Textbook Question
In Exercises 5–8, find the critical value zc necessary to construct a confidence interval at the level of confidence c.
c = 0.97
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Ch. 6 - Confidence Intervals
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
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Table of contents
- 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: 9780137493470Not the one you use?Change textbook
Chapter 6, Problem 6.2.7
In Exercises 7 and 8, find the margin of error for the values of c, s, and n.
c = 0.95, s = 5, n = 16
Verified step by step guidance1
Step 1: Understand the formula for the margin of error (ME). The formula is ME = z * (s / sqrt(n)), where z is the critical value corresponding to the confidence level (c), s is the sample standard deviation, and n is the sample size.
Step 2: Determine the critical value (z) for the given confidence level (c = 0.95). For a 95% confidence level, the z-value can be found using a z-table or statistical software. It corresponds to the area under the standard normal curve.
Step 3: Calculate the standard error (SE) of the sample mean using the formula SE = s / sqrt(n). Substitute the given values of s = 5 and n = 16 into the formula.
Step 4: Multiply the critical value (z) by the standard error (SE) to compute the margin of error (ME). This step combines the results from Step 2 and Step 3.
Step 5: Interpret the margin of error in the context of the problem. It represents the range within which the true population mean is expected to lie with 95% confidence.
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Margin of Error
The margin of error quantifies the uncertainty in a statistical estimate. It indicates the range within which the true population parameter is expected to lie, given a certain confidence level. A smaller margin of error suggests a more precise estimate, while a larger margin indicates more variability. It is commonly calculated using the formula: Margin of Error = z * (s / √n), where z is the z-score corresponding to the desired confidence level.
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Finding the Minimum Sample Size Needed for a Confidence Interval
Confidence Level
The confidence level represents the probability that the margin of error will contain the true population parameter. It is expressed as a percentage, with common levels being 90%, 95%, and 99%. A higher confidence level means a wider margin of error, reflecting greater uncertainty about the estimate. In this case, c = 0.95 indicates a 95% confidence level, meaning we can be 95% confident that the true value lies within the calculated margin of error.
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Sample Size (n)
Sample size (n) refers to the number of observations or data points collected in a study. It plays a crucial role in determining the reliability and validity of statistical estimates. A larger sample size generally leads to a smaller margin of error, enhancing the precision of the estimate. In this question, n = 16 indicates that the analysis is based on 16 observations, which will influence the calculation of the margin of error.
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Related Practice
Textbook Question
In Exercises 7–10, use the confidence interval to find the margin of error and the sample proportion.
(0.087, 0.263)
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Textbook Question
Choosing a Distribution In Exercises 35–40, use the standard normal distribution or the t-distribution to construct a 95% confidence interval for the population mean. Justify your decision. If neither distribution can be used, explain why. Interpret the results.
Body Mass Index In a random sample of 50 people, the mean body mass index (BMI) was 27.7 and the standard deviation was 6.12.
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Textbook Question
A researcher claims that 5% of people who wear eyeglasses purchase their eyeglasses online. Describe type I and type II errors for a hypothesis test of the claim. (Source: Consumer Reports)
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Textbook Question
Drug Concentration You are analyzing the times for the drug concentrations to peak in the patients in Exercise 14. The population standard deviation of the times for epinephrine concentrations to peak should be less than 10 minutes. Does the confidence interval you constructed for σ suggest that the variation in the times is at an acceptable level? Explain your reasoning.
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Textbook Question
In Exercises 13–16, find the margin of error for the values of c, σ and n.
c = 0.95, σ = 5.2, n = 30
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