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Chapter 6: Problem 2

Common tests such as the SAT, ACT, Law School Admission test (LSAT), and Medical College Admission Test (MCAT) use multiple choice test questions, each with possible answers of a, b, c, d, e, and each question has only one correct answer. We want to find the probability of getting at least 25 correct answers for someone who makes random guesses for answers to a block of 100 questions. If we plan to use the methods of this section with a normal distribution used to approximate a binomial distribution, are the necessary requirements satisfied? Explain.

Short Answer

Expert verified
The normal distribution can be used because \(n\cdot p = 20\) and \(n\cdot(1-p) = 80\) both satisfy the necessary conditions.

Step by step solution

01

Define the Binomial Distribution

Let each correct answer be a 'success', with the probability of a success denoted as \(p\). Since the answer is chosen randomly from five options (a, b, c, d, e), the probability of guessing correctly is \(p = \frac{1}{5} = 0.2\). The number of questions, \(n\), is 100.
02

Check Criteria for Normal Approximation

For a binomial distribution to be approximated by a normal distribution, the following criteria must be met:1. Both \(n \cdot p > 5\)2. \(n \cdot (1-p) > 5\)
03

Calculate \(n\cdot p\)

Calculate \(n\cdot p\) to see if it is greater than 5:\[n\cdot p = 100 \cdot 0.2 = 20\]
04

Calculate \(n\cdot(1-p)\)

Calculate \(n\cdot(1-p)\) to see if it is greater than 5:\[n\cdot(1-p) = 100 \cdot (1 - 0.2) = 100 \cdot 0.8 = 80\]
05

Compare with Criteria

Both calculations satisfy the criteria for normal approximation: \(20 > 5\) and \(80 > 5\). Therefore, we can use the normal distribution to approximate the binomial distribution in this problem.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

normal approximation
To approximate a binomial distribution using a normal distribution, you need to check whether certain conditions are met. This process is important for simplifying calculations, especially for large numbers of trials.

The conditions for using normal approximation are:
  • The expected number of successes (\( n\u00b7 p \)) should be greater than 5.
  • The expected number of failures (\( n\u00b7 (1 - p) \)) should also be greater than 5.


Once these conditions are satisfied, we can use the normal distribution to estimate probabilities in a binomial setting. This approximation is useful because it allows us to use Z-scores and other properties of the normal distribution to find probabilities.
probability
Probability is a measure of how likely an event is to occur. It ranges from 0 to 1, where 0 means the event cannot happen and 1 means it definitely will happen.
Probability in multiple-choice tests can be calculated by dividing the number of successful outcomes by the total number of possible outcomes.
  • For example, if there are 5 answer choices and only one correct answer, the probability of guessing correctly is\(\frac{1}{5}\), or 0.2.


When calculating probabilities for multiple questions, such as in a test with many questions, we often use a binomial distribution. Each question represents a trial, and the probability of success is constant across trials.
multiple choice tests
Multiple choice tests are common in standardized testing. Each question has a set of possible answers, usually 4 or 5 options, with only one correct answer. When making random guesses, the probability of getting any one question right is equal to the number of correct answers divided by the number of possible choices.

For example:
  • If there are 5 options (a, b, c, d, e), then the probability of guessing correctly is 0.2 or 20%.
When answering multiple questions, the distribution of correct answers can be modeled using a binomial distribution. If the number of questions is large, this binomial distribution may be approximated by a normal distribution for easier calculation.
SAT
The SAT is a standardized test used for college admissions in the United States.

It includes a variety of multiple choice questions that cover different subjects such as critical reading, writing, and mathematics. Each subject has several questions with multiple answer choices. If a student doesn't know the answer and guesses, the probability of getting it correct is relatively low, often around 20% if there are 5 choices per question.

Knowing the normal and binomial distributions can help students understand their expected scores when guessing randomly.
ACT
The ACT is another standardized test widely used for college admissions in the United States, similar to the SAT.

It covers English, mathematics, reading, and science subjects. Each section consists of multiple choice questions. The probability of guessing correctly on any one question is determined by the number of answer choices.

The ACT typically has 4 answer choices per question, so the probability of guessing correctly is 0.25 or 25%. Like the SAT, understanding the probability and distribution of correct answers when guessing can help estimate scores.
LSAT
The Law School Admission Test (LSAT) is used for law school admissions. It assesses reading comprehension, logical, and analytical reasoning.

The test consists of multiple choice questions, usually with 4 or 5 answer options. If a student guesses, the likelihood of a correct answer is 1 divided by the number of choices. For a 5-option question, the probability is 20%.

Using concepts of binomial and normal distributions can help estimate outcomes if a student guesses on a significant number of questions.
MCAT
The Medical College Admission Test (MCAT) is used for medical school admissions. It tests knowledge in biological sciences, physical sciences, and verbal reasoning.

Like other standardized tests, it consists of multiple choice questions. The number of answer choices varies, but there are generally 4 or 5 options. The probability of guessing correctly depends on the number of options.

For a balanced understanding, statistical tools such as the binomial and normal approximations are useful for predicting the expected number of correct answers when guessing.

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Most popular questions from this chapter

Water Taxi Safety Passengers died when a water taxi sank in Baltimore's Inner Harbor. Men are typically heavier than women and children, so when loading a water taxi, assume a worst-case scenario in which all passengers are men. Assume that weights of men are normally distributed with a mean of 189 lb and a standard deviation of 39 lb (based on Data Set 1 "Body Data" in Appendix B). The water taxi that sank had a stated capacity of 25 passengers, and the boat was rated for a load limit of 3500 lb. a. Given that the water taxi that sank was rated for a load limit of 3500 lb, what is the maximum mean weight of the passengers if the boat is filled to the stated capacity of 25 passengers? b. If the water taxi is filled with 25 randomly selected men, what is the probability that their mean weight exceeds the value from part (a)? c. After the water taxi sank, the weight assumptions were revised so that the new capacity became 20 passengers. If the water taxi is filled with 20 randomly selected men, what is the probability that their mean weight exceeds 175 lb, which is the maximum mean weight that does not cause the total load to exceed 3500 lb? d. Is the new capacity of 20 passengers safe?

Use the data in the table below for sitting adult males and females (based on anthropometric survey data from Gordon, Churchill, et al.). These data are used often in the design of different seats, including aircraft seats, train seats, theater seats, and classroom seats. (Hint: Draw a graph in each case.) $$\begin{array}{|l|l|l|l|} \hline & \text { Mean } & \text { St. Dev. } & \text { Distribution } \\ \hline \text { Males } & 23.5 \mathrm{in} . & 1.1 \mathrm{in} . & \text { Normal } \\ \hline \text { Females } & 22.7 \mathrm{in} . & 1.0 \mathrm{in} . & \text { Normal } \\ \hline \end{array}$$ Find the probability that a female has a back-to-knee length greater than 24.0 in.

Constructing Normal Quantile Plots.Use the given data values to identify the corresponding z scores that are used for a normal quantile plot, then identify the coordinates of each point in the normal quantile plot. Construct the normal quantile plot, then determine whether the data appear to be from a population with a normal distribution. Female Arm Circumferences A sample of arm circumferences (cm) of females from Data Set 1 "Body Data" in Appendix \(B: 40.7,44.3,34.2,32.5,38.5 .\)

Do the following: If the requirements of \(n p \geq 5\) and \(n q \geq 5\) are both satisfied, estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution; if \(n p < 5\) or n \(q < 5,\) then state that the normal approximation should not be used. With \(n=8\) births and \(p=0.512\) for a boy, find \(P\) (exactly 5 boys).

Do the following: If the requirements of \(n p \geq 5\) and \(n q \geq 5\) are both satisfied, estimate the indicated probability by using the normal distribution as an approximation to the binomial distribution; if \(n p < 5\) or n \(q < 5,\) then state that the normal approximation should not be used. With \(n=20\) guesses and \(p=0.2\) for a correct answer, find \(P(\text { at least } 6\) correct answers).
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