Domanda del libro di testo
In Exercises 6–11, find the indicated area under the standard normal curve. If convenient, use technology to find the area.
To the left of z = 0.72
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Ch. 5 - Normal Probability Distributions
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
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TOC.tableOfContents
- Ch. 1 - Introduction to Statistics87
- Ch. 2 - Descriptive Statistics176
- Ch. 3 - Probability184
- Ch. 4 - Discrete Probability Distributions102
- Ch. 5 - Normal Probability Distributions183
- Ch. 6 - Confidence Intervals154
- Ch. 7 - Hypothesis Testing with One Sample165
- Ch. 8 - Hypothesis Testing with Two Samples134
- Ch. 9 - Correlation and Regression87
- Ch. 10 - Chi-Square Tests and the F-Distribution87
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Non è quello che usi tu?Cambia libro di testo
Capitolo 5, Problema 5.CR.1c
A survey of adults in the United States found that 61% ate at a restaurant at least once in the past week. You randomly select 30 adults and ask them whether they ate at a restaurant at least once in the past week. (Source: Gallup)
c. Is it unusual for exactly 14 out of 30 adults to have eaten in a restaurant at least once in the past week? Explain your reasoning.
Guida verificata passo dopo passo1
Step 1: Recognize that this is a binomial probability problem. The survey indicates that the probability of success (an adult eating at a restaurant at least once in the past week) is p = 0.61, and the probability of failure is q = 1 - p = 0.39. The number of trials is n = 30, and we are interested in the probability of exactly x = 14 successes.
Step 2: Use the binomial probability formula to calculate the probability of exactly 14 successes: P(X = x) = (n choose x) * p^x * q^(n-x). Here, (n choose x) is the binomial coefficient, which can be calculated as (n! / (x! * (n - x)!)).
Step 3: Calculate the mean (μ) and standard deviation (σ) of the binomial distribution to assess whether 14 is unusual. The mean is given by μ = n * p, and the standard deviation is given by σ = sqrt(n * p * q).
Step 4: Determine the range of usual values using the rule of thumb: usual values typically fall within μ ± 2σ. Calculate the lower bound (μ - 2σ) and the upper bound (μ + 2σ).
Step 5: Compare the value of x = 14 to the range of usual values. If 14 falls outside the range, it is considered unusual. If it falls within the range, it is not unusual. Provide reasoning based on this comparison.
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Binomial Distribution
The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this context, each adult's response to whether they ate at a restaurant can be seen as a trial, with 'success' defined as having eaten at a restaurant. The parameters for this distribution are the number of trials (30 adults) and the probability of success (61% or 0.61).
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Percorso guidato
03:28
Mean & Standard Deviation of Binomial Distribution
Normal Approximation
For large sample sizes, the binomial distribution can be approximated by a normal distribution, which simplifies calculations. This approximation is valid when both np and n(1-p) are greater than 5. In this case, with n=30 and p=0.61, we can use the normal distribution to assess the likelihood of observing exactly 14 successes.
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06:23Using the Normal Distribution to Approximate Binomial Probabilities
Unusual Events
An event is considered unusual if its probability of occurrence is less than 5%. To determine if having exactly 14 out of 30 adults eating at a restaurant is unusual, we can calculate the probability of this outcome using the binomial distribution or its normal approximation. If this probability is below 0.05, we would classify it as unusual.
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05:54Probability of Multiple Independent Events
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In Exercises 6–11, find the indicated area under the standard normal curve. If convenient, use technology to find the area.
Between z = 0 and z = 2.95
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