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

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3:13 minutes

Problem 6.1.5

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

Verified step by step guidance

Identify the level of confidence, c, which is given as 0.80. This represents the proportion of the area under the standard normal curve that corresponds to the confidence interval.

Calculate the area in the tails of the standard normal distribution. Since the confidence level is 0.80, the remaining area in the tails is 1 - 0.80 = 0.20. Divide this equally between the two tails, so each tail has an area of 0.10.

Determine the cumulative area to the left of the critical value zc. Since the area in the left tail is 0.10, the cumulative area to the left of zc is 0.10 + 0.80 = 0.90.

Use a standard normal distribution table (z-table) or a statistical calculator to find the z-score that corresponds to a cumulative area of 0.90. This z-score is the critical value zc.

Verify the critical value zc by ensuring that the area between -zc and zc under the standard normal curve equals the confidence level, c = 0.80.

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Key Concepts

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

Critical Value (zc)

The critical value, denoted as zc, is a point on the standard normal distribution that corresponds to a specified level of confidence. It defines the boundaries of the confidence interval, indicating how far from the mean we can expect the true population parameter to lie with a certain level of certainty. For example, a confidence level of 80% means that 80% of the distribution lies within zc standard deviations from the mean.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence. It is calculated using the sample mean, the critical value, and the standard error. For instance, an 80% confidence interval suggests that if we were to take many samples, approximately 80% of the calculated intervals would contain the true parameter.

Standard Normal Distribution

The standard normal distribution is a probability distribution that has a mean of 0 and a standard deviation of 1. It is used to determine the critical values for various confidence levels. By converting a normal distribution to a standard normal distribution, we can easily find zc values using z-tables or statistical software, which helps in constructing confidence intervals.

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Textbook Question

In Exercises 51 and 52, a population and sample size are given. (b) List all samples (with replacement) of the given size from the population and find the mean of each. (c) Find the mean and standard deviation of the sampling distribution of sample means and compare them with the mean and standard deviation of the population.

The goals scored in a season by the four starting defenders on a soccer team are 1, 2, 0, and 3. Use a sample size of 2.

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Textbook Question

In Exercises 53 and 54, find the mean and standard deviation of the indicated sampling distribution of sample means. Then sketch a graph of the sampling distribution.

The population densities in people per square mile in the 50 U.S. states have a mean of 199.6 and a standard deviation of 265.4. Random samples of size 35 are drawn from this population, and the mean of each sample is determined.

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Textbook Question

In Exercises 53 and 54, find the mean and standard deviation of the indicated sampling distribution of sample means. Then sketch a graph of the sampling distribution.

The test scores for the Law School Admission Test (LSAT) in a recent year are normally distributed, with a mean of 151.88 and a standard deviation of 9.95. Random samples of size 40 are drawn from this population, and the mean of each sample is determined.

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Textbook Question

For the same sample statistics, which level of confidence would produce the widest confidence interval? Explain your reasoning.

a. 90%

b. 95%

c. 98%

d. 99%

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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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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

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

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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

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In Exercises 5–8, find the critical value zc necessary to construct a confidence interval at the level of confidence c.c = 0.80

Key Concepts

Critical Value (zc)

Confidence Interval

Standard Normal Distribution

Watch next

In Exercises 5–8, find the critical value zc necessary to construct a confidence interval at the level of confidence c.
c = 0.80