survey calls 500 randomly selected numbers in New York City. X is the number of times that a person is reached.

Yes, a binomial distribution is reasonable; B(500, 1/2).

Yes, a binomial distribution is reasonable; B(1/2, 500).

Yes, a binomial distribution is reasonable; B(1/12, 500).

Yes, a binomial distribution is reasonable; B(500, 1/12).

No, a binomial distribution is not reasonable.

(d) You deal 10 cards from a shuffled deck of standard playing cards and count the number X of black cards.

Yes, a binomial distribution is reasonable; B(1/2, 10).

Yes, a binomial distribution is reasonable; B(1/4, 10).

Yes, a binomial distribution is reasonable; B(10, 1/2).

Yes, a binomial distribution is reasonable; B(10, 1/4).

No, a binomial distribution is not reasonable.

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“What do you think is the ideal number of children for a family to have?” A Gallup Poll asked this question of 1016 randomly chosen

adults. Almost half (49%) thought two children was ideal.† We are supposing that the proportion of all adults who think that two children

is ideal is p = 0.49.

What is the probability that a sample proportion �� falls between 0.46 and 0.52 (that is, within ±3 percentage points of the true p) if the

sample is an SRS of size n = 200? (Round your answer to four decimal places.)

What is the probability that a sample proportion �� falls between 0.46 and 0.52 if the sample is an SRS of size n = 5000? (Round your

answer to four decimal places.)

Combine these results to make a general statement about the effect of larger samples in a sample survey.

Larger samples give a larger probability that �� will be close to the true proportion p.

Larger samples give a smaller probability that �� will be close to the true proportion p.

Larger samples have no effect on the probability that �� will be close to the true proportion p.

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The standard deviation of a sample proportion �� gets smaller as the sample size n increases. If the population proportion is p = 0.45,

how large a sample is needed to reduce the standard deviation of �� to σ = 0.004? (The 68−95−99.7 rule then says that about 95% of

all samples will have �� within 0.01 of the true p. Round your answer to up to the next whole number.)

17.

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The number of flaws per square yard in a type of carpet material varies with mean 1.6 flaws per square yard and standard deviation 1.2

flaws per square yard. This population distribution cannot be normal, because a count takes only whole-number values. An inspector

studies 167 square yards of the material, records the number of flaws found in each square yard, and calculates x, the mean number of

flaws per square yard inspected. Use the central limit theorem to find the approximate probability that the mean number of flaws exceeds

1.7 per square yard. (Round your answer to four decimal places.)

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“Durable press” cotton fabrics are treated to improve their recovery from wrinkles after washing. Unfortunately, the treatment also

reduces the strength of the fabric. The breaking strength of untreated fabric is normally distributed with mean 51.4 pounds and standard

deviation 2.7 pounds. The same type of fabric after treatment has normally distributed breaking strength with mean 17.3 pounds and

standard deviation 1.6 pounds. A clothing manufacturer tests 4 specimens of each fabric. All 8 strength measurements are independent.

(Round your answers to four decimal places.)

(a) What is the probability that the mean breaking strength of the 4 untreated specimens exceeds 50 pounds?

(b) What is the probability that the mean breaking strength of the 4 untreated specimens is at least 25 pounds greater than the

mean strength of the 4 treated specimens?

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Typographic errors in a text are either nonword errors (as when “the” is typed as “teh”) or word errors that result in a real but incorrect

word. Spell-checking software will catch nonword errors but not word errors. Human proofreaders catch 70% of word errors. You ask a

fellow student to proofread an essay in which you have deliberately made 10 word errors. What is the smallest number of misses m with

P(X ≥ m) no larger than 0.05? You might consider m or more misses as evidence that a proofreader actually catches fewer than 70% of

word errors.

misses

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The unique colors of the cashmere sweaters your firm makes result from heating undyed yarn in a kettle with a dye liquor. The pH

(acidity) of the liquor is critical for regulating dye uptake and hence the final color. There are 5 kettles, all of which receive dye liquor

from a common source. Past data show that pH varies according to a Normal distribution with μ = 4.95 and σ = 0.151. You use statistical

process control to check the stability of the process. Twice each day, the pH of the liquor in each kettle is measured, giving a sample of

size 5. The mean pH x is compared with “control limits” given by the 99.7 part of the 68−95−99.7 rule for normal distributions, namely

μ x ± 3σ x. What are the numerical values of these control limits for x? (Round your answers to three decimal places.)

4.747

(smaller value)

5.152

(larger value)

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