stats 2381 exam 3

A nationwide study of American homeowners revealed that 65% own at least one lawn mower. A lawn equipment manufacturer, located in Charlotte, feels that this estimate is too low for households in Charlotte. To support his claim, he randomly selects 497 homes in Charlotte and finds that 340 had one or more lawn mowers. Find the P-value for testing the claim that the proportion of homeowners owning lawn mowers in Charlotte is higher than 65%. Round your answer to three decimal places

.056

From the sample statistics, find the value of \hat{p}_1 - \hat{p}_2 , the point estimate of the difference of proportions. Unless otherwise indicated, round to the nearest thousandth when necessary. n_1 = 100, n_2 = 100, \hat{p}_1 = .12, \hat{p}_2 = 0.1

0.02

Determine the null and alternative hypotheses. . A cereal company claims that the mean weight of its individual serving boxes is at least 14 oz.

ho mu=14 ha mu>14

Determine the null and alternative hypotheses. . A paranormal researcher claims that the proportion of Americans that have seen a UFO, p, is less than 5 in every one thousand.

ho p=.005 ha p<.005

Use the paired t-interval procedure to obtain the required confidence interval for the mean difference. Assume that the conditions and assumptions for inference are satisfied. An agricultural company wanted to know if a new insecticide would increase corn yields. Eight test plots showed an average increase of 3.125 bushels per acre. The standard deviation of the increases was 2.911 bushels per acre. Determine a 90% confidence interval for the mean increase in yield.

(1.175, 5.075)

From the sample statistics, find the value of \hat{p}_1 - \hat{p}_2 , the point estimate of the difference of proportions. Unless otherwise indicated, round to the nearest thousandth when necessary. n_1 = 216, n_2 = 186, x_1 = 76, x_2 = 99

-0.18

Assume that a simple random sample has been selected from a normally distributed population. Find the test statistic. Round your answer to two decimal places. Test the claim that the mean age of the prison population in one city is less than 26 years. Sample data are summarized as n = 25, \bar x = 24.4 years, and s = 9.2 years. Use a significance level of \alpha = 0.05.

-0.87

Assume that a simple random sample has been selected from a normally distributed population. Find the test statistic. Round your answer to two decimal places. Test the claim that for the adult population of a certain town, the mean annual salary is given by \mu = \$30,000. Sample data are summarized as n = 17, \bar x = \$22,298, and s = \$14,200. Use a significance level of \alpha = 0.05.

-2.24

For the given sample data and null hypothesis, compute the value of the test statistic. Round your answer to two decimal places. A drug company claims that over 80% of all physicians recommend their drug. 1200 physicians were asked if they recommend the drug to their patients. 30% said yes. The null hypothesis is H_0 : p = 0.8.

-43.30

In a random sample of 140 forty-year-old women, 25% are smokers. Find the P-value for testing the claim that the percentage of forty-year-old women that smoke is 22%. Round your answer to two decimal places.

.39

Assume that a simple random sample has been selected from a normally distributed population. Find the test statistic. Round your answer to two decimal places. Test the claim that for the population of female college students, the mean weight is given by \mu = 132 lb. Sample data are summarized as n = 20, \bar x = 137 lb, and s = 14.2 lb. Use a significance level of \alpha = 0.1.

1.57

A research company claims that more than 55% of Americans regularly watch public access television. You decide to test this claim and ask a random sample of 425 Americans if they watch these programs regularly. Of the 425, 255 respond yes. Calculate the test statistic for the population proportion. Round your answer to two decimal places.

2.07

For the given sample data and null hypothesis, compute the value of the test statistic. Round your answer to two decimal places. The claim is that the proportion of drowning deaths of children occurring at beaches is more than 0.25. The sample statistics include n = 615 drowning deaths of children with 30% of them occurring at beaches.

2.86

For the given sample data and null hypothesis, compute the value of the test statistic. Round your answer to three decimal places. A research group wants to determine whether the proportion of car accidents that were caused by drivers using cell phones has changed from the previous value of 13%. They obtained 10,000 auto accident reports and found that 14% were caused by drivers using cell phones. The hypotheses are H_0 : p = 0.13, H_a : p \ne 0.13, where p is the proportion of car accidents caused by drivers using cell phones.

2.974

For the given sample data and null hypothesis, compute the value of the test statistic. Round your answer to two decimal places. 410 people were asked if they were satisfied with their jobs. 37% of the responses were affirmative. The null hypothesis is H_0 : p = 0.30.

3.09

For the given sample data and null hypothesis, compute the value of the test statistic. Round your answer to two decimal places. In a school district with 10,000 high school students, 1200 randomly selected students completed a class designed to improve their math skills. 708 of these students scored better than the district-wide median on a standardized math exam. The district would like to know whether the class improves the students' math skills. The hypotheses are H_0 : p = 0.5, H_a : p > 0.5, where p is the proportion of all those taking the special class who score better than the district-wide median.

6.23

Examine the given statement, then identify whether the statement is a null hypothesis, an alternative hypothesis, or neither. The mean amount of a certain diet soda is at least 12 oz.

Alternative hypothesis

Examine the given statement, then identify whether the statement is a null hypothesis, an alternative hypothesis, or neither. The mean income of workers who have majored in history is less than $25,000.

Alternative hypothesis

Explain what the P-value means in the given context. The federal guideline for smog is 12% pollutants per 10,000 volume of air. A metropolitan city is trying to bring its smog level into federal guidelines. The city comes up with a new policy where city employees are to use city transportation to and from work. A local environmental group does not think the city is doing enough and that no real decrease in pollution will occur. An independent agency, hired by the city, conducts a test to determine if the city's strategy lessens the smog rate and concludes that the P-value = 0.055. What is reasonable to conclude about the new strategy using \alpha=0.05?

Assuming the smog rate has not changed, there is a 5.5% chance of obtaining the results found by the agency or even more extreme results. With \alpha=0.05, there is not enough evidence to claim that the new policy is effective in reducing smog.

In the past, the mean lifetime of a certain type of radio battery has been 9.8 hours. The manufacturer has introduced a change in the production method and wants to perform a significance test to determine whether the mean lifetime has increased as a result. The hypotheses are H_0 : \mu = 9.8 hours and H_a : \mu > 9.8 hours. Explain the meaning of a Type I error.

Concluding that \mu > 9.8 hours when in fact \mu = 9.8 hours

Determine the null and alternative hypotheses. . Ten years ago, the average duration of long-distance telephone calls originating in one town was 7.2 minutes. A long-distance telephone company wants to perform a hypothesis test to determine whether the average duration of long-distance phone calls has changed from the mean of 7.2 minutes from 10 years ago.

Ho mu=7. 2 Ha mu =/ 7.2

Explain what the P-value means in the given context. A weight loss center has a 72% success rate (weight loss of 5 or more pounds). The center's CEO decides to test a new weight loss strategy to see if it is more effective and obtains a P-value of 0.23. What is reasonable to conclude about the new strategy using \alpha=0.1?

If the success rate of the new strategy is 72%, the probability of obtaining results at least as extreme as those obtained from the CEO's study is 0.23. We are unable to conclude that the new strategy provides better results.

A two-sided significance test for two population proportions is to be performed using the P-value approach. The null hypothesis is H_0 : p_1 = p_2 and the alternative is H_a : p_1 \ne p_2 . Use the given sample data to find the P-value for the significance test. Give an interpretation of the p-value. n_1 = 50, n_2 = 75, x_1 = 20, x_2 = 15

P-value = 0.015; If there is no difference in the proportions, there is about a 1.5% chance of seeing the observed difference or larger due to natural sampling variation.

The U.S. Department of Labor and Statistics wanted to compare the results of an unemployment program for the past two months in the U.S. Suppose the proportion of the unemployed two months ago is p_2 and the proportion of the unemployed one month ago is p_1. A study found a 99% confidence interval for p_2 - p_1 to be ( -0.0012,0.003). What conclusions can we draw about the unemployment percentage for the past two months?

Since 0 is included in the interval, the unemployment program was not shown to be statistically significant in lowering the unemployment rate for the past two month

A researcher wishes to determine whether people with high blood pressure can reduce their blood pressure by following a particular diet. A 95% confidence interval for \mu_1 - \mu_2 based on sample data is given by ( -26.79, -2.41) where \mu_1 is the population mean blood pressure level for the treatment group and \mu_2 is the population mean blood pressure level for the control group. Based on this confidence interval, what, if any, conclusions can we draw?

Since all of the values in the interval are negative, we can conclude that the population mean blood pressure level is lower for the treatment group than for the control group.

A survey asked respondents "Should divorce in this country be easier or more difficult to obtain than it is now?" In testing H_0 : p_M = p_F versus H_a : p_M > p_F, where p_M is the proportion of males who responded "more difficult" and p_F is the proportion of females who responded "more difficult," suppose the z statistic is 0.33. What is your conclusion using a significance level of 0.05?

Since pvalue > 0.05, we are unable to reject the null hypothesis. There is insufficient evidence to show that the proportion of males who responded "more difficult" is greater than the proportion of females who responded "more difficult".

A journal article reports that 34% of American fathers take no responsibility for child care. A researcher claims that the figure is higher for fathers in a particular town. A random sample of 233 fathers from this town yielded 96 who did not help with child care. Do the data provide sufficient evidence to conclude that in this town the proportion is higher than 0.34? Use a 0.05 significance level. The hypotheses are H_0 : p = 0.34 and H_a : p > 0.34, the test statistic is z = 2.32, and the P-value is 0.0102. State your conclusion in terms of H_a .

Since the P-value <\alpha , we can conclude that the proportion of fathers who take no responsibility for childcare is higher than 34% in this town. Since the P-value <\alpha , we can conclude that the proportion of fathers who take no responsibility for childcare is 41%. Since the P-value < 0.34, we can conclude that the proportion of fathers who take no responsibility for childcare is higher than 34% in this town. Since the P-value < 0.34, we are unable to conclude that the proportion of fathers who take no responsibility for childcare is higher than 34% in this town.

In a poll of 1556 registered voters nationwide, 43% of those polled blamed oil companies the most for the recent increase in gasoline prices. Test the claim that the percentage of registered voters nationwide who blame oil companies the most for the recent increase in gasoline prices is at least 45% using \alpha = 0.01. The test statistic is z = -1.59 and the P-value is 0.94.

Since the P-value > \alpha , we cannot conclude that the percentage of registered voters nationwide who blame oil companies the most for the recent increase in gasoline prices is at least 45%.

Assume that the assumptions and conditions for inference with a two-sample t-test are met. Test the indicated claim about the means of the two populations. State your conclusion. Two types of flares are tested for their burning times (in minutes). The hypotheses are H_0: \mu_1 - \mu_2 = 0 and H_a: \mu_1 - \mu_2 \ne 0. The p-value for the test is 0.0000735. Use a 5% significance level.

Since the P-value is approximately 0, we reject the null hypothesis and conclude that the mean burning time for the Brand X flare differs from that of the Brand Y flare.

A survey asked respondents "When a person has a disease that cannot be cured, do you think doctors should be allowed by law to end the patient's life by some painless means if the patient and his family request it?". In testing H_0 : p_M = p_F versus H_a : p_M > p_F, where p_M is the proportion of males who responded "yes" and p_F is the proportion of females who responded "yes", suppose the z statistic is 4.48. What is your conclusion using a significance level of 0.05?

Since the P-value is approximately 0, we reject the null hypothesis and conclude that the percentage of males holding this belief is greater than the percentage of females.

Assume that the assumptions and conditions for inference with a two-sample t-test are met. Test the indicated claim about the means of the two populations. State your conclusion. Researchers wanted to compare the effectiveness of a water softener used with a filtering process to that of a water softener used without filtering. Ninety locations were randomly divided into two groups of equal size. Group A locations used a water softener and the filtering process, while group B used only the water softener. At the end of three months, water samples were tested at each location for softness level. (Water softness was measured on a scale of 1 to 5, with 5 being the softest water.) The hypotheses are H_0: \mu_1 - \mu_2 = 0 and H_a: \mu_1 - \mu_2 \ne 0. The p-value for the test is 0.0014. Use a 1% significance level.

Since the P-value is less than 0.01, we reject the null hypothesis and conclude that there is a difference in the softness level achieved by the two processes.

A health insurer has determined that the "reasonable and customary" fee for a certain medical procedure is $1200. They suspect that the average fee charged by one particular clinic for this procedure is higher than $1200. The insurer performs a significance test to determine whether their suspicion is correct using \alpha = 0.05. The hypotheses are H_0 : \mu = \$1200 and H_a : \mu > \$1200. If the P-value is 0.09 and a decision error is made, what type of error is it? Explain.

The decision is a Type II error. We conclude that the average fee charged for the procedure is not higher than $1200 when it actually is higher.

A study is conducted to determine whether a new cancer drug increases the mean survival time by at least 30 days for a certain type of cancer. Explain in context the conclusion of the test if you fail to reject H_0 .

The drug was not shown to increase mean survival time by at least 30 days.

A health insurer has determined that the "reasonable and customary" fee for a certain medical procedure is $1200. They suspect that the average fee charged by one particular clinic for this procedure is higher than $1200. The insurer wants to perform a significance test to determine whether their suspicion is correct. The hypotheses are H_0 : \mu = \$1200 and H_a : \mu > \$1200. Suppose that the results of the sample lead to rejection of the null hypothesis. Classify that conclusion as a Type I error, a Type II error, or a correct decision, if in fact the average fee charged by the clinic is $1200.

Type I error

In the past, the mean lifetime for a certain type of flashlight battery has been 9.6 hours. The manufacturer has introduced a change in the production method and wants to perform a significance test to determine whether the mean lifetime has increased as a result. The hypotheses are H_0 : \mu = 9.6 hours and H_a : \mu > 9.6 hours. Suppose that the results of the sample lead to nonrejection of the null hypothesis. Classify that conclusion as a Type I error, a Type II error, or a correct decision, if in fact the mean running time has increased.

Type II error

Assume the proportion of students retained at a certain university in the Spring semester is p_1 and the proportion of students retained in the Fall semester is p_2 . Based on a recent study, a 90% confidence interval for p_1 - p_2 is ( -0.0398, 0.0262). Give an interpretation of this confidence interval.

We are 90% confident that the proportion of students retained in the Spring semester is between 3.98% less and 2.62% more than the proportion of students retained in the Fall semester.

A researcher wishes to determine whether people with high blood pressure can reduce their blood pressure by following a particular diet. Subjects were randomly assigned to either a treatment group or a control group. The mean blood pressure was determined for each group, and a 95% confidence interval for the difference in the means for the treatment group versus the control group, \mu_t - \mu_c, was found to be (-21, -6). Give an interpretation of this confidence interval.

We are 95% confident that the average blood pressure of those who follow the diet is between 6 and 21 points lower than the average for those who do not follow the diet.

A study was conducted to compare the effectiveness of two weight loss strategies for obese participants. The proportion of obese clients who lost at least 10% of their body weight was compared for the two strategies. The resulting 98% confidence interval for p_1 - p_2 is ( -0.13, 0.09). Give an interpretation of this confidence interval.

We are 98% confident that the proportion of obese clients losing weight under strategy 1 is between 13% less and 9% more than the proportion of obese clients losing weight under strategy 2.

The U.S. Department of Labor and Statistics wanted to compare the results of an unemployment program for the past two months in the U.S. Suppose the proportion of the unemployed two months ago is p_2 and the proportion of the unemployed one month ago is p_1 . A study found a 99% confidence interval for p_2 - p_1 to be ( -0.0012, 0.003). Give an interpretation of this confidence interval.

We are 99% confident that the proportion of the unemployed two months ago is between 0.12% less and 0.3% more than the proportion of the unemployed one month ago.

Determine the null and alternative hypotheses. . The percentage of viewers tuned to a certain news channel is equal to 85%.

h0 p=.85 ha p=/ .85

Determine the null and alternative hypotheses. . A health insurer has determined that the "reasonable and customary" fee for a certain medical procedure is $1200. They suspect that the average fee charged by one particular clinic for this procedure is higher than $1200. The insurer wants to perform a hypothesis test to determine whether their suspicion is correct.

ho mu=1200 ha mu>1200

Determine the null and alternative hypotheses. . In the past, the mean running time for a certain type of radio battery has been 9.6 hours. The manufacturer has introduced a change in the production method and wants to perform a hypothesis test to determine whether the mean running time has changed as a result.

ho mu=9.6 ha mu=/ 9.6

Twins Ben and Michael claim that they can communicate telepathically. To test the claim, a researcher plans to put the twins in separate rooms. Five chips of different colors (red, blue, green, yellow, and white) will be placed in a bag. Ben will be blindfolded as he selects a chip at random from the bag. He will then look at the chip and attempt to communicate the color telepathically to his twin Michael, seated in the next room, who will record the color he believes the chip to be. This process will be repeated 50 times. Let p denote the probability that Michael correctly records the color of a chip. State hypotheses for a significance test, letting the alternative hypothesis reflect the possibility that Michael does better than he would by random guessing.

ho p=.2 ha p>.2

Determine whether the samples are independent or dependent. The effectiveness of a new headache medicine is tested by measuring the amount of time before the headache is cured for patients who use the medicine and another group of patients who use a placebo drug.

independent samples

An agricultural company would like to predict cotton yield per acre in a certain area using rainfall (in inches). Identify the explanatory variable.

rainfall in inches

Assume that a simple random sample has been selected from a normally distributed population. State the final conclusion. Round the test statistic to two decimal places. Test the claim that the mean lifetime of a particular car engine is greater than 220,000 miles. Sample data are summarized as n = 23, \bar x = 226,450 miles, and s = 11,500 miles. Use a significance level of \alpha = 0.01. The hypotheses are H_0 : \mu = 220,000 and H_a : \mu > 220,000.

t = 2.69, reject H_0

A pharmaceutical company reports that in testing whether their new cancer drug increases the mean survival time for certain types of cancer patients, their results were statistically significant. Suppose that in testing H_0 : \mu = 100 versus H_a : \mu > 100 they obtain \bar x = 101 and s = 5 for a random sample of 400 patients. Find the test statistic and discuss the practical significance of the test.

t = 4; although the test is statistically significant, the increase of 1 day is not practically significant.

A survey asked respondents "When a person has a disease that cannot be cured, do you think doctors should be allowed by law to end the patient's life by some painless means if the patient and his family request it?" If 616 of the 839 male responded answered "yes" and 693 of the 1086 female respondents answered "yes," calculate a 90% confidence interval for p_M - p_F where p_M is the proportion of males who responded "yes" and p_F is the proportion of females who responded "yes". Round to the nearest hundredth.

( 0.13, 0.06)

A survey asked respondents "When a person has a disease that cannot be cured, do you think doctors should be allowed by law to end the patient's life by some painless means if the patient and his family request it?" If 616 of the 839 male responded answered "yes" and 693 of the 1086 female respondents answered "yes," calculate a 90% confidence interval for p_M - p_F where p_M is the proportion of males who responded "yes" and p_F is the proportion of females who responded "yes". Round to the nearest hundredth.

(0.06,0.13)

A soft drink company claims the mean caffeine content of its top selling soda is 40 milligrams per one 8-ounce bottle. To verify this claim, a random sample of 30 bottles is found to have a mean caffeine content of 39.2 milligrams with a standard deviation of 7.5 milligrams. Calculate the test statistic in order to test the company's claim. Round your answer to two decimal places.

-.58

Which P-value provides the strongest evidence against the null hypothesis?

.001

From the sample statistics, find the value of \hat{p}_1 - \hat{p}_2 , the point estimate of the difference of proportions. Unless otherwise indicated, round to the nearest thousandth when necessary. n_1 = 100, n_2 = 100, x_1 = 34, x_2 = 30

0.04

An airline claims that the no-show rate for passengers booked on its flights is less than 6%. Of 380 randomly selected reservations, 18 were no-shows. Find the P-value for a test of the airline's claim. Round your answer to two decimal places.

0.15

A grocery store is interested in determining whether or not a difference exists between the shelf life of Hot'n Now doughnuts and Sugar Yum doughnuts. A random sample of 100 boxes of each brand was selected and the mean shelf life in days was determined for each brand. A 90% confidence interval for the difference of the means, \mu_{hn}-\mu_{sy}, was determined to be (1.1,2.4). Give an interpretation of this confidence interval.

Based on this sample, we are 90% confident that Hot'n Now doughnuts will last on average between 1.1 and 2.4 days longer than Sugar Yum doughnuts.

A high school coach uses a new technique in training middle distance runners. He records the times for 4 different athletes to run 800 meters before and after this training. A 90% confidence interval for the difference of the means before and after the training was determined to be (2.7,4.2). Interpret this confidence interval.

Based on this sample, we are 90% confident that the average time for the 800-meter run for middle distance runners at this high school is between 2.7 and 4.2 seconds shorter after the new training.

Determine the null and alternative hypotheses. . Ten years ago, the average duration of long-distance telephone calls originating in one town was 7.2 minutes. A long-distance telephone company wants to perform a hypothesis test to determine whether the average duration of long-distance phone calls has changed from the mean of 7.2 minutes from 10 years ago

H0 mu =7.2 Ha mu =/7.2

The county health department has concerns about the chlorine level of 0.4% mg/mL at a local water park increasing to an unsafe level. The water department tests the hypothesis that the local water park's chlorine proportions have remained the same, and obtains a P-value of 0.005. Provide an appropriate conclusion.

If the chlorine level has not changed, the probability of observing a sample chlorine level as high or higher as that observed in the sample is 0.005. We conclude that the chlorine level has increased.

Explain what the P-value means in the given context. A state university wants to increase its retention rate of 4% for graduating students from the previous year. After implementing several new programs during the last two years, the university reevaluates its retention rate and comes up with a P-value of 0.075. Using \alpha = 0.05, what can we conclude?

If the retention rate is truly 4%, there is a 7.5% chance of obtaining the retention rates seen over the past two years due to chance variation. With \alpha = 0.05, there is not enough evidence to conclude that the new programs affect the retention rate for graduating students.

A manufacturer claims that the mean amount of juice in its 16 ounce bottles is 16.1 ounces. A consumer advocacy group wants to perform a significance test to determine whether the mean amount is actually less than this. The hypotheses are H_0 : \mu = 16.1 ounces and H_a : \mu < 16.1 ounces. Suppose that the results of the sample lead to rejection of the null hypothesis. Classify that conclusion as a Type I error, a Type II error, or a correct decision, if in fact the mean amount of juice, \mu , is less than 16.1 ounces.

No error

Examine the given statement, then identify whether the statement is a null hypothesis, an alternative hypothesis, or neither. The mean is \mu = 5.5.

Null hypothesis

Examine the given statement, then identify whether the statement is a null hypothesis, an alternative hypothesis, or neither. The percentage of viewers tuned to a certain news channel is equal to 85%.

Null hypothesis

A grocery store would like to determine whether there is a difference in the shelf life of two different brands of doughnuts. A random sample of 40 boxes of each brand was selected and the shelf life in days was determined for each box. A 95% confidence interval for \mu_A - \mu_B, the difference in mean shelf life between brand A and brand B, was found to be ( -0.7,0.1). Based on this confidence interval, what, if any, conclusions can we draw?

Since 0 is contained in the interval, we are unable to conclude that there is a difference in the average shelf life of the two brands of doughnuts.

Ten families are randomly selected and their daily water usage (in gallons) is measured before and after viewing a conservation video. Calculate the test statistic and state your conclusion for testing H_0 : \mu_d = 0 versus H_a : \mu_d \ne 0 where the difference is After - Before. Assume \bar{x}_d = -4.8, s_d = 5.2451 and \alpha = 0.05.

The test statistic is -2.89. Reject the null hypothesis and conclude that the mean daily water usage decreased following the viewing of the conservation video.

Determine whether the samples are independent or dependent. The effect of caffeine on taste is tested by randomly giving participants a sample of regular soda and another sample with decaffeinated soda

dependent samples

Determine whether the samples are independent or dependent. The effect of caffeine on taste is tested by randomly giving participants a sample of regular soda and another sample with decaffeinated soda.

dependent samples

Determine whether the samples are independent or dependent. The effectiveness of a headache medicine is tested by measuring the intensity of a headache in patients before and after drug treatment. The data consist of before and after intensities for each patient.

dependent samples

An agricultural company would like to predict cotton yield per acre in a certain area using rainfall (in inches). Identify the response variable.

yield per acre