A bowl contains two red balls, two white balls, and a fifth ball that is either red or

PROBLEM 2E A bowl contains two red balls, two white balls, and a fifth ball that is either red or white. Let p denote the probability of drawing a red ball from the bowl. We shall test the simple null hypothesis H0: p = 3/5 against the simple alternative hypothesis H1: p = 2/5. Draw four balls at random from the bowl, one at a time and with replacement. Let X equal the number of red balls drawn. (a) Define a critical region C for this test in terms of X. (b) For the critical region C defined in part (a), find the values of ? and ?.

PROBLEM 16E Let p be the fraction of engineers who do not understand certain basic statistical concepts. Unfortunately, in the past, this number has been high, about p = 0.73. A new program to improve the knowledge of statistical methods has been implemented, and it is expected that under this program p would decrease from the aforesaid 0.73 value. To test H0: p = 0.73 against H1: p < 0.73, 300 engineers in the new program were tested and 204 (i.e., 68%) did not comprehend certain basic statistical concepts. Compute the p-value to determine whether this result indicates progress. That is, can we reject H0 is favor of H1? Use ? = 0.05.

PROBLEM 6E It was claimed that 75% of all dentists recommend a certain brand of gum for their gum-chewing patients. A consumer group doubted this claim and decided to test H0: p = 0.75 against the alternative hypothesis H1: p < 0.75, where p is the proportion of dentists who recommend that brand of gum. A survey of 390 dentists found that 273 recommended the given brand of gum. (a) Which hypothesis would you accept if the significance level is ? = 0.05? (b) Which hypothesis would you accept if the significance level is ? = 0.01? (c) Find the p-value for this test.

PROBLEM 5E If a newborn baby has a birth weight that is less than 2500 grams (5.5 pounds), we say that the baby has a low birth weight. The proportion of babies with a low birth weight is an indicator of lack of nutrition for the mothers. For the United States, approximately 7% of babies have a low birth weight. Let p equal the proportion of babies born in the Sudan who weigh less than 2500 grams. We shall test the null hypothesis H0: p = 0.07 against the alternative hypothesis H1: p > 0.07. In a random sample of n = 209 babies, y = 23 weighed less than 2500 grams. (a) What is your conclusion at a significance level of ? = 0.05? (b) What is your conclusion at a significance level of ? = 0.01? (c) Find the p-value for this test.

PROBLEM 1E Let Y be b(100, p). To test H0: p = 0.08 against H1: p < 0.08, we rejectH0 and acceptH1 if and only if Y ? 6. (a) Determine the significance level ? of the test. (b) Find the probability of the Type II error if, in fact, p = 0.04.

PROBLEM 3E Let Y be b(192, p). We reject H0: p = 0.75 and acceptH1: p > 0.75 if and only if Y ? 152. Use the normal approximation to determine (a) ? = P(Y ? 152; p = 0.75). (b) ? = P(Y < 152) when p = 0.80.

PROBLEM 8E Let p equal the proportion of drivers who use a seat belt in a state that does not have a mandatory seat belt law. It was claimed that p = 0.14. An advertising campaign was conducted to increase this proportion. Two months after the campaign, y = 104 out of a random sample of n = 590 drivers were wearing their seat belts. Was the campaign successful? (a) Define the null and alternative hypotheses. (b) Define a critical region with an ? = 0.01 significance level. (c) What is your conclusion?

PROBLEM 7E The management of the Tigers baseball team decided to sell only low-alcohol beer in their ballpark to help combat rowdy fan conduct. They claimed that more than 40% of the fans would approve of this decision. Let p equal the proportion of Tiger fans on opening day who approved of the decision. We shall test the null hypothesis H0: p = 0.40 against the alternative hypothesis H1: p > 0.40. (a) Define a critical region that has an ? = 0.05 significance level. (b) If, out of a random sample of n = 1278 fans, y = 550 said that they approved of the new policy, what is your conclusion?

PROBLEM 11E A machine shop that manufactures toggle levers has both a day and a night shift. A toggle lever is defective if a standard nut cannot be screwed onto the threads. Let p1 and p2 be the proportion of defective levers among those manufactured by the day and night shifts, respectively. We shall test the null hypothesis, H0: p1 = p2, against a two-sided alternative hypothesis based on two random samples, each of 1000 levers taken from the production of the respective shifts. (a) Define the test statistic and a critical region that has an ? = 0.05 significance level. Sketch a standard normal pdf illustrating this critical region. (b) If y1 = 37 and y2 = 53 defectives were observed for the day and night shifts, respectively, calculate the value of the test statistic. Locate the calculated test statistic on your figure in part (a) and state your conclusion.

PROBLEM 13E Let pm and pf be the respective proportions of male and female white-crowned sparrows that return to their hatching site. Give the endpoints for a 95% confidence interval for pm ? pf if 124 out of 894 males and 70 out of 700 females returned (The Condor, 1992, pp. 117–133). Does your result agree with the conclusion of a test of H0: p1 = p2 against H1: p1 ? p2 with ? = 0.05?

PROBLEM 9E According to a population census in 1986, the percentage of males who are 18 or 19 years old and are married was 3.7%. We shall test whether this percentage increased from 1986 to 1988. (a) Define the null and alternative hypotheses. (b) Define a critical region that has an approximate significance level of ? = 0.01. Sketch a standard normal pdf to illustrate this critical region. (c) If y = 20 out of a random sample of n = 300 males, each 18 or 19 years old, were married (U.S. Bureau of the Census, Statistical Abstract of the United States: 1988), what is your conclusion? Show the calculated value of the test statistic on your figure in part (b).

Let p equal the proportion of yellow candies in a package of mixed colors. It is claimed that \(p = 0.20\). (a) Define a test statistic and critical region with a significance level of \(\alpha=0.025\) for testing \(H_{0}: p=0.20\) against a two-sided alternative hypothesis. (b) To perform the test, each of 20 students counted the number of yellow candies, \(y\), and the total number of candies, \(n\), in a \(48.1\)-gram package, yielding the following ratios, \(y / n: 8 / 56,13 / 55,12 / 58,13 / 56,14 / 57,5 / 54\), \(14 / 56,15 / 57,11 / 54,13 / 55,10 / 57,8 / 59,10 / 54,11 / 55\), \(12 / 56,11 / 57,6 / 54,7 / 58,12 / 58,14 / 58\). If each individual tests \(H_{0}: p=0.20\), what proportion of the students rejected the null hypothesis? (c) If we may assume that the null hypothesis is true, what proportion of the students would you have expected to reject the null hypothesis? (d) For each of the 20 ratios in part (b), a \(95 \%\) confidence interval for \(p\) can be calculated. What proportion of these \(95 \%\) confidence intervals contain \(p=0.20\)? (e) If the 20 results are pooled so that \(\sum_{i=1}^{20} y_{i}\) equals the number of yellow candies and \(\sum_{i=1}^{20} n_{i}\) equals the total sample size, do we reject \(H_{0}: p=0.20\)? Equation Transcription: , . Text Transcription: y n 48.1 y/n:8/56,13/55,12/58,13/56,14/57,5/5414/56,15/57,11/54,13/55,10/57,8/59,10/54,11/55, 12/56,11/57,6/54,7/58,12/58,14/58 H_0:p=0.20 95% p p=0.20 sum_i=1^20 y_i sum_i=1^20 n_i

PROBLEM 10E Because of tourism in the state, it was proposed that public schools in Michigan begin after Labor Day. To determine whether support for this change was greater than 65%, a public poll was taken. Let p equal the proportion of Michigan adults who favor a post–Labor Day start. We shall test H0: p = 0.65 against H1: p > 0.65. (a) Define a test statistic and an ? = 0.025 critical region. (b) Given that 414 out of a sample of 600 favor a post–Labor Day start, calculate the value of the test statistic. (c) Find the p-value and state your conclusion. (d) Find a 95% one-sided confidence interval that gives a lower bound for p.

PROBLEM 15E Each of six students has a deck of cards and selects a card randomly from his or her deck. (a) Show that the probability of at least one match is equal to 0.259. (b) Now let each of the students randomly select an integer from 1–52, inclusive. Let p equal the probability of at least one match. Test the null hypothesis H0: p =0.259 against an appropriate alternative hypothesis. Give a reason for your alternative. (c) Perform this experiment a large number of times. What is your conclusion?

PROBLEM 14E For developing countries in Africa and the Americas, let p1 and p2 be the respective proportions of babies with a low birth weight (below 2500 grams). We shall test H0: p1 = p2 against the alternative hypothesis H1: p1 > p2. (a) Define a critical region that has an ? = 0.05 significance level. (b) If respective random samples of sizes n1 = 900 and n2 = 700 yielded y1 = 135 and y2 = 77 babies with a low birth weight, what is your conclusion? (c) What would your decision be with a significance level of ? = 0.01? (d) What is the p-value of your test?

Let \(p\) denote the probability that, for a particular tennis player, the first serve is good. Since \(p=0.40\), this player decided to take lessons in order to increase \(p\). When the lessons are completed, the hypothesis \(H_{0}\) : \(p=0.40\) will be tested against \(H_{1}: p>0.40\) on the basis of \(n=25\) trials. Let \(y\) equal the number of first serves that are good, and let the critical region be defined by \(C=\{y: y \geq 13\}\). (a) Determine \(\alpha=P(Y \geq 13 ; p=0.40)\). Use Table II in the appendix. (b) Find \(\beta=P(Y<13)\) when \(p=0.60\); that is, \(\beta=\) \(P(Y \leq 12 ; p=0.60)\). Use Table II. Equation Transcription: : Text Transcription: p p=0.40 H_0 : p=0.40 H_1:p >0.40 n=25 y C={y: y > or =13} alpha=P(Y> or =13;p=0.40) beta=P(Y<13) p=0.60 beta= P(Y< or =12;p=0.60)

Let Y be b(100, p). To test H0: p = 0.08 against H1: p < 0.08, we reject H0 and accept H1 if and only if Y 6. (a) Determine the significance level of the test. (b) Find the probability of the Type II error if, in fact, p = 0.04.

A bowl contains two red balls, two white balls, and a fifth ball that is either red or white. Let p denote the probability of drawing a red ball from the bowl. We shall test the simple null hypothesis H0: p = 3/5 against the simple alternative hypothesis H1: p = 2/5. Draw four balls at random from the bowl, one at a time and with replacement. Let X equal the number of red balls drawn. (a) Define a critical region C for this test in terms of X. (b) For the critical region C defined in part (a), find the values of and .

Let Y be b(192, p). We reject H0: p = 0.75 and accept H1: p > 0.75 if and only if Y 152. Use the normal approximation to determine (a) = P(Y 152; p = 0.75). (b) = P(Y < 152) when p = 0.80.

Let p denote the probability that, for a particular tennis player, the first serve is good. Since p = 0.40, this player decided to take lessons in order to increase p. When the lessons are completed, the hypothesis H0: p = 0.40 will be tested against H1: p > 0.40 on the basis of n = 25 trials. Let y equal the number of first serves that are good, and let the critical region be defined by C = {y: y 13}. (a) Determine = P(Y 13; p = 0.40). Use Table II in the appendix. (b) Find = P(Y < 13) when p = 0.60; that is, = P(Y 12; p = 0.60). Use Table II.

If a newborn baby has a birth weight that is less than 2500 grams (5.5 pounds), we say that the baby has a low birth weight. The proportion of babies with a low birth weight is an indicator of lack of nutrition for the mothers. For the United States, approximately 7% of babies have a low birth weight. Let p equal the proportion of babies born in the Sudan who weigh less than 2500 grams. We shall test the null hypothesis H0: p = 0.07 against the alternative hypothesis H1: p > 0.07. In a random sample of n = 209 babies, y = 23 weighed less than 2500 grams. (a) What is your conclusion at a significance level of = 0.05? (b) What is your conclusion at a significance level of = 0.01? (c) Find the p-value for this test.

It was claimed that 75% of all dentists recommend a certain brand of gum for their gum-chewing patients. A consumer group doubted this claim and decided to test H0: p = 0.75 against the alternative hypothesis H1: p < 0.75, where p is the proportion of dentists who recommend that brand of gum. A survey of 390 dentists found that 273 recommended the given brand of gum. (a) Which hypothesis would you accept if the significance level is = 0.05? (b) Which hypothesis would you accept if the significance level is = 0.01? (c) Find the p-value for this test.

The management of the Tigers baseball team decided to sell only low-alcohol beer in their ballpark to help combat rowdy fan conduct. They claimed that more than 40% of the fans would approve of this decision. Let p equal the proportion of Tiger fans on opening day who approved of the decision. We shall test the null hypothesis H0: p = 0.40 against the alternative hypothesis H1: p > 0.40. (a) Define a critical region that has an = 0.05 significance level. (b) If, out of a random sample of n = 1278 fans, y = 550 said that they approved of the new policy, what is your conclusion?

Let p equal the proportion of drivers who use a seat belt in a state that does not have a mandatory seat belt law. It was claimed that p = 0.14. An advertising campaign was conducted to increase this proportion. Two months after the campaign, y = 104 out of a random sample of n = 590 drivers were wearing their seat belts. Was the campaign successful? (a) Define the null and alternative hypotheses. (b) Define a critical region with an \(\alpha = 0.01\) significance level. (c) What is your conclusion?

According to a population census in 1986, the percentage of males who are 18 or 19 years old and are married was 3.7%. We shall test whether this percentage increased from 1986 to 1988. (a) Define the null and alternative hypotheses. (b) Define a critical region that has an approximate significance level of = 0.01. Sketch a standard normal pdf to illustrate this critical region. (c) If y = 20 out of a random sample of n = 300 males, each 18 or 19 years old, were married (U.S. Bureau of the Census, Statistical Abstract of the United States: 1988), what is your conclusion? Show the calculated value of the test statistic on your figure in part (b).

Because of tourism in the state, it was proposed that public schools in Michigan begin after Labor Day. To determine whether support for this change was greater than 65%, a public poll was taken. Let p equal the proportion of Michigan adults who favor a postLabor Day start. We shall test H0: p = 0.65 against H1: p > 0.65. (a) Define a test statistic and an = 0.025 critical region. (b) Given that 414 out of a sample of 600 favor a post Labor Day start, calculate the value of the test statistic. (c) Find the p-value and state your conclusion. (d) Find a 95% one-sided confidence interval that gives a lower bound for p.

A machine shop that manufactures toggle levers has both a day and a night shift. A toggle lever is defective if a standard nut cannot be screwed onto the threads. Let p1 and p2 be the proportion of defective levers among those manufactured by the day and night shifts, respectively. We shall test the null hypothesis, H0: p1 = p2, against a two-sided alternative hypothesis based on two random samples, each of 1000 levers taken from the production of the respective shifts. (a) Define the test statistic and a critical region that has an = 0.05 significance level. Sketch a standard normal pdf illustrating this critical region. (b) If y1 = 37 and y2 = 53 defectives were observed for the day and night shifts, respectively, calculate the value of the test statistic. Locate the calculated test statistic on your figure in part (a) and state your conclusion

Let p equal the proportion of yellow candies in a package of mixed colors. It is claimed that p = 0.20. (a) Define a test statistic and critical region with a significance level of = 0.05 for testing H0: p = 0.20 against a two-sided alternative hypothesis.(b) To perform the test, each of 20 students counted the number of yellow candies, y, and the total number of candies, n, in a 48.1-gram package, yielding the following ratios, y/n: 8/56, 13/55, 12/58, 13/56, 14/57, 5/54, 14/56, 15/57, 11/54, 13/55, 10/57, 8/59, 10/54, 11/55, 12/56, 11/57, 6/54, 7/58, 12/58, 14/58. If each individual tests H0: p = 0.20, what proportion of the students rejected the null hypothesis? (c) If we may assume that the null hypothesis is true, what proportion of the students would you have expected to reject the null hypothesis? (d) For each of the 20 ratios in part (b), a 95% confidence interval for p can be calculated. What proportion of these 95% confidence intervals contain p = 0.20? (e) If the 20 results are pooled so that 20 i=1 yi equals the number of yellow candies and 20 i=1 ni equals the total sample size, do we reject H0: p = 0.20?

Let pm and pf be the respective proportions of male and female white-crowned sparrows that return to their hatching site. Give the endpoints for a 95% confidence interval for pm pf if 124 out of 894 males and 70 out of 700 females returned (The Condor, 1992, pp. 117133). Does your result agree with the conclusion of a test of H0: p1 = p2 against H1: p1 = p2 with = 0.05?

For developing countries in Africa and the Americas, let \(p_1\) and \(p_2\) be the respective proportions of babies with a low birth weight (below 2500 grams). We shall test \(H_0: p_1 = p_2\) against the alternative hypothesis \(H_1: p_1 > p_2\).

Each of six students has a deck of cards and selects a card randomly from his or her deck. (a) Show that the probability of at least one match is equal to 0.259. (b) Now let each of the students randomly select an integer from 152, inclusive. Let p equal the probability of at least one match. Test the null hypothesis H0: p = 0.259 against an appropriate alternative hypothesis. Give a reason for your alternative. (c) Perform this experiment a large number of times. What is your conclusion?

Let p be the fraction of engineers who do not understand certain basic statistical concepts. Unfortunately, in the past, this number has been high, about p = 0.73. A new program to improve the knowledge of statistical methods has been implemented, and it is expected that under this program p would decrease from the aforesaid 0.73 value. To test H0: p = 0.73 against H1: p < 0.73, 300 engineers in the new program were tested and 204 (i.e., 68%) did not comprehend certain basic statistical concepts. Compute the p-value to determine whether this result indicates progress. That is, can we reject H0 is favor of H1? Use = 0.05.