Using R for Introductory Statistics : John Verzani

Describing populations 155more would be expected to fail if the passing mark were moved to 23 or better? Assume anormal distribution of scores.5.17 A study found that foot lengths for Japanese women are normally distributed withmean 24.9 centimeters and standard deviation 1.05 centimeters. For this population, findthe probability that a randomly chosen foot is less than 26 centimeters long. What is the95th percentile?5.18 Assume that the average finger length for females is 3.20 inches, with a standarddeviation of 0.35 inches, and that the distribution of lengths is normal. If a glovemanufacturer makes a glove that fits fingers with lengths between 3.5 and 4 inches, whatpercent of the population will the glove fit?5.19 The term “six sigma” refers to an attempt to reduce errors to the point that thechance of their happening is less than the area more than six standard deviations from themean. What is this area if the distribution is normal?5.20 Cereal is sold by weight not volume. This introduces variability in the volumedue to settling. As such, the height to which a cereal box is filled is random. If the heightsfor a certain type of cereal and box have a Normal(12, 0.5) distribution in units of inches,what is the chance that a randomly chosen cereal box has cereal height of 10.7 inches orless?5.21 For the f height variable in the father. son (UsingR) data set, compute whatpercent of the data is within 1, 2, and 3 standard deviations from the mean. Compare tothe percentages 68%, 95%, and 99.7%.5.22 Find the quintiles of the standard normal distribution.5.23 For a Uniform(0, 1) random variable, the mean and variance are 1/2 and 1/12.Find the area within 1, 2, and 3 standard deviations from the mean and compare to 68%,95%, and 99.7%. Do the same for the Exponential(l/5) distribution with mean andstandard deviation of 5.5.24 A q-q plot is an excellent way to investigate whether a distribution isapproximately normal. For the symmetric distributions Uniform(0, 1), Normal(0, 1) and twith 3 degrees of freedom, take a random sample of size 100 and plot a quantile-normalplot using qqnorm(). Compare the three and comment on the curve of the plot as it relatesto the tail length. (The uniform is short-tailed; the t-distribution with 3 degrees offreedom is long-tailed.)5.25 For the t-distribution, we can see that as the degrees of freedom get large thedensity approaches the normal. To investigate, plot the standard normal density with thecommand> curve(dnorm(x),−4,4)and add densities for the t-distribution with k=5,10,25,50, and 100 degrees of freedom.These can be added as follows:> k=5; curve(dt(x,df=k), lty=k, add=TRUE)5.26 The mean of a chi-squared random variable with k degrees of freedom is k. Can youguess the variance? Plot the density of the chi-squared distribution for k=2, 8, 18, 32, 50,and 72, and then try to guess. The first plot can be done with curve (), as in