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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7. Sampling Distributions & Confidence Intervals: Mean
Confidence Intervals for Population Mean
1:43 minutesProblem 6.1.16
Textbook Question
In Exercises 13–16, find the margin of error for the values of c, σ and n.
c = 0.975, σ = 4.6, n = 100
Verified step by step guidance1
Step 1: Understand the formula for the margin of error (ME). The formula is ME = Z * (σ / √n), where Z is the critical value corresponding to the confidence level (c), σ is the population standard deviation, and n is the sample size.
Step 2: Determine the critical value (Z) for the given confidence level (c = 0.975). Use a Z-table or statistical software to find the Z-value that corresponds to the middle 97.5% of the standard normal distribution.
Step 3: Calculate the standard error (SE) using the formula SE = σ / √n. Substitute the given values of σ = 4.6 and n = 100 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. The margin of error represents the range within which the true population parameter is expected to lie with the given confidence level.
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Video duration:
1mKey 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 sample 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 used in survey results and statistical inference.
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Confidence Level (c)
The confidence level represents the probability that the margin of error will contain the true population parameter. In this case, a confidence level of 0.975 means that if the same sampling method were repeated multiple times, approximately 97.5% of the calculated margins of error would capture the true value. It reflects the degree of certainty in the results.
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Standard Deviation (σ) and Sample Size (n)
Standard deviation (σ) measures the dispersion of data points around the mean, indicating how spread out the values are. A larger standard deviation suggests more variability in the data. Sample size (n) refers to the number of observations in a sample; larger sample sizes generally lead to more reliable estimates and smaller margins of error, as they better represent the population.
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