Skip to main content
Ch. 8 - Integration Techniques
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 8 - Integration TechniquesProblema 8.8.64
Capitolo 8, Problema 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)?

Guida verificata passo dopo passo
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.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
12m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

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.
Video consigliato:
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.
Video consigliato:
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.
Video consigliato:
Percorso guidato
05:43
Definition of the Definite Integral
Pratica correlata
Domanda del libro di testo

108. Draining a tank Water is drained from a 3000-gal tank at a rate that starts at 100 gal/hr and decreases continuously by 5%/hr. If the drain is left open indefinitely, how much water drains from the tank? Can a full tank be emptied at this rate? 

38
views
Domanda del libro di testo

76–83. Preliminary steps The following integrals require a preliminary step such as a change of variables before using the method of partial fractions. Evaluate these integrals.

76. ∫ [cosθ / (sin³θ - 4sinθ)] dθ

44
views
Domanda del libro di testo

Visual proof Let F(x)=∫₀ˣ √(a²−t²) dt. The figure shows that F(x)= area of sector OAB+ area of triangle OBC.

a. Use the figure to prove that

F(x) = (a² sin ⁻¹(x/a))/2 + x√(a²−x²)/2

b. Conclude that ∫ √(a²−x²) dx = (a² sin ⁻¹(x/a))/2 + x√(a²−x²)/2 + C.

47
views
Domanda del libro di testo

9–40. Integration by parts Evaluate the following integrals using integration by parts.

40. ∫ e^√x dx

48
views
Domanda del libro di testo

7–84. Evaluate the following integrals.

25. ∫ [1 / (x√(1 - x²))] dx

89
views
Domanda del libro di testo

7–58. Improper integrals Evaluate the following integrals or state that they diverge.

50. ∫ (from 0 to 9) 1/(x - 1)¹ᐟ³ dx

71
views