(a) If X has distribution function F(x), we can show that F(X) is uniformly distributed

The following data represent the number of aphids per plant found in a sample of 10 plants: 17, 13, 21, 47, 3, 6, 12, 25, 0, 18 Find the median, the sample mean, and the sample variance.

The following data represent the number of seeds per flower head in a sample of nine flowering plants: 27, 39, 42, 18, 21, 33, 45, 37, 21 Find the median, the sample mean, and the sample variance.

The following data represent the age of patients in a clinical trial: 28, 45, 34, 36, 30, 42, 35, 45, 38, 27 Find the median, the sample mean, and the sample variance.

The following data represent blood cholesterol levels, in mg/dL, of patients in a clinical trial: 174, 138, 212, 203, 194, 245, 146, 149, 164, 209, 158 Find the median, the sample mean, and the sample variance.

The following data represent the frequency distribution of seed numbers per flower head in a flowering plant: Seed Number Frequency 9 37 10 48 11 53 12 49 13 61 14 42 15 31 Calculate the sample mean and the sample variance.

The following data represent the frequency distribution of the numbers of days that it took a certain ointment to clear up a skin rash: Number of Days Frequency 1 2 2 7 3 9 4 27 5 11 6 5 Calculate the sample mean and the sample variance.

The following data represent the relative frequency distribution of clutch size in a sample of 300 laboratory guinea pigs: Clutch Size Relative Frequency 2 0.05 3 0.09 4 0.12 5 0.19 6 0.23 7 0.12 8 0.13 9 0.07 Calculate the sample mean and the sample variance.

The following data represent the relative frequency distribution of clutch size in a sample of 42 mallards: Clutch Size Relative Frequency 6 0.10 7 0.24 8 0.29 9 0.21 10 0.16 Calculate the sample mean and the sample variance.

Assume that a population consists of the three numbers 1, 6, and 8. List all samples of size 2 that can be drawn from this population with replacement, and find the sample mean of each sample.

Use a graphing calculator to generate five samples, each of size 6, from a uniform distribution over the interval (0,1). Compute the sample means of each sample.

Let (X1, X2, . . . , Xn) denote a sample of size n. Show that #n k=1 (Xk X) = 0 where X is the sample mean.

Let (X1, X2, . . . , Xn) denote a sample of size n. Show that nX2 = 1 n _ #n k=1 Xk _2 where X is the sample mean.

Assume that a sample of size n has l distinct values x1, x2, . . . , xl , where xk occurs fk times in the sample. Explain why the sample mean is given by the formula X = 1 n #l k=1 xk fk

Assume that a sample of size n has l distinct values x1, x2, . . . , xl , where xk occurs fk times in the sample. Explain why the sample variance is given by the formula S2 = 1 n 1 #l k=1 x2 k fk 1 n _ #l k=1 xk fk _2

Assume that X is exponentially distributed with parameter = 3.0. (a) Assume that a sample of size 50 is taken from this population. What is the approximate distribution of the sample mean? (b) Assume now that 1000 samples, each of size 50, are taken from this population and a histogram of the sample means of each of the samples is produced. What shape will the histogram be approximately?

Assume that X is exponentially distributed with parameter = 3.0. Assume that a sample of size 50 is taken from this population and that the sample mean of this sample is calculated. How likely is it that the sample mean will exceed 0.43?

Use the random-number generator on a graphing calculator to generate three samples, each of size 10, from a uniform distribution over the interval (0, 1). (a) Compute the sample mean and the sample variance of each sample. (b) Combine all three samples, and compute the mean and the sample variance of the combined sample. (c) Compare your answers in (a) and (b) with the true values of the mean and the variance.

Suppose that X is exponentially distributed with mean 1. A computer generates the following sample of independent observations from the population X: 0.3169, 0.5531, 2.376, 1.150, 0.6174, 0.1563, 2.936, 1.778, 0.7357, 0.1024 Find the sample mean and the sample variance, and compare them with the corresponding population parameters.

Compute the sample mean and the standard error for the sample in Problem 1.

Compute the sample mean and the standard error for the sample in Problem 2.

The following data represent a sample from a normal distribution with mean 0 and variance 1: 0.68, 1.22, 1.33,0.84,0.06, 0.50, 0.03,0.13,0.29,0.47 Construct a 95% confidence interval.

The following data represent a sample from a normal distribution with mean 0 and variance 1: 1.18, 0.52, 0.36,0.16, 0.92, 0.68,0.61,0.54, 0.15, 1.04 Construct a 95% confidence interval.

Use a graphing calculator to construct a 95% confidence interval for a sample of size 30 from a uniform distribution over the interval (0, 1). Take a class poll to determine the percentage of confidence intervals that contain the true mean. Discuss the result in class.

(a) If X has distribution function F(x), we can show that F(X) is uniformly distributed over the interval (0, 1). Use this fact, a graphing calculator, and the table for the standard normal distribution to generate 15 standard normally distributed random variables. (b) Use your data from (a) to construct a 95% percent confidence interval. Take a class poll to determine the percentage of confidence intervals that contain the true mean. Discuss the result in class.

To determine the germination success of seeds of a certain plant, you plant 162 seeds. You find that 117 of the seeds germinate. Estimate the probability of germination and give a 95% confidence interval.

To test a new drug for lowering cholesterol, 72 people with elevated cholesterol receive the drug; 51 of them show reduced cholesterol levels. Estimate the probability that the drug lowers cholesterol, and construct a 95% confidence interval.

In Problems 27 and 28, fit a linear regression line through the given points and compute the coefficient of determination. (3,6.3), (2,5.6), (1,3.3), (0, 0.1), (1, 1.7), (2, 2.1)

In Problems 27 and 28, fit a linear regression line through the given points and compute the coefficient of determination. (0, 0.1), (1,1.3), (2,3.5), (3,5.7), (4,5.8)

Show that the sum of the residuals about any linear regression line is equal to 0.

Show that the last two terms in (12.48), namely 2(y a bx) # (yk y) and 2(y a bx) # (xk x) are equal to 0.

To determine whether the frequency of chirping crickets depends on temperature, the following data were obtained (Pierce, 1949): Temperature (F) Chirps/s 69 15 70 15 72 16 75 16 81 17 82 17 83 16 84 18 89 20 93 20 Fit a linear regression line to the data, and compute the coefficient of determination.

The initial velocity v of an enzymatic reaction that follows MichaelisMenten kinetics is given by v = vmaxs Km + s (12.54) where s is the substrate concentration and vmax and Km are two parameters that characterize the reaction. The following computer-generated table contains values of the initial velocity v when the substrate concentration s was varied: s v 1 4.1 2.5 6.1 5 9.3 10 12.9 20 17.1 (a) Invert (12.54) and show that 1 v = Km vmax 1 s + 1 vmax (12.55) This is the LineweaverBurk equation. If we plot 1/v as a function of 1/s, a straight line with slope Km/vmax and intercept 1/vmax results. Use (12.55) to transform the data, and fit a linear regression line to the transformed data. Find the slope and the intercept of the linear regression line, and determine Km and vmax. (b) Dowd and Riggs (1965) proposed to use the transformation v = vmax Km v s (12.56) and then plot v against v/s. The resulting straight line has slope Km and intercept vmax. Use (12.56) to transform the data, and fit a linear regression line to the transformed data. Find the slope and the intercept of the linear regression line, and determine Km and vmax.