As a clue to the amount of organic waste in Lake Macatawa

A random sample of size 8 from \(N(\mu, 72)\) yielded \(\bar{x}=85\). Find the following confidence intervals for \(\mu:\) (a) \(99 \%\) (b) \(95 \%\). (c) \(90 \%\). (d) \(80 \%\). Equation Transcription: Text Transcription: N(mu,72) Bar x=85 mu 99% 95% 90% 80%

To determine the effect of \(100 \%\) nitrate on the growth of pea plants, several specimens were planted and then watered with \(100 \%\) nitrate every day. At the end of two weeks, the plants were measured. Here are data on seven of them: 17.5 14.5 15.2 14.0 17.3 18.0 13.8 Assume that these data are a random sample from a normal distribution \(N\left(\mu, \sigma^{2}\right)\). (a) Find the value of a point estimate of \(\mu\). (b) Find the value of a point estimate of \(\sigma\). (c) Give the endpoints for a \(90 \%\) confidence interval for \(\mu\). Equation Transcription: Text Transcription: 100% N(mu,sigma^2) mu Sigma 90%

A random sample of size 16 from the normal distribution \(N(\mu, 25)\) yielded \(\bar{x}=73.8\). Find a \(95 \%\) confidence interval for \(\mu\). Equation Transcription: Text Transcription: N(mu,25) Bar x=73.8 95% mu

Let \(X\) equal the weight in grams of a "52-gram" snack pack of candies. Assume that the distribution of \(X\) is \(N(\mu, 4)\). A random sample of \(n=10\) observations of \(X\) yielded the following data: (a) Give a point estimate for \(\mu\). (b) Find the endpoints for a \(95 \%\) confidence interval for (\mu\). (c) On the basis of these very limited data, what is the probability that an individual snack pack selected at random is filled with less than 52 grams of candy? Equation Transcription: Text Transcription: X N(mu,4) n=10 Mu 95%

An automotive supplier of interior parts places several electrical wires in a harness. A pull test measures the force required to pull spliced wires apart. A customer requires that each wire spliced into the harness must withstand a pull force of 20 pounds. Let \(X\) equal the pull force required to pull 20 gauge wires apart. Assume that the distribution of \(X\) is \(N\left(\mu, \sigma^{2}\right)\). The following data give 20 observations of \(X\): 28.8 24.4 30.1 25.6 26.4 23.9 22.1 22.5 27.6 28.1 20.8 27.7 24.4 25.1 24.6 26.3 28.2 2.2 26.3 24.4 (a) Find point estimates for \(\mu\) and \(\sigma\). (b) Find a \(99 \%\) one-sided confidence interval for ? that provides a lower bound for \(\mu\). Equation Transcription: Text Transcription: X N(mu,sigma^2) Mu sigma 99%

As a clue to the amount of organic waste in Lake Macatawa (see Example 7.1-4), a count was made of the number of bacteria colonies in 100 milliliters of water. The number of colonies, in hundreds, for n = 30 samples of water from the east basin yielded 93 140 8 120 3 120 33 70 91 61 7 100 19 98 110 23 14 94 57 9 66 53 28 76 58 9 73 49 37 92 Find an approximate 90% confidence interval for the mean number (say,\(\mu_E\)) of colonies in 100 milliliters of water in the east basin.

To determine whe ther the bacteria count was lower in the west basin of Lake Macatawa than in the east basin, \(n=37\) samples of water were taken from the west basin and the number of bacteria colonies in 100 milliliters of water was counted. The sample characteristics were \(\bar{x}=11.95\) and \(s=11.80\), measured in hundreds of colonies. Find an approximate \(95 \%\) confidence interval for the mean number of colonies (say, \(\mu_{W}\) ) in 100 milliliters of water in the west basin. Equation Transcription: Text Transcription: n=37 Bar x=11.95 s=11.80 95% mu_W

During the Friday night shift, \(n=28\) mints were selected at random from a production line and weighed. They had an average weight of \(\bar{x}=21.45\) grams and a standard deviation of \(s=0.31\) grams. Give the lower endpoint of a \(90 \%\) one-sided confidence interval for \(\mu\), the mean weight of all the mints. Equation Transcription: Text Transcription: n=28 Bar x=21.45 s=0.31 90%

Thirteen tons of cheese, including "22-pound" wheels (label weight), is stored in some old gypsum mines. A random sample of \(n=9\) of these wheels yielded the following weights in pounds: 21.50 18.95 18.55 19.40 19.15 22.35 22.90 22.20 23.10 Assuming that the distribution of the weights of the wheels of cheese is \(N\left(\mu, \sigma^{2}\right)\), find a \(95 \%\) confidence interval for \(\mu\). Equation Transcription: Text Transcription: n=9 N(mu, sigma^2) 95% mu

A leakage test was conducted to determine the effectiveness of a seal designed to keep the inside of a plug airtight. An air needle was inserted into the plug, and the plug and needle were placed under water. The pressure was then increased until leakage was observed. Let \(X\) equal the pressure in pounds per square inch. Assume that the distribution of \(X\) is \(N\left(\mu, \sigma^{2}\right)\). Use the observations to (a) Find a point estimate of \(\mu\). (b) Find a point estimate of \(\sigma\). (c) Find a \(95 \%\) one-sided confidence interval for \(\mu\) that provides an upper bound for \(\mu\). Equation Transcription: Text Transcription: X N(mu, sigma^2) Mu sigma n=10 95%

Students took \(n=35\) samples of water from the east basin of Lake Macatawa (see Example 7.1-4) and measured the amount of sodium in parts per million. For their data, they calculated \(\bar{x}=24.11\) and \(s^{2}=24.44\). Find an approximate \(90 \%\) confidence interval for \(\mu\), the mean of the amount of sodium in parts per million. Equation Transcription: Text Transcription: n=35 Bar x=24.11 s^2=24.44 90% mu

In nuclear physics, detectors are often used to measure the energy of a particle. To calibrate a detector, particles of known energy are directed into it. The values of signals from 15 different detectors, for the same energy, are (a) Find a \(95 \%\) confidence interval for \(\mu\), assuming that these are observations from a \(N\left(\mu, \sigma^{2}\right)\) distribution. (b) Construct a box-and-whisker diagram of the data. (c) Are these detectors doing a good job or a poor job of putting out the same signal for the same input energy? Equation Transcription: Text Transcription: 95% Mu N(mu,sigma^2)

A study was conducted to measure (1) the amount of cervical spine movement induced by different methods of gaining access to the mouth and nose to begin resuscitation of a football player who is wearing a helmet and (2) the time it takes to complete each method. One method involves using a manual screwdriver to remove the side clips holding the face mask in place and then flipping the mask up. Twelve measured times in seconds for the manual screwdriver are 33.8 31.6 28.5 29.9 29.8 26.0 35.7 27.2 29.1 32.1 26.1 24.1 Assume that these are independent observations of a normally distributed random variable that is \(N\left(\mu, \sigma^{2}\right)\). (a) Find point estimates of \(\mu\) and \(\sigma\). (b) Find a \(95 \%\) one-sided confidence interval for \(\mu\) that provides an upper bound for . (c) Does the assumption of normality seem to be justified? Why? Equation Transcription: Text Transcription: N(mu,sigma^2) Mu sigma 95%

Let \(X_{1}, X_{2^{\prime} \cdots}, X_{n}\) be a random sample of size \(n\) from the normal distribution \(N\left(\mu, \sigma^{2}\right)\) . Calculate the expected length of a confidence interval for \(\mu\), assuming that \(n=5\) and the variance is (a) known. (b) unknown. HINT: To find \(E(S)\), first determine \(E\left[\sqrt{(n-1) S^{2} / \sigma^{2}}\right]\), recalling that \((n-1) S^{2} / \sigma^{2}\) is \(\chi^{2}(n-1)\). (See Exercise 6.4-14.) Equation Transcription: Text Transcription: X_1,X_2,...,X_n n N(mu,sigma^2) mu n=5 E(S) E[sqrt(n-1)S^2/sigma^2 (n-1)S2/2 chi^2(n-1)

PROBLEM 8E Assume that the yield per acre for a particular variety of soybeans is N(?, ?2). For a random sample of n = 5 plots, the yields in bushels per acre were 37.4, 48.8, 46.9, 55.0, and 44.0. (a) Give a point estimate for ?. (b) Find a 90% confidence interval for ?.

A random sample of size 16 from the normal distribution N(, 25) yielded x = 73.8. Find a 95% confidence interval for .

A random sample of size 8 from N(, 72) yielded x = 85. Find the following confidence intervals for : (a) 99%. (b) 95%. (c) 90%. (d) 80%.

To determine the effect of 100% nitrate on the growth of pea plants, several specimens were planted and then watered with 100% nitrate every day. At the end of two weeks, the plants were measured. Here are data on seven of them: 17.5 14.5 15.2 14.0 17.3 18.0 13.8 Assume that these data are a random sample from a normal distribution N(, 2). (a) Find the value of a point estimate of . (b) Find the value of a point estimate of . (c) Give the endpoints for a 90% confidence interval for .

Let X equal the weight in grams of a 52-gram snack pack of candies. Assume that the distribution of X is N(, 4). A random sample of n = 10 observations of X yielded the following data: 55.95 56.54 57.58 55.13 57.48 56.06 59.93 58.30 52.57 58.46 (a) Give a point estimate for . (b) Find the endpoints for a 95% confidence interval for . (c) On the basis of these very limited data, what is the probability that an individual snack pack selected at random is filled with less than 52 grams of candy?

As a clue to the amount of organic waste in Lake Macatawa (see Example 7.1-4), a count was made of the number of bacteria colonies in 100 milliliters of water. The number of colonies, in hundreds, for \(n = 30\) samples of water from the east basin yielded \(\begin{array}{rrrrrrrrrr} 93 & 140 & 8 & 120 & 3 & 120 & 33 & 70 & 91 & 61 \\ 7 & 100 & 19 & 98 & 110 & 23 & 14 & 94 & 57 & 9 \\ 66 & 53 & 28 & 76 & 58 & 9 & 73 & 49 & 37 & 92 \end{array}\) Find an approximate 90% confidence interval for the mean number (say, \(\left.\mu_{E}\right)\)) of colonies in 100 milliliters of water in the east basin

To determine whether the bacteria count was lower in the west basin of Lake Macatawa than in the east basin, n = 37 samples of water were taken from the west basin and the number of bacteria colonies in 100 milliliters of water was counted. The sample characteristics were x = 11.95 and s = 11.80, measured in hundreds of colonies. Find an approximate 95% confidence interval for the mean number of colonies (say, W) in 100 milliliters of water in the west basin.

Thirteen tons of cheese, including 22-pound wheels (label weight), is stored in some old gypsum mines. A random sample of n = 9 of these wheels yielded the following weights in pounds: 21.50 18.95 18.55 19.40 19.15 22.35 22.90 22.20 23.10 Assuming that the distribution of the weights of the wheels of cheese is N(, 2), find a 95% confidence interval for .

Assume that the yield per acre for a particular variety of soybeans is N(, 2). For a random sample of n = 5 plots, the yields in bushels per acre were 37.4, 48.8, 46.9, 55.0, and 44.0. (a) Give a point estimate for . (b) Find a 90% confidence interval for .

During the Friday night shift, n = 28 mints were selected at random from a production line and weighed. They had an average weight of x = 21.45 grams and a standard deviation of s = 0.31 grams. Give the lower endpoint of a 90% one-sided confidence interval for , the mean weight of all the mints.

A leakage test was conducted to determine the effectiveness of a seal designed to keep the inside of a plug airtight. An air needle was inserted into the plug, and the plug and needle were placed under water. The pressure was then increased until leakage was observed. Let X equal the pressure in pounds per square inch. Assume that the distribution of X is N(, 2). The following n = 10 observations of X were obtained: 3.1 3.3 4.5 2.8 3.5 3.5 3.7 4.2 3.9 3.3 Use the observations to (a) Find a point estimate of . (b) Find a point estimate of . (c) Find a 95% one-sided confidence interval for that provides an upper bound for .

Students took n = 35 samples of water from the east basin of Lake Macatawa (see Example 7.1-4) and measured the amount of sodium in parts per million. For their data, they calculated x = 24.11 and s2 = 24.44. Find an approximate 90% confidence interval for , the mean of the amount of sodium in parts per million.

In nuclear physics, detectors are often used to measure the energy of a particle. To calibrate a detector, particles of known energy are directed into it. The values of signals from 15 different detectors, for the same energy, are 260 216 259 206 265 284 291 229 232 250 225 242 240 252 236 (a) Find a 95% confidence interval for , assuming that these are observations from a N(, 2) distribution. (b) Construct a box-and-whisker diagram of the data. (c) Are these detectors doing a good job or a poor job of putting out the same signal for the same input energy?

A study was conducted to measure (1) the amount of cervical spine movement induced by different methods of gaining access to the mouth and nose to begin resuscitation of a football player who is wearing a helmet and (2) the time it takes to complete each method. One method involves using a manual screwdriver to remove the side clips holding the face mask in place and then flipping the mask up. Twelve measured times in seconds for the manual screwdriver are 33.8 31.6 28.5 29.9 29.8 26.0 35.7 27.2 29.1 32.1 26.1 24.1 Assume that these are independent observations of a normally distributed random variable that is N(, 2). (a) Find point estimates of and . (b) Find a 95% one-sided confidence interval for that provides an upper bound for . (c) Does the assumption of normality seem to be justified? Why?

Let X1, X2, ... , Xn be a random sample of size n from the normal distribution N(, 2). Calculate the expected length of a 95% confidence interval for , assuming that n = 5 and the variance is (a) known. (b) unknown. Hint: To find E(S), first determine E[  (n 1)S2/2 ], recalling that (n 1)S2/2 is 2(n 1). (See Exercise 6.4-14.)

An automotive supplier of interior parts places several electrical wires in a harness. A pull test measures the force required to pull spliced wires apart. A customer requires that each wire spliced into the harness must withstand a pull force of 20 pounds. Let X equal the pull force required to pull 20 gauge wires apart. Assume that the distribution of X is \(N(\mu, \sigma^2)\). The following data give 20 observations of X: 28.8 24.4 30.1 25.6 26.4 23.9 22.1 22.5 27.6 28.1 20.8 27.7 24.4 25.1 24.6 26.3 28.2 22.2 26.3 24.4 (a) Find point estimates for \(\mu\) and \(\sigma\). (b) Find a 99% one-sided confidence interval for ? that provides a lower bound for \(\mu\).

Let \(S^2\) be the variance of a random sample of size n from \(N(\mu, \sigma^2)\). Using the fact that \((n ? 1)S^2/\sigma^2\) is \(\chi^2(n-1)\), note that the probability \(P\left[a \leq \frac{(n-1) S^{2}}{\sigma^{2}} \leq b\right]=1-\alpha\), where \(a=\chi_{1-\alpha / 2}^{2}(n-1)\) and \(b=\chi_{\alpha / 2}^{2}(n-1)\). Rewrite the inequalities to obtain \(P\left[\frac{(n-1) S^{2}}{b} \leq \sigma^{2} \leq \frac{(n-1) S^{2}}{a}\right]=1-\alpha\), If n = 13 and \(12 S^{2}=\sum_{i=1}^{13}\left(x_{i}-\bar{x}\right)^{2}=128.41\), show that [6.11, 24.57] is a 90% confidence interval for the variance \(\sigma^2\). Accordingly, [2.47, 4.96] is a 90% confidence interval for \(\sigma\).

Let X be the mean of a random sample of size n from N(, 9). Find n so that P( X1 << X+1) = 0.90