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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 6.2.8
Chapter 6, Problem 6.2.8

In Exercises 7 and 8, find the margin of error for the values of c, s, and n.
c = 0.99, s = 3, n = 6

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Step 1: Understand the formula for the margin of error (E). The margin of error is calculated using the formula: E = z \(\cdot\) \(\frac{s}{\sqrt{n}\)}, 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 = 0.99). Use a z-table or statistical software to find the z-value that corresponds to a 99% confidence level. For a two-tailed test, this is the z-value where the area in the tails is 0.01 (0.005 in each tail).
Step 3: Plug in the given values for s and n into the formula. Here, s = 3 and n = 6. Substitute these values into the formula: E = z \(\cdot\) \(\frac{3}{\sqrt{6}\)}.
Step 4: Simplify the denominator by calculating the square root of the sample size (n). Compute \(\sqrt{6}\) and substitute it into the formula.
Step 5: Multiply the critical value (z) by the fraction \(\frac{3}{\sqrt{6}\)} to calculate the margin of error (E). This will give you the final result.

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Key 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 higher margin of error suggests less precision in the estimate, while a lower margin indicates greater confidence in the results.
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Confidence Level

The confidence level represents the probability that the margin of error will capture the true population parameter. Common confidence levels include 90%, 95%, and 99%. In this case, a confidence level of 0.99 means that if the same sampling method were repeated multiple times, 99% of the calculated intervals would contain the true parameter.
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Sample Size (n)

Sample size refers to the number of observations or data points collected in a study. A larger sample size generally leads to a smaller margin of error, enhancing the reliability of the estimate. In this context, with n = 6, the sample size is relatively small, which may result in a larger margin of error compared to larger samples.
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Related Practice
Textbook Question

Graphical Analysis In Exercises 9–12, use the values on the number line to find the sampling error.

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Textbook Question

Which statistic is the best unbiased estimator for μ?

a. s

b. xbar

c. the median

d. the mode

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Textbook Question

In Exercises 9–12, construct the indicated confidence intervals for (a) the population variance and (b) the population standard deviation . Assume the sample is from a normally distributed population.

c = 0.99, s^2 = 0.64, n = 7

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Textbook Question

In Exercises 7–10, the statement represents a claim. Write its complement and state which is Ho and which is Ha.


μ<33

49
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Textbook Question

In Exercises 21–24, construct the indicated confidence interval for the population mean μ.

c = 0.95, xbar = 31.39, σ = 0.80, n = 82.

129
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Textbook Question

In Exercises 29–32, determine the minimum sample size n needed to estimate for the values of c, σ, and E.

c = 0.95, σ = 2.5, E = 1.

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