The acidity of soils is measured by a quantity called the pH$,$ which may range from $0($high acidity$)$ to $14($high alkalinity$).$ A soil scientist wants to estimate the average pH for a large field by randomly selecting n core samples and measuring the pH in each sample. Although the population standard deviation of pH

measurements is not known, past experience indicates that most soils have a pH value of between $5$ and $8.$

a$.$ If the scientist selects n $=40$ samples, find the approximate probability that the sample mean of the $40$ pH measurements will be within $0\mathrm{.}2$ unit of the true average pH for the field. b$).$The soil scientist would like the sample mean to be within $\mathrm{.}1$ of the true mean with probability $\mathrm{.}90.$

How many core samples should the scientist take?