Domanda del libro di testo
When all other quantities remain the same, how does the indicated change affect the width of a confidence interval? Explain.
c. Increase in the population standard deviation
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Ch. 6 - Confidence Intervals
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 6, Problema 6.1d
In a survey of 2096 U.S. adults, 1740 think football teams of all levels should require players who suffer a head injury to take a set amount of time off from playing to recover. (Adapted from The Harris Poll)
d. Find the minimum sample size needed to estimate the population proportion at the 99% confidence level to ensure that the estimate is accurate within 3% of the population proportion.
Guida verificata passo dopo passo1
Step 1: Identify the key components of the problem. The confidence level is 99%, the margin of error (E) is 3% or 0.03, and the sample proportion (p̂) is calculated as the number of favorable responses divided by the total sample size: p̂ = 1740 / 2096.
Step 2: Recall the formula for the minimum sample size needed to estimate a population proportion: n = (Z² * p̂ * (1 - p̂)) / E². Here, Z is the critical value corresponding to the 99% confidence level.
Step 3: Determine the critical value (Z) for a 99% confidence level. Using a Z-table or standard normal distribution, the Z-value for 99% confidence is approximately 2.576.
Step 4: Substitute the values into the formula. Use p̂ from Step 1, Z = 2.576, and E = 0.03. The formula becomes: n = (2.576² * p̂ * (1 - p̂)) / 0.03².
Step 5: Simplify the expression to calculate the minimum sample size. Ensure that the result is rounded up to the nearest whole number, as sample size must be an integer.
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Population Proportion
The population proportion refers to the fraction of a population that exhibits a certain characteristic. In this context, it represents the percentage of U.S. adults who believe that football players should take time off after a head injury. Understanding this concept is crucial for estimating how representative a sample is of the larger population.
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Percorso guidato
02:35
Finding a Confidence Interval for a Population Proportion Using a TI84
Sample Size Calculation
Sample size calculation is a statistical method used to determine the number of observations or replicates needed to ensure that the results of a survey or experiment are reliable. It takes into account the desired confidence level, margin of error, and the estimated population proportion. In this case, it helps to find the minimum number of respondents needed to accurately estimate the population proportion within a specified margin of error.
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05:11Sampling Distribution of Sample Proportion
Confidence Level
The confidence level indicates the degree of certainty that the population parameter lies within the confidence interval calculated from the sample data. A 99% confidence level means that if the same survey were conducted multiple times, 99% of the calculated intervals would contain the true population proportion. This high level of confidence is important for making informed decisions based on survey results.
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06:33Introduction to Confidence Intervals
Pratica correlata
Domanda del libro di testo
Finite Population Correction Factor In Exercises 57 and 58, use the information below.
In this section, you studied the construction of a confidence interval to estimate a population mean. In each case, the underlying assumption was that the sample size n was small in comparison to the population size N. When n ≥ 0.05N however, the formula that determines the standard error of the mean needs to be adjusted, as shown below.
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Recall from the Section 5.4 exercises that the expression sqrt[(N-n)/(n-1)] is called a finite population correction factor. The margin of error is
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Use the finite population correction factor to construct each confidence interval for the population mean.
c. c = 0.95, xbar = 40.3, σ = 0.5, N = 300, n = 68.
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Domanda del libro di testo
Constructing a Confidence Interval In Exercises 25–28, use the data set to (c) construct a 99% confidence interval for the population mean. Assume the population is normally distributed.
Homework The weekly time spent (in hours) on homework for 18 randomly selected high school students
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