Chapter 6

A pharmacologist measured the concentration of dopamine in the brains of several rats. The mean concentration was $1,269\mathrm{ng}/\mathrm{gm}$ and the standard deviation was $145\mathrm{ng}/\mathrm{gm}{.}^4$ What was the standard error of the mean if (a) 8 rats were measured? (b) 30 rats were measured?

An experiment is being planned to compare the effects of several diets on the weight gain of beef cattle, measured over a 140 -day test period. ${}^{20}$ In order to have enough precision to compare the diets, it is desired that the standard error of the mean for each diet should not exceed $5\mathrm{kg}$. (a) If the population standard deviation of weight gain is guessed to be about $20\mathrm{kg}$ on any of the diets, how many cattle should be put on each diet in order to achieve a sufficiently small standard error? (b) If the guess of the standard deviation is doubled, to $40\mathrm{kg}$, does the required number of cattle double? Explain.

SGOT is an enzyme that shows elevated activity when the heart muscle is damaged. In a study of 31 patients who underwent heart surgery, serum levels of SGOT were measured 18 hours after surgery. 30 The mean was $49.3\mathrm{U}/\mathrm{l}$ and the standard deviation was $68.3\mathrm{U}/\mathrm{l}$. If we regard the 31 observations as a sample from a population, what feature of the data would cause one to doubt that the population distribution is normal?

To study the conversion of nitrite to nitrate in the blood, researchers injected four rabbits with a solution of radioactively labeled nitrite molecules. Ten minutes after injection, they measured for each rabbit the percentage of the nitrite that had been converted to nitrate. The results were as follows ${}^{49}$ : $$\begin{array}{llll}51.1& 55.4& 48.0& 49.5\end{array}$$ (a) For these data, calculate the mean, median, and the standard error of the mean. (b) Assuming the percentage of nitrite converted to nitrate follows a normal distribution, construct a $90\mathrm{\%}$ confidence interval for the population mean percentage. (c) Without doing any calculations, would a $95\mathrm{\%}$ confidence interval be wider, narrower, or the same width as the confidence interval you found in part (b)? Why?

compute the standard error of $({\overline{Y}}_1-{\overline{Y}}_2)$ for the following data: $$\begin{array}{|lcc|}\hline & \text{ Sample 1 }& \text{ Sample 2 }\\ n& 10& 10\\ \overline{y}& 125& 217\\ s& 44.2& 28.7\\ \hline\end{array}$$

An agronomist measured the heights of $n$ corn plants. ${}^5$ The mean height was $220\mathrm{cm}$ and the standard deviation was $15\mathrm{cm}$. Calculate the standard error of the mean if (a) $n=25$ (b) $n=100$

The diameter of the stem of a wheat plant is an important trait because of its relationship to breakage of the stem, which interferes with harvesting the crop. An agronomist measured stem diameter in eight plants of the Tetrastichon cultivar of soft red winter wheat. All obser-- $-$ wheat vations were made 3 weeks after flowering of the plant. The stem diameters (mm) were as follows ${}^{50}$. $$\begin{array}{llllllll}2.3& 2.6& 2.4& 2.2& 2.3& 2.5& 1.9& 2.0Themeanofthesedatais2.275andthestandarddeviationis\text{(}0.238.\text{)}(a)Calculatethestandarderrorofthemean.(b)Constructa\text{(}95\mathrm{\%}\text{)}confidenceintervalforthepopulationmean.(c)Defineinwordsthepopulationmeanthatyouestimatedinpart(b).(SeeExample6.1.1.)\end{array}$$

Ferulic acid is a compound that may play a role in disease resistance in corn. A botanist measured the con- $-$ centration of soluble ferulic acid in corn seedlings grown in the dark or in a light/dark photoperiod. The results (nmol acid per gm tissue) were as shown in the table. ${}^{41}$ $$\begin{array}{|ccc|}\hline & \text{ Dark }& \text{ Photoperiod }\\ n& 4& 4\\ \overline{y}& 92& 115\\ s& 13& 13\\ \hline\end{array}$$ (a) Construct a $95\mathrm{\%}$ confidence interval for the difference in ferulic acid concentration under the two lighting conditions. (Assume that the two populations from which the data came are normally distributed.) [Note: Formula (6.7.1) yields 6 degrees of freedom for these data. (b) Repeat part (a) for a $90\mathrm{\%}$ level of confidence.

As part of a study of the development of the thymus gland, researchers weighed the glands of five chick embryos after 14 days of incubation. The thymus weights (mg) were as follows ${}^{12}$ $$\begin{array}{lllll}29.6& 21.5& 28.0& 34.6& 44.9\end{array}$$ For these data, the mean is 31.7 and the standard deviation is 8.7 (a) Calculate the standard error of the mean. (b) Construct a $90\mathrm{\%}$ confidence interval for the population mean.

Compute the standard error of $({\overline{Y}}_1-{\overline{Y}}_2)$ for the following data: $$\begin{array}{|ccc|}\hline & \text{ Sample 1 }& \text{ Sample 2 }\\ n& 5& 7\\ \overline{y}& 44& 47\\ s& 6.5& 8.4\\ \hline\end{array}$$