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Statistical Reasoning Questions

I need some help with these statistic problems, with step-by-step detailed answers for my understanding:
Item 1

Two students are in the same mathematics class. On 14 out of 15 quizzes, student A has outscored student B by at least 10 points. Which of the following statements best describes this situation? Explain your reasoning

There is ample evidence to prove that by the end of the year, student A will have outscored student B on at least 28 of the 30 quizzes.

There is ample evidence to prove that by the end of the year, student A will have outscored student B on at least 20 of the 30 quizzes.

There is ample evidence to prove that by the end of the year, student A will have outscored student B on at least 15 of the 30 quizzes.

There is not sufficient evidence to prove that student A will outscore student B on any set number of additional quizzes.

Item 2
There are only hens and mice in a barnyard. There are, altogether, 50 heads and 140 legs among the animals. How many hens and mice are there in the barnyard? Explain your reasoning

30 mice; 20 hens

20 mice; 30 hens

25 mice; 25 hens

It cannot be determined from the information given.

Item 3
An economist claims that the price of the average family car has been increasing at a rate of about 5% per year for the last 5 years, substantially more than the rate of inflation. If the economist is correct, and if the cost of an average family car this year is $32,000, the cost of the average family car 5 years ago was closest to

$27,000 , $25,000, $24,000 $22,000 Explain your reasoning

Item 4
There is an interesting study on the genders of children born to families with exactly 12 children in a certain region of France. When we calculate the numbers of families we would expect to have, say, 8 boys and 4 girls, we usually say that the situation is like tossing a fair coin 12 times and figuring out the probability of getting 4 heads and 8 tails. This reasoning is an example of problem restatement, working backwards, arguing by analogy, indirect reasoning
Explain your reasoning

Item 5
It is estimated that the peak available energy available in the foreseeable future for New Zealand is approximately 90 terawatts. Currently, peak usage is 45 terawatts but is increasing about 7% per year. In how many years will New Zealand's peak demand for electricity exceed its peak available supply? Explain your reasoning

Item 6
Vodka is 45% alcohol by volume. If a quart of vodka is mixed with orange juice, how many quarts of orange juice are needed to create a mixture that is only 10% alcohol by volume? Explain your reasoning

Item 7
Which of the following is the inverse of the function f(x) = 3x + 7 ? Explain your reasoning

Item 8
Police use the square of the speed in mph of a car divided by 24 to determine the length of skid marks in feet from where the brakes are first applied. If a car leaves skid marks 60 feet long, what is the speed of the car when the brakes were first applied? Explain your reasoning

Item 9
In 1985, research and development expenses in the United States were $102 billion. Following 1985, expenses increased at a linear rate of $1.6 billion per year. What is the the U.S. research and development expenditures for the year 1988? Explain your reasoning

Item 10
A state is developing a funding formula for the three community colleges in the state. The total amount of money available to be distributed among the three schools is $190 million per year. Colleges B and C have the same enrollment, and so it is deemed that they should receive the same funding. College A also has the same enrollment as colleges B and C, but because of a need for capital improvements, it is determined that college A should get $10 million more than either colleges B or C. How much will each community college receive per year? Explain your reasoning

Item 11
Which of the following values of a, b and c prove the following statement is NOT true for all values of a, b and c? Explain your reasoning

a = b = c = 0

a = b = c = 1

a = 0, b = c = 1

a = 4, b = 2, c = 3

Item 12
The PTA is selling drinks and candy at a high school football game to raise $500 for the purchase of calculators. The PTA makes a profit of 75 cents on each beverage sold and 45 cents on each candy item. If the PTA sells 400 beverages, what is the minimum number of candy items that must sell to reach the $500 goal? Explain your reasoning

Item 13
In a quality control experiment, the mean, standard deviation, and median of 25 measurements on an industrial process are computed. The values of these three statistics are (rounded to 2 decimal places): Mean = 11.50; Median = 7.77; Standard Deviation = 20.47. When the data is examined, it becomes apparent that the largest value has been recorded as 109.2 instead of its true value of 10.92. When the data is corrected and the three statistics are recomputed, which of the following statements is true concerning the mean, median, and standard deviation? Explain your reasoning

All three statistics will be smaller.

The mean and the median will be smaller, and the standard deviation will be the same as before.

The mean and the median will be the same as before, and the standard deviation will be smaller.

The mean and the standard deviation will be smaller, and the median will be the same as before.

Item 14

In a large introductory statistics class, the average score on the midterm was 60 out of 100, and the standard deviation was 12. The 10 students with the lowest scores on the first midterm (all scored about 30) are invited to participate in a new intensive tutoring program. In the second midterm, the class average is 68 and the standard deviation is 12. The correlation between the first and second midterm scores is about 0.7. The students who got the special tutoring averaged 78 out of 100 on the second midterm, and all of them passed the second midterm. The tutoring program was declared a spectacular success. Which of the following statements is correct? Explain your reasoning

The improvement in the tutored students' scores is due only to the higher class average on the second midterm.

The improvement on the second midterm for the tutored students is an illustration of the regression effect that one always observes in test-retest situations.

The improvement in the scores may be explained by chance.

The tutoring program really did help the participants.

Item 15
What is the probability of randomly choosing a green marble from a bag containing 2 blue marbles, 5 green marbles, and 7 yellow marbles? Explain your reasoning

Item 16
If 2 fair dice are rolled, what is the probability that at least 1 die will display 6 spots? Explain your reasoning

Item 17
If a fair coin is tossed, and the probability of this coin landing with "heads" up is 50%, which of the following is the best interpretation of this percentage using the Law of Large Numbers? Explain your reasoning

In 10 tosses of the coin, 5 heads will be obtained.

As the coin is tossed over and over again, the percentage of heads will eventually get closer and closer to 50% of the outcomes.

In 100 tosses of the coin, there is a 50% chance that one half of the outcomes will be a head.

For a large number of tosses, the percentage of heads will never exceed 50%.

Item 18
A researcher wants to collect information from high school children randomly selected from four high schools within a school district. Within each school, the researcher randomly chooses two English classes in each of the 10th, 11th, and 12th grades. From each class selected, the researcher obtains information from every student in the class. Which of the following sampling techniques is being utilized? Explain your reasoning



Simple random

Complex random

Item 19
Which of the following values represents the probability of getting between 45 and 55 heads (both endpoints included) in 100 tosses of a fair coin? Explain your reasoning





Item 20

A random sample of 500 households was selected from a large city. The average household income of the sample was $25,000 per year. Based on this sample, which of the following options is a reasonable conclusion? Explain your reasoning

The average household income for the entire city is exactly $25,000 per year.

All possible samples of size 500 households will have an average household income of $25,000 per year.

The average household income for the city is over $20,000 per year.

A random sample of 1,000 households will have an average household income of $50,000 per year.

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Solution Summary

This solution contains detailed step-by-step calculations and reasoning to the statistical reasoning problems. Clear explanations are included for further understanding.