Textbook Question
In Exercises 5–8, find the critical value zc necessary to construct a confidence interval at the level of confidence c.
c = 0.80
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7. Sampling Distributions & Confidence Intervals: Mean
Confidence Intervals for Population Mean
1:47 minutesProblem 6.1.13
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
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 = 0.95). For a 95% confidence level, the z-value can be found using a z-table or standard normal distribution. It is approximately 1.96.
Step 3: Plug in the given values into the formula. Here, σ = 5.2 and n = 30. The formula becomes: ME = 1.96 * (5.2 / √30).
Step 4: Simplify the denominator by calculating the square root of the sample size (n). Compute √30.
Step 5: Divide the population standard deviation (σ = 5.2) by the square root of the sample size (√30), then multiply the result by the critical value (z = 1.96) to find the margin of error.
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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 in the data.
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Confidence Level (c)
The confidence level represents the probability that the margin of error will contain the true population parameter. A confidence level of 0.95, for example, means that if the same sampling procedure were repeated multiple times, approximately 95% of the calculated margins of error would capture the true parameter.
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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. The sample size (n) refers to the number of observations in the sample. Both σ and n are critical in calculating the margin of error, as they influence the precision of the estimate; larger samples generally yield smaller margins of error.
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