**“Is the Normal Body Temperature (Average) Really 98.6°F?”**

*The establishment of normal body temperature comes from the research of Carl Wunderlich. In 1861 he released data on the temperatures of 25,000 people. It was Carl Wunderlich who established 98.6 *° F* as the norm. Up until 1990, very little was done to refute his established norms. It was not until the publication of “A Critical Appraisal of 98.6” (see below) appeared that the medical community began to make a fuss. Let’s see what the fuss is all about. *

In a1996 article in the *Journal of Statistics Education* (vol. 4, no. 2), entitled “A Critical Appraisal of 98.6”, Allen Shoemaker describes a study that was reported in the Journal of the American Medical Association (JAMA). * It is generally accepted that the mean body temperature of an adult human is 98.6 ° F. In his article, Shoemaker uses the data from the JAMA article to test this hypothesis. You are given the data below. Note: The temperatures given are the peoples “normal” body temperature.

Temperature (in °F) |
# of Adults |
---|---|

96.0-96.4 |
2 |

96.5-96.9 |
4 |

97.0-97.4 |
13 |

97.5-97.9 |
21 |

98.0-98.4 |
38 |

98.5-98.9 |
33 |

99.0-99.4 |
15 |

99.5-99.9 |
2 |

100.0-100.4 |
1 |

100.5-100.9 |
1 |

*Data for the JAMA article were collected from **healthy **men and women, ages 18 to 40, at the University of Maryland Center for Vaccine Development, Baltimore.

1. How many adults did Allen Shoemaker sample in his study on “normal” body temperature?

________________

Note: The sample size from #1 should remain the sample size in your calculator (for all TESTS) for the rest of the lab!

2. Enter the midpoints in L1 and the frequencies in L2, and use 1-VarStats to calculate the mean of the frequency distribution (see section 3.2 if needed).

The sample mean of Shoemaker data: ________________________ (2 decimal places)

For the remaining parts, assume the standard deviation (σ) of normal body temperatures is 0.62º F.

3. It is commonly believed (claimed) that the mean body temperature for healthy adults is 98.6F, so the null hypothesis to test that claim would be that the mean temperature is 98.6F. In hypothesis testing about a mean, µ, we assume the null hypothesis is true. We then use the sample mean, x ̅, to test it.

(a) In hypothesis testing about a mean, µ, we assume the null hypothesis is true. We then use the sample mean, x ̅, to test it. We can use a z-test if the Central Limit Theorem (CLT) holds. (see section 7.5 if needed) The CLT holds if one of the following two conditions is met.

(1) sample size is _________________________, or

(2) population is _____________________________________________.

Would the Central Limit Theorem apply here? _______ Explain why/why not.

(b) The 3 “pieces” of the CLT are given below. Fill in the blanks.

1. μ_x ̅ =μ = _________ (assuming the null hypothesis true)

2. σ_x ̅ =σ⁄√n = ____________ (2 decimal places).

3. The shape of the sampling distribution of x ̅ will be _________________________________.

(c)

(d) Shoemaker’s sample mean (from #2) was __________. Based on the graph above, would this be an unusual sample mean? __________ Without formally performing the test, does our sample mean seem to indicate that the mean temperature is in fact not 98.6°F?________ ( unusual is 2.5 or more standard deviations from the mean.)

#4-6 Rounding Rules

• Round standardized test statistics to 2 decimal places.

• Round p-values to 4 decimal places.

4.

(a)Perform a hypothesis test to test the claim that the mean body temperature for healthy adults is now different than 98.6F. Use a 0.01 level of significance.

Hypotheses:

Standardized Test Statistic: __________

(show formula and your work)

(b) On the sketch below, show the claimed mean and your sample mean, and shade the region for the p-value. Compute the p-value “manually” (show/explain your work; should not be using TESTS).

p-value = _____________________

(c)Conclude the test: Since the p-value is ___________________ than the ________ (enter the number) level of significance, then we _______________________ the null hypothesis and conclude that the data __________________ statistically significant.

(d) Interpretation (in context):

5. P-Value Approach. Use a TEST on your calculator to test the same hypothesis as above, using the P-value approach. You don’t need to state hypotheses again, nor re-write the interpretation.

(a) TEST used and Inputs:

(b) Standardized Test Statistic __________

p-value ____________________

Conclude the test: Since the p-value is ___________________ than the ________ (enter the number) level of significance, then we _______________________ the null hypothesis and conclude that the data __________________ statistically significant.

6. In the 1996 Shoemaker article, it was suggested that 98.2F is actually the adult mean body temperature. At the 0.01 level of significance, can 98.2F be rejected as the mean? Test this hypothesis using the P-value approach (just as you did in #5).

(a) Hypotheses:

(b) Test and Inputs:

(c) Standardized Test Statistic: _________________

(d) P-Value: ____________________

(e) Conclude the test: Since the p-value is ___________________ than the ________ (enter the number) level of significance, then we _______________________ the null hypothesis and conclude that the data __________________ statistically significant.

Interpretation (in context)

7.

(a) Using the sample mean from #2, construct a 99% confidence interval for µ, the mean body temperature for all healthy adults (2 decimal places). Show all work, or state the TEST and inputs used on the calculator, to get the answer.

(b) Interpret the interval in context in a complete sentence.

8.

(a) Confidence Interval (from #7) __________________________________________________

(b) In #4 and #5 above, you tested the null hypothesis that µ = 98.6°F.

What was your conclusion? _______________________________

Is your conclusion consistent with the confidence interval? ___________ Explain.

(c) In #6 above, you tested the null hypothesis that µ = 98.2.

What was your conclusion? ______________________________________

Is your conclusion consistent with the confidence interval? _________________ Explain.

9. Suppose you’re asked to write short for a publication. The title of the article is: “Is the Average Normal Body Temperature Really 98.6°F?”

This should be written as though you were sharing this as a professional article. Summarize in a few sentences defending your position using your work from the lab as evidence.

You may assume your readers understand P-values, confidence intervals, etc.