Question

What is normal, when it comes to people's body temperatures? A random sample of 130 human body temperatures, provided by Allen Shoemaker \(^{9}\) in the Journal of Statistical Education, had a mean of 98.25 degrees and a standard deviation of 0.73 degrees. a. Construct a \(99 \%\) confidence interval for the average body temperature of healthy people. b. Does the confidence interval constructed in part a contain the value 98.6 degrees, the usual average temperature cited by physicians and others? If not, what conclusions can you draw?

Short answer

Answer: No, the 99% confidence interval for the average body temperature of healthy people is (98.085, 98.415) and does not contain the value 98.6 degrees.

Step-by-step solution

a. Constructing a 99% confidence interval for the average body temperature of healthy people

To construct a 99% confidence interval, we first need to determine the critical z-value, \(z_{\frac{\alpha}{2}}\), for a 99% confidence level. We can find this using a z-table or from a calculator. Since this is a two-tailed confidence interval, we want to find the z-score that corresponds to the area between the z-score and -z-score, which is 0.995 (as half of the remaining area, i.e., 0.01, is split between the two tails). Using a z-table, we find the closest value to 0.995, which is 2.576. Now we can plug all the information into the formula for the confidence interval: \(\bar{x} \pm z_{\frac{\alpha}{2}} \times \frac{s}{\sqrt{n}}\) Where: \(\bar{x} = 98.25 \) (the sample mean) \(s = 0.73 \) (the sample standard deviation) \(n = 130 \) (the sample size) \(z_{\frac{\alpha}{2}} = 2.576 \) Calculating the confidence interval, we get: \(98.25 \pm 2.576 \times \frac{0.73}{\sqrt{130}}\) Calculate the margin of error: \(2.576 \times \frac{0.73}{\sqrt{130}} = 0.165\) So the 99% confidence interval for the average body temperature of healthy people is: \((98.25 - 0.165, 98.25 + 0.165) = (98.085, 98.415)\)

b. Checking if the confidence interval contains the value 98.6 degrees

Now we need to check if the confidence interval we calculated in part a contains the value 98.6 degrees. The confidence interval we calculated is \((98.085, 98.415)\). Since 98.6 is not within this interval, we can conclude that based on this sample, the usual average temperature cited by physicians and others (98.6 degrees) is not within the 99% confidence interval of the average body temperature for healthy people. This means that there is significant evidence at the 1% level of significance to reject the hypothesis that the true average body temperature is 98.6 degrees.

Key concepts

Standard Deviation

Standard deviation is a statistic that measures the amount of variation or dispersion in a set of values. It is often denoted by the letter 's' for a sample or 'σ' (sigma) for a population. In simple terms, a low standard deviation indicates that the values tend to be close to the mean (average) of the set, while a high standard deviation indicates that the values are spread out over a wider range.

For example, consider a random sample of human body temperatures with a standard deviation of 0.73 degrees. This indicates that most of the temperatures in our sample are within 0.73 degrees of the average temperature. Understanding standard deviation is crucial for interpreting the spread of data points and for calculating confidence intervals, which reflect the reliability of an estimate.

Sample Mean

The sample mean, often denoted as \(\bar{x}\), is the average of the values in a sample. It is calculated by summing all the values and then dividing by the number of values. The sample mean is a commonly used estimate of the population mean, but since it is based on a sample, it comes with uncertainty. This uncertainty can be quantified using a confidence interval.

Continuing with our example from the introduction, the sample mean of the body temperatures is 98.25 degrees. This is our best estimate for the average body temperature of healthy people based on the sample data. However, to make inferences about the average temperature for all healthy people, we need to calculate a confidence interval that will give us a range of plausible values for the population mean.

Z-Score

The z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, all measured in terms of standard deviations. It indicates how many standard deviations an element is from the mean. A z-score can be positive or negative, with a positive value indicating that the score is above the mean, and a negative score indicating it's below the mean.

In the context of confidence intervals, the critical z-value (denoted as \(z_{\frac{\alpha}{2}}\)) corresponds to the desired confidence level. This z-score is used to determine the range within which a certain percentage of the data is expected to fall. For a 99% confidence level, the z-score is high, reflecting the fact that we expect 99% of all possible sample means to fall within the interval defined by this z-score.

Normal Distribution

The normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In graph form, this distribution is represented by a bell-shaped curve, known as the 'bell curve.'

The normal distribution is extremely important in statistics because of its widespread appearance in natural phenomena. Many statistical tests assume that the data follows a normal distribution. For confidence interval calculations, if we assume that the population distribution is normal, then the sampling distribution of the sample mean is also normal. This property is what allows us to use z-scores and the standard deviation of the sample mean to calculate confidence intervals for population parameters like the average body temperature.