Ch. 7 - Estimating Parameters and Determining Sample Sizes

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 7 - Estimating Parameters and Determining Sample Sizes

Problem 7.RE.10b

Chapter 7, Problem 7.RE.10b

Arm Circumferences Listed below are arm circumferences (cm) of randomly selected women (based on Data Set 1 “Body Data” from Appendix B). Also shown is the normal quantile plot of those measurements.

b. Are the requirements for constructing a 95% confidence interval estimate of the population standard deviation satisfied? If so, construct that confidence interval.

Verified step by step guidance

Step 1: Verify the requirements for constructing a confidence interval for the population standard deviation. The requirements include: (a) the data must come from a simple random sample, and (b) the population from which the sample is drawn should follow a normal distribution. Use the normal quantile plot provided to assess whether the data appears approximately normal.

Step 2: If the normality assumption is satisfied, calculate the sample standard deviation (s) using the formula:

\sqrt{\frac{\sum {(x}_{i} - x¯)^{2}}{n - 1}}

, where

x¯

is the sample mean,

n

is the sample size, and

x_{i}

are the individual data points.

Step 3: Identify the degrees of freedom (df) for the chi-square distribution, which is equal to

n - 1

, where

n

is the sample size. Use the chi-square distribution table or software to find the critical values

χ_{1} ²

and

χ_{2} ²

for a 95% confidence level.

Step 4: Use the formula for the confidence interval of the population standard deviation:

[\sqrt{\frac{n - 1}{s_{2}}}, \sqrt{\frac{n - 1}{s_{2}}}]

, where

s_{2}

is the sample variance.

Step 5: Interpret the confidence interval. The resulting interval provides a range of plausible values for the population standard deviation at the 95% confidence level. Ensure the interpretation aligns with the context of the problem.

Verified video answer for a similar problem:

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

Video duration:

Was this helpful?

Key Concepts

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

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, typically 95%. It provides an estimate of uncertainty around the sample mean or standard deviation, allowing researchers to infer about the population from which the sample was drawn.

Normal Distribution

The normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. Many statistical methods, including confidence intervals, assume that the data follows a normal distribution, especially when sample sizes are small.

Sample Size and Random Sampling

Random sampling involves selecting individuals from a population in such a way that each individual has an equal chance of being chosen. The sample size affects the reliability of the confidence interval; larger samples tend to provide more accurate estimates of the population parameters and help meet the assumptions necessary for constructing confidence intervals.

Recommended video:

05:11

Sampling Distribution of Sample Proportion

Related Practice

Textbook Question

Space Mountain Use the following wait times (minutes) for the Space Mountain ride at Disney World (from Data Set 33 “Disney World Wait Times” in Appendix B). Construct a 95% confidence interval estimate of the mean of all wait times. Write a brief statement that interprets that confidence interval.

40 35 40 40 25 80 50 30 35 40

112

views

Textbook Question

Controversial Song The song “Baby It’s Cold Outside” generated much controversy because of its lyrics and tone. CBS New York conducted a survey by asking viewers to use the Internet to respond to a question asking whether that song was really too offensive to play. Among 1043 Internet users who chose to respond, 986 said that the song was not too offensive, and 57 of the respondents said that the song was too offensive.

b. Based on the result from part (a), is it safe to say that the majority of the population does not feel that the song is too offensive.

116

views

Textbook Question

Distributions Identify the distribution (normal, Student t, chi-square) that should be used in each of the following situations. If none of the three distributions can be used, what other method could be used?

a. In constructing a confidence interval of , you have 75 sample values and they appear to be from a population with a skewed distribution. The population standard deviation is not known.

188

views

Textbook Question

Critical Thinking. In Exercises 17–28, use the data and confidence level to construct a confidence interval estimate of p, then address the given question.

Job Interviews In a Harris poll of 514 human resource professionals, 45.9% said that body piercings and tattoos were big personal grooming red flags.

a. Among the 514 human resource professionals who were surveyed, how many of them said that body piercings and tattoos were big personal grooming red flags?

128

views

Textbook Question

Astrology A sociologist plans to conduct a survey to estimate the percentage of adults who believe in astrology. How many people must be surveyed if we want a confidence level of 99% and a margin of error of four percentage points?

a. Assume that nothing is known about the percentage to be estimated.

views

Textbook Question

c. What is wrong with this survey? Based on this survey, what do we really know about the population?

112

views