Chapter 5

In an agricultural experiment, a large field of wheat was divided into many plots (each plot being $7\times$ $100\mathrm{ft}$ ) and the yield of grain was measured for each plot. These plot yields followed approximately a normal distribution with mean $88\mathrm{lb}$ and standard deviation $7\mathrm{lb}$ (as in Exercise 4.3.5). Let $\overline{Y}$ represent the mean yield of five plots chosen at random from the field. Find $$\begin{gathered} \mathrm{Pr} \\ \overline{Y}>80 \end{gathered}$$.

Consider interviewing a random sample of $n=50$ adults. Let $\hat{P}$ denote the proportion of the 50 sampled adults who drink coffee. If the population proportion of coffee drinkers is $0.80,$ what is the appropriate approximate model for the distribution of $\hat{P}$ over many such samples of size $50?$ That is, what type of distribution is this, what is the mean, and what is the standard deviation?

A fair coin is to be tossed 20 times. Find the probability that 10 of the tosses will fall heads and 10 will fall tails, (a) using the binomial distribution formula. (b) using the normal approximation with the continuity correction.

Consider taking a random sample of size 20 from the population of students at a certain college and measuring the diastolic blood pressure each of the 20 students. In the context of this setting, explain what is meant by the sampling distribution of the sample mean.

In the United States, $44\mathrm{\%}$ of the population has type O blood. Suppose a random sample of 12 persons is taken. Find the probability that 6 of the persons will have type $\mathrm{O}$ blood (and 6 will not) (a) using the binomial distribution formula. (b) using the normal approximation.

The serum cholesterol levels of a population of 12 to 14-year-olds follow a normal distribution with mean $155\mathrm{mg}/\mathrm{dl}$ and standard deviation $27\mathrm{mg}/\mathrm{dl}$ (as in Example 4.1.1). (a) What percentage of the 12 - to 14 -year-olds have serum cholesterol values between 145 and $165\mathrm{mg}/\mathrm{dl}?$ (b) Suppose we were to choose at random from the population a large number of groups of nine 12 - to 14-year-olds each. In what percentage of the groups would the group mean cholesterol value be between 145 and $165\mathrm{mg}/\mathrm{dl}?$ (c) If $\overline{Y}$ represents the mean cholesterol value of a random sample of nine 12 - to 14 -year-olds from the population, what is $$\begin{gathered} \mathrm{Pr} \\ 145\le \overline{Y}\le 165 \\ ? \end{gathered}$$

The heights of men in a certain population follow a normal distribution with mean 69.7 inches and standard deviation 2.8 inches ${}^{15}$. (a) If a man is chosen at random from the population, find the probability that he will be more than 76 inches tall. (b) If two men are chosen at random from the population, find the probability that (i) both of them will be more than 76 inches tall; (ii) their mean height will be more than 76 inches.

Consider a hypothetical population of dogs in which there are four possible weights, all of which are equally likely: $42,48,52,$ or 58 pounds. If a sample of size $n=2$ is drawn from this population, what is the sampling distribution of the total weight of the two dogs selected? That is, what are the possible values for the total and what are the probabilities associated with each of those values?

An important indicator of lung function is forced expiratory volume (FEV), which is the volume of air that a person can expire in one second. Dr. Hernandez plans to measure FEV in a random sample of $n$ young women from a certain population, and to use the sample mean $\overline{y}$ as an estimate of the population mean. Let $E$ be the event that Hernandez's sample mean will be within $\pm 100\mathrm{ml}$ of the population mean. Assume that the population distribution is normal with mean $3,000\mathrm{ml}$ and standard deviation $400{\mathrm{ml}}^3$ Find $$\begin{gathered} \mathrm{Pr} \\ E \end{gathered}$$ if (a) $n=15$ (b) $n=60$ (c) How does $$\begin{gathered} \mathrm{Pr} \\ E \end{gathered}$$ depend on the sample size? That is, as $n$ increases, does $$\begin{gathered} \mathrm{Pr} \\ E \end{gathered}$$ increase, decrease, or stay the same?

Consider random sampling from a dichotomous population with $p=0.3,$ and let $E$ be the event that $\hat{P}$ is within ±0.05 of $p.$ Use the normal approximation (without the continuity correction) to calculate $$\begin{gathered} \mathrm{Pr} \\ E \end{gathered}$$ for a sample of size $n=400$.