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Chapter 8: Problem 91

When calculating a confidence interval for the population mean \(\mu\) with a known population standard deviation \(\sigma\), describe the effects of the following two changes on the confidence interval: (1) doubling the sample size, (2) quadrupling (multiplying by 4) the sample size. Give two reasons why this relationship does not hold true if you are calculating a confidence interval for the population mean \(\mu\) with an unknown population standard deviation.

Short Answer

Expert verified
Doubling and quadrupling the sample size narrows the confidence interval, given that the population standard deviation is known. This is because a larger sample size reduces the standard error, leading to a more precise estimate of the population mean. However, when the population standard deviation is unknown, the discussed relationship does not hold true due to the increased uncertainty from using the sample standard deviation and the use of t-distribution instead of z-distribution.

Step by step solution

01

Understanding Confidence Interval

A confidence interval estimates the range within which a population parameter lies with a certain confidence level. In this context, we are considering the population mean \(\mu\), and we assume the population standard deviation \(\sigma\) is known. The confidence interval is typically defined as \( \mu \pm z \times (\sigma/\sqrt{n}) \), where \(z\) is the z-score, \(\sigma\) is the standard deviation, and \(n\) is the sample size.
02

Effect of Increasing Sample Size

Doubling the sample size will effectively halve the standard error, i.e., the quantity \(\sigma/\sqrt{n}\) will decrease, therefore narrowing the confidence interval. This happens because we have more data points which can give us a better estimate of the population mean. Similarly, quadrupling the sample size will reduce the standard error by half again, causing the confidence interval to narrow further.
03

Exception When Standard Deviation is Unknown

The relationship discussed above does not hold when the population standard deviation is unknown. The reasons are: 1) When we don't know \(\sigma\), we have to use the sample standard deviation (\(s\)) as an estimate, which introduces more uncertainty. 2) We also have to use the t-distribution instead of the z-distribution in this case. The shape of the t-distribution depends on the degrees of freedom, which is related to sample size, introducing another layer of complexity to the confidence interval calculation.

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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, often denoted by the symbol \( \mu \), represents the average value of a group or entire population. In a statistical context, the population mean is crucial as it serves as a central parameter for measuring the common or typical value within a dataset.

When we calculate statistical measures, we're often interested in estimating this population mean based on a smaller set of observed data, known as a sample. It's important to understand that:
  • The population mean is a fixed value, representing the true average of the population.
  • Our goal with statistical analysis is usually to estimate this value accurately using sample data.
  • The larger the sample, the more reliable our estimate of the population mean will be.
By working with confidence intervals, statisticians can estimate the range within which the true population mean is likely to fall, offering insight into the data's overall behavior.
Sample Size
Sample size, represented as \( n \), is the number of observations or data points collected from a larger population for analysis. Deciding on an appropriate sample size is pivotal in research and statistics because it affects the precision of the estimate and the width of the confidence interval.

Increasing the sample size offers several benefits:
  • It reduces sampling error by providing more information about the population.
  • A larger sample size narrow downs the confidence interval, making the estimation of the population mean more precise.
  • It enhances the stability and reliability of the statistics calculated from the sample.
This is why statisticians often prefer larger samples; they reduce the effect of variability, and hence, decisions made from the data are generally more accurate.
Standard Error
The standard error is a measure of the statistical accuracy of an estimate, indicating the amount of variability in a sample statistic. For a mean, it is calculated as \( \sigma/\sqrt{n} \), where \( \sigma \) is the standard deviation of the population and \( n \) is the sample size.

The standard error informs us how much the sample mean is likely to differ from the true population mean, with lower values indicating more precise estimates. Key points include:
  • As the sample size increases, the standard error decreases, reflecting more reliable estimates.
  • The formula uses the population standard deviation, and thus presumes it is known.
  • Reducing the standard error by increasing sample size tightens the confidence interval, improving estimate precision.
It's essential for interpreting statistical results correctly, particularly when evaluating the confidence intervals of a population mean.
Z-Score
A Z-score is a statistical metric that measures the number of standard deviations a data point is from the mean of a set. It's particularly important when determining confidence intervals with a known population standard deviation.

Z-scores:
  • Help identify how unusual or typical a value is within a data set.
  • Are used with the normal distribution to compute confidence intervals.
  • Reflect a standard normal distribution, simplifying the process of statistical inference.
In confidence interval calculations, the z-score defines the margin of error, allowing us to estimate the range that captures the population mean with a certain degree of confidence.
T-Distribution
The t-distribution is a type of probability distribution that is symmetric and bell-shaped, similar to the normal distribution but with heavier tails. This means it puts more probability in the tails of the distribution, accounting for variability when estimating population parameters.

The t-distribution is crucial when:
  • The population standard deviation is unknown, and we rely on sample data.
  • The sample size is relatively small, making reliance on normal distribution less accurate.
  • We need to account for degrees of freedom, which become more significant with smaller samples.
When constructing confidence intervals with unknown standard deviation, the t-distribution helps provide a more accurate estimation than using a normal distribution.

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Most popular questions from this chapter

Determine the most conservative sample size for the estimation of the population proportion for the following. a. \(E=.025\), confidence level \(=95 \%\) b. \(E=.05, \quad\) confidence level \(=90 \%\) c. \(E=.015\), confidence level \(=99 \%\)

A random sample of 300 female members of health clubs in Los Angeles showed that they spend, on average, \(4.5\) hours per week doing physical exercise with a standard deviation of . 75 hour. Find a \(98 \%\) confidence interval for the population mean.

In an online poll conducted by the St. Louis Post-Dispatch during September 2014, people were asked about their favorite sports to watch on television. Of the respondents, \(42 \%\) selected baseball, \(18 \%\) mentioned hockey, \(36 \%\) liked football, and \(4 \%\) selected basketball (www.stltoday.com). Using these results, find a \(98 \%\) confidence interval for the population percentage that corresponds to each response. Write a one-page report to present your results to a group of college students who have not taken statistics. Your report should answer questions such as the following: (1) What is a confidence interval? (2) Why is a range of values (interval) more informative than a single percentage (point estimate)? (3) What does \(98 \%\) confidence mean in this context? (4) What assumptions, if any, are you making when you construct each confidence interval?

Briefly explain the difference between a confidence level and a confidence interval.

A random sample of 36 mid-sized cars tested for fuel consumption gave a mean of \(26.4\) miles per gallon with a standard deviation of \(2.3\) miles per gallon. a. Find a \(99 \%\) confidence interval for the population mean, \(\mu\). b. Suppose the confidence interval obtained in part a is too wide. How can the width of this interval be reduced? Describe all possible alternatives. Which alternative is the best and why?
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