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Chapter 8: Problem 5

Watching TV In response to the GSS question in 2008 about the number of hours daily spent watching \(\mathrm{TV}\), the responses by the five subjects who identified themselves as Hindu were 3,2,1,1,1 . a. Find a point estimate of the population mean for Hindus. b. The margin of error at the \(95 \%\) confidence level for this point estimate is 0.7 . Explain what this represents.

Short Answer

Expert verified
a. The point estimate of the population mean is 1.6. b. It means the true mean is likely between 0.9 and 2.3 hours.

Step by step solution

01

Determine the Sample Mean

To find the point estimate of the population mean, we first calculate the sample mean. Given the hours spent watching TV by the subjects are 3, 2, 1, 1, and 1, we sum these values and divide by the number of subjects.Sum: \(3 + 2 + 1 + 1 + 1 = 8\)Number of subjects: 5Sample mean: \(\bar{x} = \frac{8}{5} = 1.6\)
02

Understand the Margin of Error

The margin of error at the 95% confidence level is given as 0.7. This figure represents the maximum expected difference between the sample mean and the actual population mean. It tells us that we are 95% confident that the true population mean falls within 0.7 units of our point estimate.
03

Apply the Confidence Interval

With a sample mean of 1.6 and a margin of error of 0.7, the 95% confidence interval can be calculated. This interval indicates where the true population mean is likely to fall.Lower limit: \(1.6 - 0.7 = 0.9\)Upper limit: \(1.6 + 0.7 = 2.3\)Thus, we can say that the population mean is likely between 0.9 and 2.3 hours.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Population Mean
The population mean is a fundamental concept in statistics. It is the average of a set of values from the entire population. However, it is not always feasible to measure the population mean directly due to constraints like time and cost. Therefore, samples are instead used to estimate it.
Imagine the population mean as the true average hours Hindus watch TV daily, which in practice, could be challenging to measure precisely. Instead, by taking a smaller group of individuals鈥攍ike the five subjects in our exercise鈥攚e can calculate a sample mean, which serves as a practical estimate for what this true population mean might be.
Margin of Error
When estimating a population mean from a sample, there's always some level of uncertainty. The margin of error quantifies the amount of error that might be present in the sample's estimation of the true population mean.
In our exercise, the margin of error is given as 0.7 at a 95% confidence level. This means that the sample mean (1.6 hours of TV watched) is likely within 0.7 hours of the actual population mean. Essentially, this tells us how much we can expect our sample mean to differ from the real population mean due to random sampling variability.
  • A margin of error of 0.7 suggests a relatively narrow range鈥攊ndicating reasonable precision鈥攂ut the notion of uncertainty remains.
  • Larger samples typically result in smaller margins of error, thus more precise estimates.
Confidence Interval
A confidence interval offers a range of values within which the true population mean is likely to fall. This range reflects the degree of uncertainty or confidence in the sample mean as a representation of the population mean.
In this example, with a sample mean of 1.6 and a margin of error of 0.7, the confidence interval is calculated as 0.9 to 2.3. This essentially means we are 95% certain the actual population mean is between 0.9 and 2.3 hours daily.
  • The middle point of this interval is the sample mean.
  • The endpoints are determined by subtracting and adding the margin of error to the sample mean.
Confidence intervals are crucial as they provide a way to express the uncertainty associated with the sample estimates.
Point Estimate
A point estimate is a single value derived from sample data that serves as a best guess for an unknown population parameter, like the population mean.
In this exercise, the point estimate is the sample mean calculated from the hours spent watching TV. With values being 3, 2, 1, 1, and 1, the sample mean is 1.6 hours. This is our point estimate for the population mean of daily TV hours watched by Hindus.
  • Point estimates are straightforward but provide no indication of the estimate's reliability or margin for error.
  • Hence, they are usually supplemented with confidence intervals to give a more complete picture of the potential range of the true population parameter.

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Most popular questions from this chapter

Fear of breast cancer A recent survey of 1000 American women between the ages of 45 and 64 asked them what medical condition they most feared. Of those sampled, \(61 \%\) said breast cancer, \(8 \%\) said heart disease, and the rest picked other conditions. By contrast, currently about \(3 \%\) of female deaths are due to breast cancer, whereas \(32 \%\) are due to heart disease. \(^{5}\) a. Construct a \(90 \%\) confidence interval for the population proportion of women who most feared breast cancer. Interpret. b. Indicate the assumptions you must make for the inference in part a to be valid.

A Gallup poll taken during June 2011 estimated that \(8.8 \%\) of U.S. adults were unemployed. The poll was based on the responses of 30,000 U.S. adults in the workforce. Gallup reported that the margin of error associated with the poll is 卤0.3 percentage points. Explain how they got this result. (Source: www.gallup.com/poll/125639/Gallup-Daily-Workforce aspx.)

Why bootstrap? Explain the purpose of using the bootstrap method.

Alternative therapies The Department of Public Health at the University of Western Australia conducted a survey in which they randomly sampled general practitioners in Australia. \(^{10}\) One question asked whether the GP had ever studied alternative therapy, such as acupuncture, hypnosis, homeopathy, and yoga. Of 282 respondents, 132 said yes. Is the interpretation, "We are \(95 \%\) confident that the percentage of all GPs in Australia who have ever studied alternative therapy equals \(46.8 \%^{n}\) correct or incorrect? Explain.

Multiple choice: CI property Increasing the confidence level causes the margin of error of a confidence interval to \((\) a) increase, \((b)\) decrease, \((c)\) stay the same.
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