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Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 8 - Integration Techniques문제 8.8.64
8장, 문제 8.8.64

64. (Use of Tech) Normal distribution of movie lengths
A study revealed that the lengths of U.S. movies are normally distributed with a mean of 110 minutes and a standard deviation of 22 minutes. This means that the fraction of movies with lengths between a and b minutes (with a < b) is given by the integral:
(1/(22√(2π))) ∫[a to b] e^(-((x-110)/22)²/2) dx.
What percentage of U.S. movies are between 1 hr and 1.5 hr long (60-90 min)?

검증된 단계별 안내
1
Step 1: Recognize that the problem involves calculating the percentage of movies within a certain range using the normal distribution formula. The integral provided represents the probability density function for a normal distribution.
Step 2: Convert the given time range (60 to 90 minutes) into the integral bounds 'a' and 'b'. Here, 'a' = 60 and 'b' = 90.
Step 3: Standardize the bounds using the z-score formula: z = (x - μ) / σ, where μ is the mean (110 minutes) and σ is the standard deviation (22 minutes). Compute the z-scores for both bounds: z₁ = (60 - 110) / 22 and z₂ = (90 - 110) / 22.
Step 4: Use a technology tool (such as a graphing calculator or software like Python, MATLAB, or WolframAlpha) to evaluate the integral of the normal distribution function between the standardized z-scores. Alternatively, use the cumulative distribution function (CDF) of the standard normal distribution to find the probabilities corresponding to z₁ and z₂, and subtract P(z₁) from P(z₂).
Step 5: Multiply the resulting probability by 100 to convert it into a percentage. This percentage represents the fraction of U.S. movies with lengths between 60 and 90 minutes.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, depicting that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve, defined by its mean and standard deviation. In this context, the lengths of U.S. movies follow a normal distribution with a mean of 110 minutes and a standard deviation of 22 minutes, which allows for the calculation of probabilities for different ranges of movie lengths.
추천 영상:
06:11
Limits of Rational Functions: Denominator = 0

Standard Normal Variable

A standard normal variable is a normal random variable that has been standardized to have a mean of 0 and a standard deviation of 1. This is achieved through the transformation z = (x - μ) / σ, where μ is the mean and σ is the standard deviation. In the context of the movie lengths, converting the lengths to a standard normal variable allows us to use standard normal distribution tables or software to find probabilities associated with specific ranges of movie lengths.
추천 영상:
07:15
Separation of Variables

Definite Integral

A definite integral calculates the accumulation of quantities, such as area under a curve, between two specified limits. In this case, the integral represents the probability of movie lengths falling between 60 and 90 minutes. Evaluating this integral involves finding the area under the normal distribution curve between these two points, which corresponds to the percentage of movies that fall within this length range.
추천 영상:
05:43
Definition of the Definite Integral
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