Question

The average body temperature for healthy adults is 98.6 °F. Is this statement true? Do all healthy people have exactly the same body temperature? A study was conducted a few years go to examine this belief.

The body temperatures of n = 130 healthy adults were measured (half male and half female). The average temperature from the sample was found to be x̄ = 98.249 with standard deviation s = 0.7332.

Do these statistics contradict the belief that the average body temperature is 98.6? If the true average temperature is indeed 98.6 °F and we obtain a sample of n = 130 healthy adults, we would not expect the sample mean to come out exactly equal to 98.6 °F. We observed x̄ = 98.249- can this deviation from 98.6 be explained by chance or is it unlikely we would observe a value this different from 98.6?

Two people debating this issue could come to different conclusions.

Using the methods introduced in this module, discuss how you would determine if the data contradicts the hypothesis that the average body temperature is 98.6°F.

Compare and discuss your methods with your classmates.

Answer 1

Body Temperatures. It is widely believed that the average body temperature for healthy
adults is 98.6 degrees Fahrenheit. Is this true? Where did the 98.6 degree value come from? Do all healthy people have exactly the same body temperature? A study was conducted a few years go to examine this belief. The body temperatures of n = 130 healthy adults were measured (half male and half female). The average temperature from the sample was found to be ¯x = 98.249 with standard deviation s = 0.7332. Do these statistics contradict the belief that the average body temperature is 98.6? If the true average
temperature is indeed 98.6 and we obtain a sample of n = 130 healthy adults, we would not expect the sample mean to come out exactly equal to 98.6. We observed ¯x = 98.249 – can this deviation from 98.6 be explained by chance or is it unlikely we would observe a value this different from 98.6? Two people debating this issue could come to different conclusions. What is needed is an objective method to determine if the data contradict the hypothesis that the average body temperature is 98.6.

In this example, the parameter of interest is µ, the mean temperature of healthy adults. We want to test a hypothesis about µ. The way hypothesis testing is done is that a hypothetical value is proposed for µ which we denote by µ0. The null hypothesis, denoted H0, specifies that µ = µ0:
H0 : µ = µ0.
In the body temperature example, µ0 = 98.6 and the null hypothesis is
H0 : µ = 98.6.
Typically, the null hypothesis represents the “status quo.” The purpose of many studies is to determine if the data leads us to reject the null hypothesis. The alternative hypothesis, denoted Ha, is set up to represent the research goal:
Ha : µ not = µ0.

In the body temperature example, we have Ha : µ not = 98.6. This is an example of a two-sided alternative because we will reject the null hypothesis if there is evidence that the true mean lies to either side (greater or less than) of the hypothesized mean value of 98.6

The way the alternative hypothesis is set up depends on the scientific objective at hand. Many examples of alternative hypotheses are one-sided. For example, if we want to determine if an environmental toxin such as PCB reduces the mean eggshell thickness of pelican birds, then we would set up our hypotheses as: H0 : µ = µ0 versus Ha : µ < µ0, where µ0 is the mean thickness for birds not exposed to PCB.

On the other hand, if we want to test if a new drug increases the mean survival time for people suffering from a particular type of cancer, then we would set up our hypotheses as: H0 : µ = µ0 versus Ha : µ > µ0,

where µ0 is the mean survival time without the medication. Once the data is collected and analyzed, a decision has to be made. Should we reject H0 and accept Ha? Or is there insufficient evidence to reject H0? When making decisions, there are four possible scenarios and two of them involve errors:

1. Accept H0 when in fact H0 is true (good decision).

2. Accept H0 when in fact H0 is false (an error).

3. Reject H0 when in fact H0 is true (an error).

4. Reject H0 when H0 is false (good decision).

The two possible errors above have names:

Definition. A Type I Error is rejecting the null hypothesis H0 when H0 is true. The probability of committing a type I error is denoted by α and is called the significance level of the test. Definition.

A Type II Error is accepting H0 when H0 is false. The probability of a type II error is denoted by β. Definition. The power of a statistical test is 1 − β, which is the probability of rejecting H0 when H0 is false.

When testing hypotheses, we would like the test to have high power which means the ability to conclude the null hypothesis is false when it really is false with high probability. We would also like the probability of a type I error, α, of our tests to be small. Unfortunately, making the probability of a type I error smaller makes the test less powerful; making the test more powerful leads to a higher type I error. Therefore, a compromise is needed between these competing goals when performing hypothesis testing.

Generally, tests are set up so as to minimize the probability of committing a type I error. Typical values for the significance level α, the probability of a type I error, used in practice are 0.05, 0.01, or 0.10. We do not want to reject a null hypothesis that is true. In the body temperature, committing a type I error means that one would conclude the average body temperature differs from 98.6 when in fact the average body temperature is 98.6. Most thermometers for humans are marked at 98.6. Imagine throwing all these thermometers out because a scientific study says they are marked wrong and then realizing later that they were actually marked correctly

A useful analogy for hypothesis testing is a court of law. The defendant is assumed innocent till proven guilty. Thus, the null hypothesis is that the defendant is innocent and the alternative hypothesis is that the defendant is guilty:

H0 : Innocent versus Ha : Guilty.

The trial starts and evidence is presented. In the statistical setting, the data is the evidence. Does the data allow us to reject H0 and conclude Ha? Convicting an innocent man is committing a type I error: rejecting H0 when it is true. We certainly do not want to convict innocent people, so we set up hypothesis tests to minimize the probability of committing this error. On the other hand, we do not want to let a guilty defendant go free (i.e. commit a type II error). Note that there are two reasons a defendant will not be convicted in practice:

1. The defendant is innocent (H0 is true).

2. The defendant is guilty (H0 is false), but we lack enough evidence to convict (a type II error). In statistics, lack of evidence corresponds to lack of data. If we do not have much data (i.e. the sample size is too small), then we will lack the evidence needed to reject H0 when it is false.

We do not necessarily say a defendant is innocent (accept the null hypothesis) if we fail to convict because the failure could be due to insufficient evidence (reasonable doubt remains). Similarly, in hypothesis testing, if we do not reject the null hypothesis, we generally refrain from saying that we accept the null hypothesis (guard against a type II error); instead we may say that we “fail to reject H0.”