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Chapter 16: Problem 13

You want a \(99 \%\) confidence interval for the true weight of this specimen. The margin of error for this interval will be Fill weIGHTs (a) smaller than the margin of error for \(95 \%\) confidence. (b) greater than the margin of error for \(95 \%\) confidence. (c) about the same as the margin of error for \(95 \%\) confidence.

Short Answer

Expert verified
(b) Greater than the margin of error for 95% confidence.

Step by step solution

01

Understanding Confidence Interval

A confidence interval gives a range of values which is likely to contain the true parameter of interest, such as the true weight in this case. The margin of error is a measure of how much this interval covers (how wide the interval is).
02

Confidence Level and Margin of Error

The margin of error for a confidence interval increases as the confidence level increases. This is because a higher confidence level means we want to be more certain that we are capturing the true parameter, thus requiring a wider interval.
03

Comparing to 95% Confidence Interval

With a 95% confidence interval, we have a smaller margin of error when compared to a 99% confidence interval. Therefore, the 99% confidence interval will have a greater margin of error.

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Key Concepts

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

Confidence Level
The confidence level represents how certain we are that a particular confidence interval contains the true population parameter. It is typically expressed as a percentage, such as 95% or 99%. These percentages indicate the likelihood that the results from our sample contain the truth from the overall population.
  • A 95% confidence level tells us that if we were to take 100 different samples and compute a confidence interval from each one, we would expect about 95 of those intervals to encompass the true population parameter.
  • A higher confidence level, like 99%, provides more certainty in our conclusions. However, it comes at the expense of a wider confidence interval.
For instance, increasing the confidence level from 95% to 99% is akin to broadening our safety net, ensuring more certainty in capturing the true parameter. But remember, this broader interval implies more uncertainty about the exact position of the parameter within that range.
Margin of Error
The margin of error is a crucial part of the confidence interval concept. It essentially tells us how much we expect our sample mean might differ from the true population mean. The margin of error gives the range above and below the sample statistic.
  • For instance, if the mean weight of a specimen from our sample is 10 kg with a margin of error of 1 kg, our confidence interval is (9 kg, 11 kg).
  • The margin of error is influenced by the sample size, variability of the data, and the confidence level we choose. A larger sample size tends to reduce the margin while increasing the confidence level tends to widen it.
With higher confidence levels, we demand more certainty which results in a larger margin of error, as seen when comparing a 99% confidence level with a 95% one. To sum up, a larger margin of error generally signifies greater uncertainty in pinpointing the exact location of the true parameter.
Statistical Parameter Estimation
Statistical parameter estimation involves determining the values of parameters within a model from sample data. In practical terms, it鈥檚 all about using sample data to gauge and infer about unknown population parameters.
  • Parameters typically estimated include means, proportions, and variances of the population.
  • Common methods include point estimates and interval estimates. While a point estimate gives one value such as an average, interval estimates provide a range where the parameter is likely to lie at a given confidence level.
Estimating parameters accurately requires understanding the role of sample size and variability. Larger samples generally provide more reliable estimates. In statistical reports, you often hear about terms like *"estimating the mean weight of a population"*, which revolves around this concept. Through statistical parameter estimation, we can make educated guesses about the characteristics of large groups using manageable sample data.

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Most popular questions from this chapter

This wine stinks, Sulfur compounds cause "off-odors" in wine, so winemakers want to know the odor threshold, the lowest concentration of a compound that the human nose can detect. The odor threshold for dimethyl sulfide (DMS) in trained wine tasters is about 25 micrograms per liter of wine \((\mu \mathrm{g} / \mathrm{L})\). The untrained noses of consumers may be less sensitive, however. Here are the DMS odor thresholds for 10 untrained students: II WnNer \(\begin{array}{llllllllll}30 & 30 & 42 & 35 & 22 & 33 & 31 & 29 & 19 & 23\end{array}\) (a) Assume that the standard deviation of the odor threshold for untrained noses is known to be \(\sigma=7 \mu \mathrm{g} / \mathrm{L}\). Briefly discuss the other two "simple conditions," using a stemplot to verify that the distribution is roughly symmetric with no outliers. (b) Give a \(95 \%\) confidence interval for the mean DMS odor threshold among all students.

Why are larger samples better? Statisticians prefer large samples. Describe briefly the effect of increasing the size of a sample on the margin of error of a \(95 \%\) confidence interval.

Sample Size and Margin of Error. Example \(16.1\) (page 377 ) described NHANES survey data on the body mass index (BMI) of 654 young women. The mean BMI in the sample was \(x^{-} \bar{x}=26.8\). We treated these data as an SRS from a Normally distributed population with standard deviation \(\sigma=7.5\). (a) Suppose that we had an SRS of just 100 young women. What would be the margin of error for \(95 \%\) confidence? (b) Find the margins of error for \(95 \%\) confidence based on SRSs of 400 young women and 1600 young women. (c) Compare the three margins of error. How does increasing the sample size change the margin of error of a confidence interval when the confidence level and population standard deviation remain the same?

Pulling wood apart. How heavy a load (in pounds) is needed to pull apart p盲eces of Douglas fir 4 inches long and \(1.5\) inches square? Here are data from students doing a laboratory exercise: wood $$ \begin{array}{lllll} 33,190 & 31,860 & 32,590 & 26,520 & 33,280 \\ 32,320 & 33,020 & 32,030 & 30,460 & 32,700 \\ 23,040 & 30,930 & 32,720 & 33,650 & 32,340 \\ 24,050 & 30,170 & 31,300 & 28,730 & 31,920 \end{array} $$ (a) We are willing to regard the wood pleces prepared for the lab session as an SRS of all similar pieces of Douglas fir. Engineers also commonly assume that characteristics of materials vary Normally. Make a graph to show the shape of the distribution for these data. Does it appear safe to assume that the Normality condition is satisfied? Suppose that the strength of pieces of wood like these follows a Normal distribution with standard deviation 3000 pounds. (b) Give a \(95 \%\) confidence interval for the mean load required to pull the wood apart.

To give a \(99.9 \%\) confidence interval for a population mean \(\mu\), you would use the critical value (a) \(z^{*}=1.960\). (b) \(z^{*}=2.576\). (c) \(x^{+}=3.291\). Use the following information for Exercises \(16.12\) through 16.14. A laboratory scale is known to have a standard deviation of \(\sigma=0.001\) gram in repeated weighings. Scale readings in repeated weighings are Normally distributed, with mean equal to the true weight of the specimen. Three weighings of a specimen on this scale give \(3.412,3.416\), and \(3.414\) grams.
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