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Ch. 6 - Confidence Intervals
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 6 - Confidence IntervalsProblem 6.4.26
Chapter 6, Problem 6.4.26

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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Step 1: Understand the problem. The goal is to determine if the confidence interval for the population standard deviation (σ) suggests that the variation in the times for epinephrine concentrations to peak is less than 10 minutes. This involves interpreting the confidence interval in the context of the given threshold (10 minutes).
Step 2: Recall the formula for constructing a confidence interval for the population standard deviation. The confidence interval is based on the chi-square distribution and is given by: \( \left( \sqrt{\frac{(n-1)s^2}{\chi^2_{\alpha/2}}}, \sqrt{\frac{(n-1)s^2}{\chi^2_{1-\alpha/2}}} \right) \), where \( n \) is the sample size, \( s \) is the sample standard deviation, and \( \chi^2 \) are the critical values from the chi-square distribution.
Step 3: Verify the confidence interval you constructed for \( \sigma \). Ensure that the lower and upper bounds of the interval are correctly calculated using the formula above. If the confidence interval is already provided, identify its bounds.
Step 4: Compare the confidence interval bounds to the threshold of 10 minutes. If the entire confidence interval lies below 10 minutes, it suggests that the variation in the times is at an acceptable level. If any part of the interval exceeds 10 minutes, it indicates that the variation may not be acceptable.
Step 5: Conclude your reasoning. Based on the comparison in Step 4, explain whether the confidence interval supports the claim that the population standard deviation is less than 10 minutes. Provide a clear justification for your conclusion.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Population Standard Deviation

The population standard deviation (σ) measures the dispersion of a set of values in a population. It indicates how much individual data points deviate from the mean. In this context, a standard deviation of less than 10 minutes suggests that the times for drug concentrations to peak are relatively consistent among patients, which is crucial for evaluating the effectiveness of the drug.
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Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence. In this case, the confidence interval for the population standard deviation helps determine if the observed variation in drug peak times is statistically acceptable. If the interval includes values greater than 10 minutes, it may indicate excessive variability.
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Statistical Significance

Statistical significance refers to the likelihood that a result or relationship is caused by something other than mere random chance. In evaluating the confidence interval for the population standard deviation, determining whether the variation in peak times is statistically significant helps assess if the observed data supports the hypothesis that the drug's effect is consistent across patients.
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Related Practice
Textbook Question

In Exercise 35, would it be unusual for the population mean to be over \$1500? Explain.

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

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.

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)

Textbook Question

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

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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