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Chapter 7: Problem 18

Use technology to create the large number of bootstrap samples. Listed below are measured amounts of caffeine (mg per 12 oz of drink) obtained in one can from each of 20 brands. Using the bootstrap method with 1000 bootstrap samples, construct a \(99 \%\) confidence interval estimate of \(\mu\). How does the result compare to the confidence interval found in Exercise 22 "Caffeine in Soft Drinks" in Section \(7-2\) on page \(330 ?\) $$\begin{array}{rrrrrrrrrrrr}0 & 0 & 34 & 34 & 34 & 45 & 41 & 51 & 55 & 36 & 47 & 41 & 0 & 0 & 53 & 54 & 38 & 0 & 41 & 47\end{array}$$

Short Answer

Expert verified
Generate 1000 bootstrap samples, calculate the mean for each, determine the confidence interval from percentiles, and compare to Exercise 22, Section 7-2 on page 330.

Step by step solution

01

Organize the Data

Begin with the provided caffeine amounts for each brand: 0, 0, 34, 34, 34, 45, 41, 51, 55, 36, 47, 41, 0, 0, 53, 54, 38, 0, 41, 47.
02

Generate Bootstrap Samples

Use a technology tool like Excel, R, or Python to generate 1000 bootstrap samples. Each bootstrap sample should be created by resampling (with replacement) from the original dataset.
03

Calculate the Mean for Each Sample

For each of the 1000 bootstrap samples, calculate the mean caffeine amount.
04

Determine the Bootstrap Distribution

Create a distribution of the 1000 means calculated in Step 3. This will be the bootstrap distribution of the sample means.
05

Calculate the Confidence Interval

Identify the 0.5th percentile and the 99.5th percentile of the bootstrap distribution. These percentiles will form the bounds of the 99% confidence interval for \(\mu\).
06

Compare With The Given Confidence Interval

Finally, compare the calculated confidence interval with the one found in Exercise 22 in Section 7-2 on page 330 to note any differences or similarities.

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Key Concepts

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

confidence interval
A confidence interval helps us estimate the true mean of a population based on sample data. It's a range of values that is likely to contain the population mean with a specified level of confidence, such as 99%. To calculate a 99% confidence interval, the lower and upper bounds are determined so that there's a 99% certainty the true mean lies within this interval. In our exercise, we use the bootstrap method to find these bounds.
mean calculation
The mean or average is a basic statistical measure of central tendency. It represents the sum of all observed values divided by the number of observations. To calculate the mean caffeine amount in our provided data, you sum all caffeine measurements and then divide by the total number of measurements:
\[ \text{Mean} = \frac{0 + 0 + 34 + \text{...} + 41 + 47}{20} \] In the context of our example, we compute the mean for each of the 1000 bootstrap samples to understand the distribution of these means.
caffeine measurement
Caffeine measurement in various drinks can vary significantly. In our exercise, we consider 20 different brands of 12 oz drinks and record the caffeine content in milligrams (mg). The values range from 0 mg to 55 mg. Understanding this variability is crucial, as it reflects on the reliability and usability of calculating the confidence interval. Accurate measurements are necessary to gain meaningful statistical insights.
bootstrap distribution
Bootstrap distribution is a key concept in statistical analysis. It involves generating many resampled datasets (bootstrap samples) from the original data. In our case, we create 1000 bootstrap samples by resampling the provided caffeine measurements. For each bootstrap sample, we calculate the mean caffeine amount. These means form the bootstrap distribution.
The bootstrap distribution gives us an empirical approximation of the sampling distribution of the mean. By using this distribution, we can calculate percentiles and construct confidence intervals to estimate the true mean caffeine content in the population.

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Most popular questions from this chapter

Use the data and confidence level to construct a confidence interval estimate of \(p,\) then address the given question. A random sample of 860 births in New York State included 426 boys. Construct a \(95 \%\) confidence interval estimate of the proportion of boys in all births. It is believed that among all births, the proportion of boys is \(0.512 .\) Do these sample results provide strong evidence against that belief?

Use the data and confidence level to construct a confidence interval estimate of \(p,\) then address the given question. A study of 420,095 Danish cell phone users found that \(0.0321 \%\) of them developed cancer of the brain or nervous system. Prior to this study of cell phone use, the rate of such cancer was found to be 0.0340 \(\$$ for those not using cell phones. The data are from the Joumal of the National Cancer Institute. a. Use the sample data to construct a \)90 \%$ confidence interval estimate of the percentage of cell phone users who develop cancer of the brain or nervous system. b. Do cell phone users appear to have a rate of cancer of the brain or nervous system that is different from the rate of such cancer among those not using cell phones? Why or why not?

Finding Confidence Intervals. In Exercises \(9-16,\) assume that each sample is a simple random sample obtained from a population with a normal distribution. Highway Speeds Listed below are speeds (mi/h) measured from southbound traffic on I-280 near Cupertino, California (based on data from SigAlert). This simple random sample was obtained at 3: 30 PM on a weekday. Use the sample data to construct a \(95 \%\) confidence interval estimate of the population standard deviation. Does the confidence interval describe the standard deviation for all times during the week? \(\begin{array}{rrrrrrrrr}57 & 61 & 54 & 59 & 58 & 59 & 69 & 60 & 67\end{array}\) \(62 \quad 61 \quad 61\)

Use the data and confidence level to construct a confidence interval estimate of \(p,\) then address the given question. In clinical trials of the drug Lipitor (atorvastatin), 270 subjects were given a placebo and 7 of them had allergic reactions. Among 863 subjects treated with \(10 \mathrm{mg}\) of the drug. 8 experienced allergic reactions. Construct the two \(95 \%\) confidence interval estimates of the percentages of allergic reactions. Compare the results. What do you conclude?

Use the given data to find the minimum sample size required to estimate a population proportion or percentage. An epidemiologist plans to conduct a survey to estimate the percentage of women who give birth. How many women must be surveyed in order to be \(99 \%\) confident that the estimated percentage is in error by no more than two percentage points? a. Assume that nothing is known about the percentage to be estimated. b. Assume that a prior study conducted by the U.S. Census Bureau showed that \(82 \%\) of women give birth. c. What is wrong with surveying randomly selected adult women?
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