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Chapter 13: Problem 48

Many people believe that healthy people typically have a body temperature of \(98.6^{\circ} \mathrm{F}\). We took a random sample of 10 people and found the following temperatures: $$ 98.4,98.8,98.7,98.7,98.6,97.2,98.4,98.0,98.3, \text { and } 98.0 $$ Use the sign test to test the hypothesis that the median is not \(98.6\).

Short Answer

Expert verified
To answer the question, we need to perform a sign test based on the comparisons of given temperature values against the hypothesized median of 98.6. After getting the signs, the test is applied against the null hypothesis that the median is 98.6. The results then determine whether the actual median is significantly different from the proposed median.

Step by step solution

01

Compare each individual value with the median

Firstly, compare each body temperature measurement in the sample against the hypothesized median, which is \(98.6^{\circ} F\). Let's note the number of values that are greater than, equal to, or less than 98.6.
02

Count the Number of Positive and Negative Signs

Once comparisons are finished, count the number of positive signs (those greater than 98.6) and negative signs (those less than 98.6). The individuals with temperatures exactly equal to 98.6 are not incorporated into the count.
03

Conduct the Sign Test

Upon having the number of positive and negative signs, we have to carry out the sign test. The base assumption (null hypothesis) for the sign test is that the median is 98.6, which implies that we'd anticipate an equal number of positive and negative signs (equal to and less than signs are not considered). The sign test concentrates on the least common sign. Binomial probability calculation is conducted with an expectation of 0.5 chance for each sign under null hypothesis. If the probability is low, null hypothesis would be rejected.
04

Decision Making

Finally, interpret the results. If the calculated probability is less than the threshold significance level (commonly 0.05), it would be concluded that the median is significantly different from 98.6. Otherwise, we would not reject the null hypothesis.

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