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Biostats Questions Midterm 3
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A survey of Internet users reported that 20% downloaded music onto their computers. The filing of lawsuits by the recording industry may be a reason why this percent has decreased from the estimate of 27% from a survey taken two years before. Assume that the sample sizes are both 1381. Using a significance test, evaluate whether or not there has been a change in the percent of Internet users who download music. Provide all details for the test. (Round your value for z to two decimal places. Round your P-value to four decimal places.) Summarize your conclusion. Also report a 95% confidence interval for the difference in proportions. (Round your answers to four decimal places.) Explain what information is provided in the interval that is not in the significance test results.
8.2
A survey of Internet users reported that 21% downloaded music onto their computers. The filing of lawsuits by the recording industry may be a reason why this percent has decreased from the estimate of 31% from a survey taken two years before. Suppose we are not exactly sure about the sizes of the samples. Perform the calculations for the significance tests and 95% confidence intervals under each of the following assumptions. (Use previous − recent. Round your test statistics to two decimal places and your confidence intervals to four decimal places.)
(i) Both sample sizes are 1000.
(ii) Both sample sizes are 1600.
(iii) The sample size for the survey reporting 31% is 1000 and the sample size for the survey reporting 21% is 1600.Summarize the effects of the sample sizes on the results.
8.2
The proportion of drivers who use seat belts depends factors such as age (young people are more likely to go unbelted) and sex (women are more likely to use belts). It also depends on local law. In New York City, police can stop a driver who is not belted. In Boston at the time of the survey, police could cite a driver for not wearing a seat belt only if the driver had been stopped for some other violation. Here are data from observing random samples of female Hispanic drivers in these two cities.
City Drivers Belted
New York 224 190
Boston 113 65
(a) Is this an experiment or an observational study? Why?
(b) Comparing local laws suggests the hypothesis that a smaller proportion of drivers wear seat belts in Boston than in New York. Do the data give good evidence that this is true for female Hispanic drivers? (Use α = 0.05. Round your test statistic to two decimal places and your P-value to four decimal places.)Conclusion
8.2
Most alpine skiers and snowboarders do not use helmets. Do helmets reduce the risk of head injuries? A study in Norway compared skiers and snowboarders who suffered head injuries with a control group who were not injured. Of 578 injured subjects, 96 had worn a helmet. Of the 2992 in the control group, 662 wore helmets. Is helmet use less common among skiers and snowboarders who have head injuries? Follow the four-step process as illustrated in Examples 23.4 and 23.5. (Note that this is an observational study that compares injured and uninjured subjects. An experiment that assigned subjects to helmet and no-helmet groups would be more convincing.)
Plan: Let p1 and p2 be (respectively) the proportions of injured skiers and injured snowboarders who wear helmets. State the hypotheses for your test.
Solve: Do the samples satisfy the conditions for the significance testing procedures?
Solve (continue): Give the value of the test statistic z. (Round your answer to two decimal places.)
Solve (continue): What is the P-value for the test? (Round your answer to four decimal places.)
Conclude: What do you conclude? (Use α = 0.01.)
8.2
A new AIDS prevention drug was tried on a group of 208 HIV positive patients. Forty-two patients developed AIDS after four years. In a control group of 208 HIV positive patients, 64 developed AIDS after four years. We want to test whether the method of treatment reduces the proportion of patients that develop AIDS after four years or if the proportions of the treated group and the untreated group stay the same.
If the p-value is 0.0067, what is the conclusion (use α = 0.05)?
8.2
Question Part
Points
Submissions Used
Subjects with pre-existing cardiovascular symptoms who were receiving subitramine, an appetite suppressant, were found to be at increased risk of cardiovascular events while taking the drug. The study included 9805 overweight or obese subjects with pre-existing cardiovascular disease and/or type 2 diabetes. The subjects were randomly assigned to subitramine (4908 subjects) or a placebo (4897 subjects) in a double-blind fashion. The primary outcome measured was the occurrence of any of the following events: nonfatal myocardial infarction or stroke, resuscitation after cardiac arrest, or cardiovascular death. The primary outcome was observed in 562 subjects in the subitramine group and 493 subjects in the placebo group.
Do the data give good reason to think that there is a difference between the proportions of treatment and placebo subjects who experienced the primary outcome? (Note that sibutramine has not been available in the United States since the end of 2010 due to its manufacturer's concerns over increased risk of heart attack or stroke, although at the present time it can still be purchased in other countries.)
State hypotheses, find the test statistic, and the P-value. (Round your test statistic to two decimal places and your P-value to four decimal places.) State your conclusion. (Use α = 0.05.)
8.2
In each of the following circumstances state whether you would use the large-sample confidence interval.
(a)
n = 70, X = 50
(b)
n = 90, X = 15
(c)
n = 10, X = 4
(d)
n = 50, X = 45
(e)
n = 23, X = 15
8.2
Explain what is wrong with each of the following.
(a) An approximate 99% confidence interval for an unknown proportion p is p̂ plus or minus its standard error.
(b) When performing a large-sample significance test for a population proportion, the t distribution is used to compute the P-value.
(c) A significance test is used to evaluate H0: p̂ = 0.2 versus the two-sided alternative.
...
Gambling is an issue of great concern to those involved in intercollegiate athletics. Because of this concern, the National Collegiate Athletic Association (NCAA) surveyed student-athletes concerning their gambling-related behaviors. 1411 out of a total of 3409 female student-athletes reported participation in some gambling activity.
(a) Use the large-sample methods to find an estimate of the true proportion with a 99% confidence interval. (Round your answers to three decimal places.)
(b) The margin of error for this sample is not the same as the margin of error for male athletes. There were 5594 males in the study, and p̂ was found to be 0.6341. Explain why the margin of error differs.
...
A survey of 1269 student loan borrowers found that 443 had loans totaling more than $20,000 for their undergraduate education. Give a 99% confidence interval for the proportion of all student loan borrowers who have loans of $20,000 or more for their undergraduate education. (Give answers accurate to 3 decimal places.)
...
A matched pairs experiment compares the taste of instant versus fresh-brewed coffee. Each subject tastes two unmarked cups of coffee, one of each type, in random order and states which he or she prefers. Of the 40 subjects who participate in the study, 14 prefer the instant coffee. Let p be the probability that a randomly chosen subject prefers fresh-brewed coffee to instant coffee. (In practical terms, p is the proportion of the population who prefer fresh-brewed coffee.)
(a) Test the claim that a majority of people prefer the taste of fresh-brewed coffee. Report the large-sample z statistic. (Round your answer to two decimal places.)
(b) Draw a sketch of a standard Normal curve and mark the location of your z statistic. Shade the appropriate area that corresponds to the P-value.
(c) Is your result significant at the 5% level?
What is your practical conclusion?
...
In a study of the relationship between pet ownership and physical activity in older adults, 584 subjects reported that they owned a pet, while 1941 reported that they did not. Give a 98% confidence interval for the proportion of older adults in this population who are pet owners.
...
Suppose approximately 900 people die in bicycle accidents each year. One study examined the records of 1611 bicyclists aged 15 or older who were fatally injured in bicycle accidents in a five-year period and were tested for alcohol. Of these, 642 tested positive for alcohol (blood alcohol concentration of 0.01% or higher).
(a) Summarize the data with appropriate descriptive statistics. (Round your answer to four decimal places.)
(b) To do statistical inference for these data, we think in terms of a model where p is a parameter that represents the probability that a tested bicycle rider is positive for alcohol. Find a 99% confidence interval for p. (Round your answers to four decimal places.)
(c) Can you conclude from your analysis of this study that alcohol causes fatal bicycle accidents? Explain.
(d) In this study 600 bicyclists had blood alcohol levels above 0.10%, a level defining legally drunk at the time. Give a 99% confidence interval for the proportion who were legally drunk according to this criterion. (Round your answers to four decimal places.)
...
One of your employees has suggested that your company develop a new product. You decide to take a random sample of your customers and ask whether or not there is interest in the new product. The response is on a 1 to 5 scale with 1 indicating "definitely would not purchase"; 2, "probably would not purchase"; 3, "not sure"; 4, "probably would purchase"; and 5, "definitely would purchase." For an initial analysis, you will record the responses 1, 2, and 3 as "No" and 4 and 5 as "Yes." What sample size would you use if you wanted the 90% margin of error to be 0.1 or less? (Round your answer up to the nearest whole number.)
...
An automobile manufacturer would like to know what proportion of its customers are dissatisfied with the service received from their local dealer. The customer relations department will survey a random sample of customers and compute a 95% confidence interval for the proportion that are dissatisfied. From past studies, they believe that this proportion will be about 0.22. Find the sample size needed if the margin of error of the confidence interval is to be no more than 0.01. (Round your answer up to the nearest whole number.)
...
PTC is a substance that has a strong bitter taste for some people and is tasteless for others. The ability to taste PTC is inherited. About 78% of people from a certain country can taste PTC, for example. You want to estimate the proportion of people with at least one grandparent from this country who can taste PTC. (Round your answers up to the nearest person.)
(a) Starting with the 78% estimate, how large a sample must you collect in order to estimate the proportion of PTC tasters within ±0.04 with 90% confidence?
(b) Estimate the sample size required if you made no assumptions about the value of the proportion who could taste PTC.
How much has the required sample size changed?
...
Over the past few decades, public health officials have examined the link between weight concerns and teen girls smoking. Researchers surveyed a group of 273 randomly selected teen girls living in Massachusetts (between 12 and 15 years old). After four years the girls were surveyed again. Sixty-three (63) said they smoked to stay thin. Is there good evidence that less than thirty percent of the teen girls smoke to stay thin?
What is the alternate hypothesis?
...
A multimedia program designed to improve dietary behavior among low-income women was evaluated by comparing women who were randomly assigned to intervention and control groups. The intervention was a 30-minute session in a computer kiosk in the Food Stamp office. One of the outcomes was the score on a knowledge test taken about 2 months after the program. Here is a summary of the data:
Group n x s
Intervention 163 5.15 1.15
Control 214 4.33 1.16
(a) The test had six multiple-choice items that were scored as correct or incorrect, so the total score was an integer between 0 and 6. Do you think that these data are Normally distributed? Explain why or why not.
(b) Is it appropriate to use the two-sample t procedures that we studied in this section to analyze these data? Give reasons for your answer.
(c) Describe appropriate null and alternative hypotheses for evaluating the intervention.
Some people would prefer a two-sided alternative in this situation while others would use a one-sided significance test. Give reasons for each point of view.
(d) Carry out the significance test using a one-sided alternative. Report the test statistic with the degrees of freedom and the P-value. (Round your test statistic to three decimal places, your degrees of freedom to the nearest whole number, and your P-value to four decimal places.)
Write a short summary of your conclusion.
(e) Find a 95% confidence interval for the difference between the two means. Compare the information given by the interval with the information given by the significance test. (Round your answer to two decimal places.)
(f) The women in this study were all residents of Durham, North Carolina. To what extent do you think the results can be generalized to other populations?
...
A study that looked at beverage consumption used sample sizes that were much smaller than previous national surveys. One part of this study compared 20 children who were 7 to 10 years old with 5 who were 11 to 13. The younger children consumed an average of 8.2 oz of sweetened drinks per day while the older ones averaged 14.2 oz. The standard deviations were 10.7 oz and 8.2 oz respectively.
(a) Do you think that it is reasonable to assume that these data are Normally distributed? Explain why or why not. (Hint: Think about the 68−95−99.7 rule.)
(b) Using the methods in this section, test the null hypothesis that the two groups of children consume equal amounts of sweetened drinks versus the two-sided alternative. Report all details of the significance-testing procedure with your conclusion. (Round your answers to three decimal places.)
(c) Give a 95% confidence interval for the difference in means. (Round your answers to four decimal places.)
(d) Do you think that the analyses performed in parts (b) and (c) are appropriate for these data? Explain why or why not.
(e) The children in this study were all participants in an intervention study at a summer day camp at a particular university. To what extent do you think that these results apply to other groups of children?
...
In each of the following situations explain what is wrong and why.
(a) A researcher wants to test
H0: x1 = x2
versus the two-sided alternative
Ha: x1 ≠ x2.
(b) A study recorded the IQ scores of 50 college freshmen. The scores of the 24 males in the study were compared with the scores of all 50 freshmen using the two-sample methods of this section.
(c) A two-sample t statistic gave a P-value of 0.93. From this we can reject the null hypothesis with 90% confidence.
(d) A researcher is interested in testing the one-sided alternative
Ha: μ1 < μ2.
The significance test gave
t = 2.25.
Since the P-value for the two-sided alternative is 0.04, he concluded that his P-value was 0.02.
...
Assume x1 = 98, x2 = 121, s1 = 10, s2 = 12, n1 = 50, and n2 = 50. Find a 95% confidence interval for the difference in the corresponding values of μ. (Round your answers to three decimal places.)
Does this interval include more or fewer values than a 99% confidence interval? Explain your answer.
...
C-reactive protein (CRP) is a substance that can be measured in the blood. Values increase substantially within 6 hours of an infection and reach a peak within 24 to 48 hours after. In adults, chronically high values have been linked to an increased risk of cardiovascular disease. In a study of apparently healthy children aged 6 to 60 months in Papua New Guinea, CRP was measured in 90 children. The units are milligrams per liter (mg/l). Here are the data from a random sample of 40 of these children.
0.00 12.37 13.14 18.53 0.00 15.45 13.91 16.99 29.43 0.00
75.67 0.00 47.59 0.00 0.00 10.06 19.30 0.00 0.00 26.23
0.00 0.00 23.15 20.07 10.06 0.00 10.83 0.00 61.59 23.92
23.15 0.00 0.00 0.00 0.00 26.23 10.06 26.23 20.84 9.29
(a) Look carefully at the data above. Do you think that there are outliers or is this a skewed distribution?
Now use a histogram or stemplot to examine the distribution. (Do this on paper. Your instructor may ask you to turn this in.)
(b) Do you think that the mean is a good characterization of the center of this distribution? Explain why or why not.
(c) Find a 98% confidence interval for the mean CRP.
Discuss the appropriateness of using this methodology for these data.
7.2
A friend has performed a significance test of the null hypothesis that two means are equal. His report states that the null hypothesis is rejected in favor of the alternative that the first mean is larger than the second. In a presentation on his work, he notes that the first sample mean was larger than the second mean and this is why he chose this particular one-sided alternative.
(a) Explain what is wrong with your friend's procedure and why.
(b) Suppose he reported t = 1.70 with a P-value of 0.06. What is the correct P-value that he should report? (Round your answer to two decimal places.)
7.2
To apply the two-sample t procedures, use Option 1 if you have technology that implements that method. Otherwise, use Option 2.
Researchers gave 40 index cards to a waitress at an Italian restaurant in New Jersey. Before delivering the bill to each customer, the waitress randomly selected a card and wrote on the bill the same message that was printed on the index card. Twenty of the cards had the message "The weather is supposed to be really good tomorrow. I hope you enjoy the day!" Another 20 cards contained the message "The weather is supposed to be not so good tomorrow. I hope you enjoy the day anyway!" After the customers left, the waitress recorded the amount of the tip (percent of bill) before taxes. Here are the tips for those receiving the good-weather message.
21.0 18.8 20.0 20.9 22.1 23.6 23.1 24.9 22.4 20.5
25.4 22.6 27.2 20.7 22.5 24.3 21.3 22.1 22.5 23.1
The tips for the 20 customers who received the bad-weather message are below.
18.1 19.1 19.5 18.8 18.9 19.4 18.9 16.5 17.3 14.0
17.3 13.6 17.5 20.4 20.6 19.0 18.0 23.7 18.5 19.9
Give a 95% confidence interval for the difference between the mean percent tips for the two different messages. (Round your answers to two decimal places.)
7.2
In a study of children with a particular disorder, parents were asked to rate their child on a variety of items related to how well their child performs different tasks. One item was "Has difficulty organizing work," rated on a five-point scale of 0 to 4 with 0 corresponding to "not at all" and 4 corresponding to "very much." The mean rating for 284 boys with the disorder was reported as 2.31 with a standard deviation of 1.12. (Round your answers to four decimal places.)
Compute the 90% confidence interval.
Compute the 95% confidence interval.
Compute the 99% confidence interval.
Explain the effect of the confidence level on the width of the interval.
7.1
The level of various substances in the blood of kidney dialysis patients is of concern because kidney failure and dialysis can lead to nutritional problems. A researcher performed blood tests on several dialysis patients on 6 consecutive clinic visits. One variable measured was the level of phosphate in the blood. Phosphate levels for an individual tend to vary normally over time. The data on one patient, in milligrams of phosphate per deciliter (mg/dl) of blood, are given below.
5.0 5.0 5.5 4.8 5.6 4.5
(a) Calculate the sample mean x and its standard error.
(b) Use the t procedures to find the margin of error for a 90% confidence interval for this patient's mean phosphate level.
(c) Use the t procedures to give a 90% confidence interval for this patient's mean phosphate level.
7.1
Dual-energy X-ray absorptiometry (DXA) is a technique for measuring bone health. One of the most common measures is total body bone mineral content (TBBMC). A highly skilled operator is required to take the measurements. Recently, a new DXA machine was purchased by a research lab and two operators were trained to take the measurements. TBBMC for eight subjects was measured by both operators. The units are grams (g). A comparison of the means for the two operators provides a check on the training they received and allows us to determine if one of the operators is producing measurements that are consistently higher than the other. Here are the data:
Subject
Operator 1 2 3 4 5 6 7 8
1 1.326 1.342 1.074 1.225 0.938 1.006 1.182 1.287
2 1.323 1.322 1.073 1.233 0.934 1.019 1.184 1.304
(a) Take the difference between the TBBMC recorded for Operator 1 and the TBBMC for Operator 2. (Operator 1 minus Operator 2. Round your answers to four decimal places.)
Describe the distribution of these differences using words.
(b) Use a significance test to examine the null hypothesis that the two operators have the same mean. Give the test statistic. (Round your answer to three decimal places.)
Give your conclusion.
(c) The sample here is rather small, so we may not have much power to detect differences of interest. Use a 95% confidence interval to provide a range of differences that are compatible with these data. (Round your answers to four decimal places.)
(d) The eight subjects used for this comparison were not a random sample. In fact, they were friends of the researchers whose ages and weights were similar to the types of people who would be measured with this DXA. Comment on the appropriateness of this procedure for selecting a sample, and discuss any consequences regarding the interpretation of the significance testing and confidence interval results.
7.1
A study of commuting times reports the travel times to work of a random sample of 1008 employed adults in Chicago. The mean is x = 34.0 minutes and the standard deviation is s = 53.9 minutes. What is the standard error of the mean? (Round your answer to four decimal places.)
7.1
Researchers from the United Kingdom studied the effect of two breathing frequencies on performance times and on several physiological parameters in front crawl swimming. The breathing frequencies were one breath every second stroke (B2) and one breath every fourth stroke (B4). Subjects were 14 male collegiate swimmers. Each subject swam 200 meters using each breathing frequency: once with breathing frequency B2, and once on a different day with breathing frequency B4. A paper states that the results are expressed as mean plus or minus the standard deviation. One result reported in the paper states that the immediate postexercise heart rate for subjects when using breathing frequency B2 was 178 ± 17 beats per minute. What are x and the standard error of the mean for these subjects? (This exercise is also a warning to read carefully: that 178 ± 17 is not a confidence interval, yet summaries in this form are common in scientific reports. Round your standard error to three decimal places.)
7.1
Athletes performing in bright sunlight often smear black eye grease under their eyes to reduce glare. Does eye grease work? In one study, 16 student subjects took a test of sensitivity to contrast after three hours facing into bright sun, both with and without eye grease. (Greater sensitivity to contrast improves vision and glare reduces sensitivity to contrast.) This is a matched pairs design. Here are the differences in sensitivity, with eye grease minus without eye grease.
0.07
0.65
−0.13
−0.05
−0.17
0.16
−0.16
0.03
0.05
0.02
0.44
0.23
−0.11
0.27
0.06
0.29
How much more sensitive to contrast are athletes with eye grease than without eye grease? Give a 99% confidence interval to answer this question. (Round your answer to three decimal places.)
7.1
State the appropriate null hypothesis H0 and alternative hypothesis Ha in each of the following cases.
(a) A study reported that 80% of students owned a cell phone. You plan to take an SRS of students to see if the percent has increased.
(b) The examinations in a large freshman chemistry class are scaled after grading so that the mean score is 78. The professor thinks that students who attend morning recitation sections will have a higher mean score than the class as a whole. Her students this semester can be considered a sample from the population of all students she might teach, so she compares their mean score with 78.
(c) The student newspaper at your college recently changed the format of their opinion page. You take a random sample of students and select those who regularly read the newspaper. They are asked to indicate their opinions on the changes using a five-point scale: −2 if the new format is much worse than the old, −1 if the new format is somewhat worse than the old, 0 if the new format is the same as the old, +1 if the new format is somewhat better than the old, and +2 if the new format is much better than the old.
6.2
State the null hypothesis H0 and the alternative hypothesis Ha in each case. Be sure to identify the parameters that you use to state the hypotheses.
(a) A university gives credit in French language courses to students who pass a placement test. The language department wants to know if students who get credit in this way differ in their understanding of spoken French from students who actually take the French courses. Experience has shown that the mean score of students in the courses on a standard listening test is 26. The language department gives the same listening test to a sample of 35 students who passed the credit examination to see if their performance is different.
(b) Experiments on learning in animals sometimes measure how long it takes a mouse to find its way through a maze. The mean time is 20 seconds for one particular maze. A researcher thinks that playing rap music will cause the mice to complete the maze faster. She measures how long each of 18 mice takes with the rap music as a stimulus.
(c) The average square footage of one-bedroom apartments in a new student-housing development is advertised to be 440 square feet. A student group thinks that the apartments are smaller than advertised. They hire an engineer to measure a sample of apartments to test their suspicion.
6.2
A test of the null hypothesis
H0: μ = μ0
gives test statistic z = 0.88. (Round your answers to four decimal places.)
(a) What is the P-value if the alternative is
Ha: μ > μ0?
(b) What is the P-value if the alternative is
Ha: μ < μ0?
(c) What is the P-value if the alternative is
Ha: μ ≠ μ0?
6.2
A test of the null hypothesis
H0: μ = μ0
gives test statistic
z = −0.40.
(Round your answers to four decimal places.)
(a) What is the P-value if the alternative is
Ha: μ > μ0?
(b) What is the P-value if the alternative is
Ha: μ < μ0?
(c) What is the P-value if the alternative is
Ha: μ ≠ μ0?
6.2
The P-value for a two-sided test of the null hypothesis
H0: μ = 45
is 0.02.
(a) Does the 95% confidence interval include the value 45? Why?
(b) Does the 90% confidence interval include the value 45? Why?
6.2
Suppose the level of calcium in the blood in healthy young adults varies with mean about 9.1 milligrams per deciliter and standard deviation about σ = 0.3. A clinic in rural Guatemala measures the blood calcium level of 170 healthy pregnant women at their first visit for prenatal care. The mean is
x = 9.16.
Is this an indication that the mean calcium level in the population from which these women come differs from 9.1?
(a) State H0 and Ha.
(b) Carry out the test and give the P-value, assuming that σ = 0.3 in this population. (Round your answer to four decimal places.)
(c) Give a 95% confidence interval for the mean calcium level μ in this population. We are confident that μ lies quite close to 9.1. This illustrates the fact that a test based on a large sample (n = 170 here) will often declare even a small deviation from H0 to be statistically significant. (Round your answers to three decimal places.)
6.2
Computers in some vehicles calculate various quantities related to performance. One of these is the fuel efficiency, or gas mileage, usually expressed as miles per gallon (mpg). For one vehicle equipped in this way, the mpg were recorded each time the gas tank was filled, and the computer was then reset. In addition to the computer computing mpg, the driver also recorded the mpg by dividing the miles driven by the number of gallons at each fill-up. The following data are the differences between the computer's and the driver's calculations for a random sample of 20 records. The driver wants to determine if these calculations are different. Assume the standard deviation of a difference to be σ = 3.0.
6.0 5.5 −0.6 1.6 3.7 4.5 8.0 2.2 4.6 3.0
4.4 0.5 3.0 1.3 1.3 5.0 2.1 3.5 −0.6 −4.2
(a) State the appropriate H0 and Ha to test this suspicion.
(b) Carry out the test. Give the P-value. (Round your answer to four decimal places.)
6.2
According to data, a particular brand of cigarettes contain an average of 1.4 milligrams of nicotine. An advocacy group commissions an independent test to see if the mean nicotine content is higher than the industry laboratory claims.
(a) What are H0 and Ha?
(b) Suppose that the test statistic is z = 2.39. Is this result significant at the 5% level?
(c) Is the result significant at the 1% level?
6.2
The Survey of Study Habits and Attitudes (SSHA) is a psychological test that measures students' study habits and attitudes toward school. The survey yields several scores, one of which measures student attitudes toward studying. The mean student attitude score for college students is about 60, and standard deviation is about 13. A researcher in the Philippines is concerned about the declining performance of college graduates on professional licensure and board exams. She suspects that poor attitudes of students are partly responsible for the decline and that the mean for college seniors who plan to take professional licensure or board exams is less than 60. She gives the SSHA to an SRS of 169 college seniors in the Philippines who plan to take professional licensure or board exams. Suppose we know that the student attitude scores in the population of such students are Normally distributed with standard deviation σ = 13.
(a) We seek evidence against the claim that μ = 60. What is the sampling distribution of the mean score x of a sample of 169 students if the claim is true?
Draw the density curve of this distribution. (Sketch a Normal curve, then mark on the axis the values of the mean and one, two, and three standard deviations of the sampling distribution on either side of the mean.)
(b) Suppose that the sample data give x = 59.2. Mark this point on the axis of your sketch. In fact, the result was x = 57.2. Mark this point on your sketch. Using your sketch, explain in simple language why one result is good evidence that the mean score of all college seniors in the Philippines who plan to take professional licensure or board exams is less than 60 and why the other outcome is not.
6.2
The Survey of Study Habits and Attitudes (SSHA) is a psychological test that measures students' study habits and attitude toward school. The survey yields several scores, one of which measures student attitudes toward studying. The mean student attitude score for college students is about 50, and the standard deviation is about 15. A researcher in the Philippines is concerned about the declining performance of college graduates on professional licensure and board exams. She suspects that poor attitudes of students are partly responsible for the decline and that the mean for college seniors who plan to take professional licensure or board exams is less than 50. She gives the SSHA to an SRS of 225 college seniors in the Philippines who plan to take professional licensure or board exams. Suppose we know that the student attitude scores in the population of such students are Normally distributed with standard deviation
σ = 15.
(Use a left-tailed test.)
(a) One sample of 225 students had mean student attitude score
x = 49.6.
Enter this x, along with the other required information, into the P-Value of a Test of Significance Applet. What is the P-value? (Round your answer to four decimal places.)
Is this outcome statistically significant at the
α = 0.05
level? At the
α = 0.01
level?
(b) Another sample of 225 students had
x = 47.6.
Use the applet to find the P-value for this outcome. (Round your answer to four decimal places.)
Is it statistically significant at the
α = 0.05
level? At the
α = 0.01
level?
(c) Explain briefly why these P-values tell us that one outcome is strong evidence against the null hypothesis and that the other outcome is not.
6.2
Emissions of sulfur dioxide by industry set off chemical changes in the atmosphere that result in "acid rain." The acidity of liquids is measured by pH on a scale of 0 to 14. Distilled water has pH 7.0, and lower pH values indicate acidity. Normal rain is somewhat acidic, so acid rain is sometimes defined as rainfall with a pH below 5.0. Suppose that pH measurements of rainfall on different days in a Canadian forest follow a Normal distribution with standard deviation
σ = 0.6.
A sample of n days finds that the mean pH is x = 4.8. Is this good evidence that the mean pH μ for all rainy days is less than 5.0? The answer depends on the size of the sample. Either by hand or using the P-Value of a Test of Significance applet, carry out four tests of
H0: μ = 5.0
Ha: μ < 5.0.
Use σ = 0.6 and x = 4.8 in all four tests. But use four different sample sizes:
n = 4,
n = 36,
n = 49,
and
n = 64.
(a) What are the P-values for the four tests? (Round your answers to four decimal places.)
(b) For each test, sketch the Normal curve for the sampling distribution of x when H0 is true. This curve has mean 5.0 and standard deviation
0.6/
n
.
Mark the observed
x = 4.8
on each curve. What happens to the significance of the result
x = 4.8
as the sample size increases?
6.2
A study based on a sample of size 36 reported a mean of 91 with a margin of error of 17 for 95% confidence.
(a) Give the 95% confidence interval.
(b) If you wanted 99% confidence for the same study, would your margin of error be greater than, equal to, or less than 17? Explain your answer.
6.1
You want to rent an unfurnished one-bedroom apartment in Boston next year. The mean monthly rent for a random sample of 11 apartments advertised in the local newspaper is $1500. Assume that the standard deviation is $210. Find a 95% confidence interval for the mean monthly rent for unfurnished one-bedroom apartments available for rent in this community. (Round your answers to two decimal places.)
6.1
You want to rent an unfurnished one-bedroom apartment in Boston next year. The mean monthly rent for a random sample of 11 apartments advertised in the local newspaper is $1700. Assume that the standard deviation is $240. Find a 95% confidence interval for the mean monthly rent for unfurnished one-bedroom apartments available for rent in this community. (Round your answers to two decimal places.)
Will the 95% confidence interval include approximately 95% of the rents of all unfurnished one-bedroom apartments in this area? Explain why or why not.
6.1
You want to rent an unfurnished one-bedroom apartment in Boston next year. The mean monthly rent for a random sample of 16 apartments advertised in the local newspaper is $1400. Assume that the standard deviation is $250. How large a sample of one-bedroom apartments would be needed to estimate the mean μ within ±$50 with 95% confidence? (Round your answer up to the next whole number.)
6.1
The National Institute of Standards and Technology (NIST) supplies "standard materials" whose physical properties are supposed to be known. For example, you can buy from NIST an iron rod whose electrical conductivity is supposed to be 10.1 at 293 kelvin. (The units for conductivity are microsiemens per centimeter. Distilled water has conductivity 0.5.) Of course, no measurement is exactly correct. NIST knows the variability of its measurements very well, so it is quite realistic to assume that the population of all measurements of the same rod has the Normal distribution with mean μ equal to the true conductivity and standard deviation σ = 0.1. Here are six measurements on the same standard iron rod, which is supposed to have conductivity 10.1.
10.08 9.86 10.02 10.12 10.21 10.11
NIST wants to give the buyer of this iron rod a 90% confidence interval for its true conductivity. What is this interval? (Round your answers to three decimal places.)
6.1
We have the survey data on the body mass index (BMI) of 643 young women. The mean BMI in the sample was
x = 27.2.
We treated these data as an SRS from a Normally distributed population with standard deviation
σ = 7.7.
Give confidence intervals for the mean BMI and the margins of error for 90%, 95%, and 99% confidence. (Round your answers to two decimal places.)
How does increasing the confidence level change the margin of error of a confidence interval when the sample size and population standard deviation remain the same?
6.1
The National Assessment of Educational Progress (NAEP) includes a mathematics test for eighth-graders. Scores on the test range from 0 to 500. Suppose that you give the NAEP test to an SRS of 900 eighth-graders from a large population in which the scores have mean μ = 286 and standard deviation σ = 127. The mean x will vary if you take repeated samples.
The sampling distribution of x is approximately Normal. It has mean μ = 286. What is its standard deviation? (Round your answer to three decimal places.)
6.1
Suppose that scores on the mathematics part of a test for eighth-grade students follow a Normal distribution with standard deviation σ = 120. You want to estimate the mean score within ±19 with 90% confidence. How large an SRS of scores must you choose? (Round your answer up to the next whole number.)
6.1
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