Table of contents

1. Intro to Stats and Collecting Data55m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically1h 45m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables2h 33m
6. Normal Distribution and Continuous Random Variables1h 38m
7. Sampling Distributions & Confidence Intervals: Mean1h 53m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample2h 19m
10. Hypothesis Testing for Two Samples3h 22m
11. Correlation1h 6m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

7. Sampling Distributions & Confidence Intervals: Mean

Confidence Intervals for Population Mean

Statistics

7. Sampling Distributions & Confidence Intervals: Mean

Confidence Intervals for Population Mean

Previous problemNext problem

1:43 minutes

Problem 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 guidance

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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

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 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.

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.

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.

Watch next

Master Population Standard Deviation Known with a bite sized video explanation from Patrick

Start learning

Related Videos

Related Practice

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

views

Textbook Question

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

views

Textbook Question

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

views

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

views

Textbook Question

Matching In Exercises 17–20, match the level of confidence c with the appropriate confidence interval. Assume each confidence interval is constructed for the same sample statistics.

c = 0.88

views

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.

views

Textbook Question

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

c = 0.80, xbar = 20.6, σ = 4.7, n = 100.

views

Textbook Question

In Exercises 25–28, use the confidence interval to find the margin of error and the sample mean.

(21.61, 30.15)

views

Show more practices

Download the Mobile app

Privacy

Do not sell my personal information

Accessibility

Patent notice

Sitemap

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

Key Concepts

Margin of Error

Confidence Level (c)

Standard Deviation (σ) and Sample Size (n)

Watch next

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