Q 8.99. - Elementary Statistics

Question

Body Temperature. A study by researchers at the University of Maryland addressed the question of whether the mean body temperature of humans is $98.6°\mathrm{F}$ The results of the study by $\mathrm{P}$ Mackowiak et al. appeared in the article "A Critical Appraisal of $98.6$ F. the Upper Limit of the Normal Body Temperature, and Other Legacies of Carl Reinhold August Wunderlich" (Journal of the American Medical Association, Vol. $268$, pp. $1578-1580$). Among other data, the researchers obtained the body temperatures of 93 healthy humans, as provided on the Weiss Stats site. Use the technology of your choice to do the following.

a. Obtain a normal probability plot, boxplot, histogram, and stem-and-leaf diagram of the data.

b. Based on your results from part (a), can you reasonably apply the z-interval procedure to the data? Explain your reasoning.

c. Find and interpret a $99\%$ confidence interval for the mean body temperature of all healthy humans. Assume that $\sigma =0.63°\mathrm{F}$ Does the result surprised you? Why?

Step-by-Step Solution

Answer

Part (a) 

Stem-and-leaf of TEMP $\mathrm{N}=93$

Leaf Unit $=0.10$

$$\begin{gathered} \begin{array}{lll}1& 96& 7\\ 3& 96& 89\\ 8& 97& 00001\\ 13& 97& 22233\\ 19& 97& 444444\\ 26& 97& 6666777\\ 31& 97& 88889\\ 45& 98& 00000000000111\\ 10)& 98& 2222222233\\ 38& 98& 4444445555\\ 28& 98& 66666666677\end{array} \\ 17 98 8888888 \\ 10 99 00001 \\ 5 99 2233 \\ 1 99 4 \end{gathered}$$

Part (b) The applying $z-$interval procedure to the given data is reasonable.

Part (c) The $99\%$ confidence interval for the mean body temperature of all healthy humans is $(97.9554,98.2919)$ according to the MINITAB output.

Part (a) Step 1: Given information

we are $95\%$ certain that the mean additional hours of sleep for all patients using laevohysocymine hydrobromide is between $0.90$and $3.76$ hours.

Part (a) Step 2: Explanation

Draw the boxplot for body temperature with MINITAB.

Procedure for MINITAB:

Step 1: Select Graph>  Statistic or Boxplot> EDA >Boxplot from the menu bar.

Step 2: Select Simple under One$Y\text{'}s$ Click OK.

Step 3: In the Graph variables, fill in the Temp data.

Step 4: Click the OK button.

OUTPUT FROM MINITAB: 

Draw the normal probability plot for body temperature with MINITAB.

Procedure for MINITAB:

Step 1: Select Probability Plot from the Graph menu.

Step 2: Click OK after selecting Single.

Step 3: In the Graph variables section, type Temp in the column.

Step 4: Click the OK button.

OUTPUT FROM MINITAB: 

Part (a) Step 3: Explanation

Draw the histogram for body temperature with MINITAB.

Procedure for MINITAB:

Step 1: Select Histogram from the Graph menu.

Step 2: Click OK after selecting Simple.

Step 3: In the Graph variables section, input the Temp column.

Step 4: Click the OK button. 

Draw the stem-and-leaf diagram for body temperature with MINITAB.

Procedure for MINITAB:

Select Graph$>$Stem and leaf in the first step.

Step 2: In Graph variables, select the Temp column of variables.

Step 3: Choose OK.

OUTPUT FROM MINITAB: 

Stem-and-Leaf Display: TEMP

Stem-and-leaf of TEMP $\mathrm{N}=93$

Leaf Unit $=0.10$

$$\begin{gathered} \begin{array}{lll}1& 96& 7\\ 3& 96& 89\\ 8& 97& 00001\\ 13& 97& 22233\\ 19& 97& 444444\\ 26& 97& 6666777\\ 31& 97& 88889\\ 45& 98& 00000000000111\\ 10)& 98& 2222222233\\ 38& 98& 4444445555\\ 28& 98& 66666666677\end{array} \\ 17 98 8888888 \\ 10 99 00001 \\ 5 99 2233 \\ 1 99 4 \end{gathered}$$

Part (b) Step 1: Explanation

Check whether using the $z-$interval technique on the given data is appropriate.

The following are the conditions for using the $z-$interval procedure:

Small Sample size: 

The $z-$interval approach is employed when the sample size is less than $15$ and the variable is normally distributed or extremely near to being normally distributed.

Moderate Sample size: 

When the sample size is between $15$ and $30$ and the variable is not normally distributed or there is no outlier in the data, the $z-$interval technique is performed.

Large Sample size: 

The $z-$interval technique is utilized without restriction if the sample size is bigger than $30$

It is evident from section (a) that the data does not contain any outliers. The sample size is also $(=93)$ greater. As a result, the body temperature distribution is roughly typical. As a result, using the $z-$interval technique to analyze the data seems plausible.

Part (c) Step 1: Explanation

Using MINITAB, calculate the $99\%$ confidence interval for the mean body temperature of all healthy humans.

Procedure for MINITAB:

Step 1: Select Stat$>$ Basic Statistics $>$ 1-Sample $Z$from the drop-down menu.

Step 2: Select the Temp column from Samples in Column.

Step 3: Type $0.63$ in the Standard deviation box.

Step 4: Select Options and enter $99$ as the level of confidence.

Step 5: In the alternative, select not equal.

Step 6: In all dialogue boxes, click OK.

OUTPUT FROM MINITAB: 

One-Sample Z: TEMP

The assumed standard deviation $=0.63$

$\begin{array}{lrrrrr}\text{ Variable }& \mathrm{N}& \text{ Mean }& \text{ StDev }& \text{ SE Mean }& 998\text{ CI }\\ \text{ TEMP }& 93& 98.1237& 0.6468& 0.0653& (97.9554,98.2919)\end{array}$

The $99\%$ confidence interval for the mean body temperature of all healthy humans is $(97.9554,98.2919)$, according to the MINITAB output.

The mean body temperature of all healthy humans is estimated to be between $97.9554$ and $98.2919$