Problem 22 - An Introduction to Mathematical Statistics and Its Applications

Question

Cell phones emit radio frequency energy that is absorbed by the body when the phone is next to the ear and may be harmful. The table in the next column gives the absorption rate for a sample of twenty high-radiation cell phones. (The Federal Communication Commission sets a maximum of \(1.6\) watts per kilogram for the absorption rate of such energy.) Construct a \(90 \%\) confidence interval for the true average cell phone absorption rate. $$ \begin{array}{ll} \hline 1.54 & 1.41 \\ 1.54 & 1.40 \\ 1.49 & 1.40 \\ 1.49 & 1.39 \\ 1.48 & 1.39 \\ 1.45 & 1.39 \\ 1.44 & 1.38 \\ 1.42 & 1.38 \\ 1.41 & 1.37 \\ 1.41 & 1.33 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The 90% confidence interval for the true average cell phone absorption rate is approximately from \(1.4035\) to \(1.4515\) watts/kg.

Step 1: Calculating the mean

The mean absorption rate is calculated by adding all the values together and dividing by the number of values. After performing this operation, we get a mean of approximately \(1.4275\) watts/kg.

Step 2: Calculating the standard deviation

The standard deviation is a measure of the amount of variation or dispersion of a set of values. Here, it is calculated as the square root of the average of the squared deviations from the mean. With the given data, we can compute the standard deviation to be approximately \(0.06494\) watts/kg.

Step 3: Determining the z-score

For a 90% confidence interval, we use a z-score of \(1.645\) (this value comes from the fact that 90% of the area of a normal distribution is within 1.645 standard deviations of the mean).

Step 4: Calculating the standard error

The standard error of the mean is the standard deviation of the sampling distribution. It is calculated as standard deviation divided by the square root of the sample size. Therefore, the standard error is approximately \(0.01452\) watts/kg.

Step 5: Constructing the 90% confidence interval

The formula for a confidence interval is mean ± (z-value * standard error). Using this formula, we calculate the lower and upper bounds of the 90% confidence interval to be approximately \(1.4035\) and \(1.4515\) watts/kg respectively.

Key Concepts

Mean Calculation

The mean is essentially the average value of a dataset. To find the mean absorption rate for the sample of twenty high-radiation cell phones, we need to follow a simple arithmetic process. We add up all the absorption rate values from the data. Once you have the total sum, you divide it by the number of values, which in this case is 20. This calculation gives you the mean.
It's important to know that the mean provides a central value, giving us an idea about the overall tendency of the dataset. For our problem, the calculated mean is approximately \(1.4275\) watts per kilogram. This means that, on average, the cell phone absorption rates hover around this value.

Standard Deviation

The standard deviation indicates how spread out the values in a dataset are from the mean. Higher standard deviation means data points are more spread out. To calculate the standard deviation, you first need to find the deviation of each value from the mean. You then square these deviations to handle any negative numbers, and find the average of the squared deviations.
Once you have the average of the squared deviations, take the square root. This value is the standard deviation, which tells us about the variability of the data. In our example, the calculated standard deviation is \(0.06494\) watts per kilogram.
This relatively low value indicates that the absorption rates do not vary greatly from the mean.

Z-Score

The z-score is a statistical measurement that describes a value's position relative to the mean of a group of values. For constructing confidence intervals, it tells us how many standard deviations a value is from the mean. For a 90% confidence interval, we commonly use a z-score of \(1.645\), reflecting that 90% of values lie within this range under a normal distribution.
The chosen z-score allows us to calculate the margin of error, which, combined with the mean, specifies the range in which we expect the true mean of the population to fall. Knowing the z-score is pivotal in determining the reliability of our interval estimates.

Standard Error

The standard error measures how much the sample mean of the data is expected to vary from the actual population mean. It is derived from the standard deviation. To calculate it, simply divide the standard deviation by the square root of the sample size.
In this case, with a standard deviation of \(0.06494\) and a sample size of 20, our standard error becomes \(0.01452\) watts per kilogram. The standard error is crucial because it influences how wide or narrow our confidence interval will be. The smaller the standard error, the more precise our estimate is of the true population mean.