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Chapter 7: Problem 16
Construct the confidence interval estimate of the mean. Listed below are amounts of arsenic \((\mu \mathrm{g},\) or micrograms, perserving) in samples of brown rice from California (based on data from the Foodand Drug Administration). Use a \(90 \%\) confidence level. The Food and DrugAdministration also measured amounts of arsenic in samples of brown rice fromArkansas. Can the confidence interval be used to describe arsenic levels inArkansas? $$\begin{array}{cccccccccc} 5.4 & 5.6 & 8.4 & 7.3 & 4.5 & 7.5 & 1.5 & 5.5 & 9.1 & 8.7 \end{array}$$
Short Answer
Expert verified
No, the confidence interval for California cannot describe arsenic levels in Arkansas. The intervals are specific to California brown rice.
Step by step solution
01
- Calculate the Sample Mean
First, find the mean (average) of the given data. Add all the amounts of arsenic together and then divide by the number of data points. \[ \text{Mean} = \frac{5.4 + 5.6 + 8.4 + 7.3 + 4.5 + 7.5 + 1.5 + 5.5 + 9.1 + 8.7}{10} \]
02
- Calculate the Sample Standard Deviation
Next, calculate the standard deviation of the sample. This measures the amount of variation in the arsenic levels. Use the formula for sample standard deviation, where \( \bar{x} \) is the sample mean: \[ s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2} \]
03
- Determine the Standard Error
Calculate the standard error of the mean, which provides an estimate of the precision of the sample mean. The standard error (SE) is given by: \[ \text{SE} = \frac{s}{\sqrt{n}} \] where \( s \) is the sample standard deviation and \( n \) is the sample size.
04
- Find the Critical Value
Determine the critical value for the 90% confidence level. Since we are working with a sample and not the whole population, use a t-distribution with \( n-1 \) degrees of freedom. For a 90% confidence interval and 9 degrees of freedom (since \( n = 10 \)), the critical value \( t_{\frac{\alpha}{2}} \) can be found using t-tables or statistical software.
05
- Construct the Confidence Interval
Use the sample mean, the critical value, and the standard error to construct the confidence interval for the mean arsenic level. The confidence interval is given by: \[ \bar{x} \pm t_{\frac{\alpha}{2}} \times \text{SE} \] Where \( \bar{x} \) is the sample mean, \( t_{\frac{\alpha}{2}} \) is the critical value, and \( \text{SE} \) is the standard error.
06
- Determine the Confidence Interval
Calculate the lower and upper bounds of the confidence interval using the values obtained: \[ \left(\bar{x} - t_{\frac{\alpha}{2}} \times \text{SE}, \bar{x} + t_{\frac{\alpha}{2}} \times \text{SE}\right) \]
07
- Answer the Question about Arkansas
Evaluate if the confidence interval calculated can be applied to arsenic levels in brown rice from Arkansas. Since the data we have is specific to California, and arsenic levels could vary based on geographical and environmental factors, it is not appropriate to use this confidence interval to describe arsenic levels in Arkansas.
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Sample Mean
The sample mean, often represented as \(\bar{x}\) or mu, is the average value of a set of data points. It's calculated by adding together all of the values in the sample and then dividing by the number of values. For example, for the arsenic data \(5.4, 5.6, 8.4, 7.3, 4.5, 7.5, 1.5, 5.5, 9.1 \text{and} 8.7\), you sum all these values and divide by 10 (since there are 10 data points). The formula is: \[\text{Mean} = \frac{5.4 + 5.6 + 8.4 + 7.3 + 4.5 + 7.5 + 1.5 + 5.5 + 9.1 + 8.7}{10} = 6.35\] This gives the average amount of arsenic in the brown rice samples from California. The sample mean is a critical piece because it serves as the center point for constructing the confidence interval.
Standard Deviation
The standard deviation measures how spread out the data points are around the mean. It indicates the amount of variation or dispersion of a set of values. For a sample, the formula to calculate the standard deviation is: \[s = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2}\] In this formula, \(n\) is the sample size, \(x_i\) are the individual data points, and \(\bar{x}\) is the sample mean. High standard deviation means that the data points are spread out over a larger range of values, whereas a low standard deviation indicates that the data points are close to the mean. In our case, once we calculate the sample deviation, we have more insight into how the arsenic levels vary in the brown rice samples.
T-Distribution
The t-distribution is a type of probability distribution that is symmetric and bell-shaped, much like the normal distribution, but has heavier tails. This means it is more prone to producing values that fall further from its mean. It is particularly useful when dealing with small sample sizes (typically n < 30). The t-distribution is used instead of the normal distribution to estimate population parameters when the sample size is small and the population standard deviation is unknown. For our purpose, we use the t-distribution to find the critical value when creating the confidence interval. The degrees of freedom for the t-distribution are determined by the sample size minus one (n - 1). In this case, with 10 samples, we use 9 degrees of freedom.
Confidence Level
The confidence level represents the percentage of all possible samples that can be expected to include the true population parameter. A 90% confidence level means that if we were to take 100 different samples and compute a confidence interval for each sample, we would expect that 90 of the intervals will contain the population mean. It does not imply that the probability that our specific calculated interval contains the population mean is 90%. To construct a 90% confidence interval for the mean arsenic level in brown rice, we will use the sample mean, the standard error, and the critical value from the t-distribution with 9 degrees of freedom. Combining these elements appropriately helps estimate the range within which the true mean of arsenic levels in California brown rice samples lies, giving us confidence in the precision of our results.
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