stats unit nine review (final)
In a test of H0: μ = 8 versus Ha: μ ≠ 8, a sample of size of 220 leads to a p-value of 0.034. Which of the following must be true?
A 95% confidence interval for μ calculated from these data will not include μ = 8.
A business analyst is investigating whether the mean amount of purchases made by customers at an online department store is greater than $100. The analyst obtained a random sample of 56 orders and calculated a sample mean of $102.30 and a sample standard deviation of $5.30. Which of the following is an appropriate test for the investigation?
A one-sample t-test for a population mean
An experimenter conducted a two-tailed hypothesis test on a set of data and obtained a p-value of 0.44. If the experimenter had conducted a one-tailed test on the same set of data, which of the following is true about the possible p-value(s) that the experimenter could have obtained?
The possible p-values are 0.22 and 0.78.
Consider a population with population proportion p, and a sample from the population with sample proportion p(hat). Which of the following describes the purpose of the one-sample z-test?
To estimate the probability of observing a value as extreme as p(hat) given p
Which of the following is the best interpretation of the power of a significance test?
Power is the probability of detecting an effect if an effect exists.
The p-value for a one-sided t-test is 0.10. If the test had been two-sided, what would the p-value have been?
0.20
A survey of a random sample of 1,045 young adults found that 60 percent do not have a landline telephone number. A hypothesis test will be used to determine whether the data provide convincing statistical evidence that more than 50 percent of all young adults do not have a landline telephone number. Which of the following is the test statistic for the appropriate test?
0.60−0.50(0.50)(0.50)1,0451,045(0.50)(0.50)0.60−0.50
The distribution of mass for United States pennies minted since 1982 is approximately normal with mean 2.5 grams. A random sample of 10 pennies minted since 1982 was selected. The sample had a mean mass of 2.47 grams and a standard deviation of 0.04 gram. The test statistic for the population mean has which of the following distributions?
A t-distribution with 9 degrees of freedom
On their birthdays, employees at a large company are permitted to take a 60-minute lunch break instead of the usual 30 minutes. Data were obtained from 10 randomly selected company employees on the amount of time that each actually took for lunch on his or her birthday. The company wishes to investigate whether these data provide convincing evidence that the mean time is greater than 60 minutes. Of the following, which information would NOT be expected to be a part of the process of correctly conducting a hypothesis test to investigate the question, at the 0.05 level of significance?
Given that the p-value is greater than 0.05, rejecting the null hypothesis and concluding that the mean time was not greater than 60 minutes
Medical researchers are testing a new surgical procedure designed to minimize the side effects of surgery. The null hypothesis is that the procedure is not effective in minimizing side effects. For the researchers, the more consequential error would be that the procedure actually is effective in minimizing the side effects, but the test does not detect the effectiveness of the procedure. Which of the following should the researchers do to avoid the more consequential error?
Increase the significance level to increase the probability of a Type I error.
A matched-pairs t-test is NOT an appropriate way to analyze data consisting of which of the following?
Measurements of annual income for both individuals in pairs formed by assigning 100 people to pairs at random
A one-sample z-test for a population proportion will be conducted using a simple random sample selected without replacement from a population. Which of the following is a check for independence?
The population size is more than 10 times the sample size.
Most dermatologists recommend that the ideal shower lasts approximately 10 minutes. A researcher suspects that the average shower length of high school students is greater than 10 minutes. To test the belief, the researcher surveyed 125 randomly selected high school students and found that their average shower length was 14.7 minutes. With all conditions for inference met, a hypothesis test was conducted at the significance level of α=0.05,α=0.05, and the test produced a p-value of 0.0000. Which of the following is an appropriate conclusion?
The researcher has statistical evidence to conclude that the population mean shower length for high school students is greater than 10 minutes.
If all else is constant, which of the following results in an increase in the probability of a Type II error?
The sample size is decreased.
The germination rate is the rate at which plants begin to grow after the seed is planted. A seed company claims that the germination rate for their seeds is 90 percent. Concerned that the germination rate is actually less than 90 percent, a botanist obtained a random sample of seeds, of which only 80 percent germinated. What are the correct hypotheses for a one-sample z-test for a population proportion pp?
H0:p=0.90 Ha:p<0.90Ha:p<0.90
Educators are testing a new program designed to help children improve their reading skills. The null hypothesis of the test is that the program does not help children improve their reading skills. For the educators, the more consequential error would be that the program does not help children improve their reading skills but the test indicated that it does help. Which of the following should the researchers do to avoid the more consequential error?
Decrease the significance level to decrease the probability of Type I error
researcher t-test of the hypothesis p-value alternative hypothesis
12(0.0627)21(0.0627) one over two
Studies indicate that about 10 percent of polar bears weigh more than 1,000 pounds. A biologist studying the bears thinks that percent might be too high. From a random sample of polar bears, the biologist found only 8 percent of the sample weighing over 1,000 pounds. Which of the following is the most appropriate method for the biologist's study?
A one-sample z-test for a population proportion
A national health study reported that the proportion of students with elevated blood pressure is 0.15. The principal of a local high school believes that the proportion of students in the school with elevated blood pressure is greater than 0.15. If a large random sample is used, which of the following is the most appropriate test to investigate the principal's belief?
A z-test for a proportion
A fast-food restaurant claims that a small order of french fries contains 120 calories. A nutritionist is concerned that the true average calorie count is higher than that. The nutritionist randomly selects 35 small orders of french fries and determines their calories. The resulting sample mean is 155.6 calories, and the p-value for the hypothesis test is 0.00093. Which of the following is a correct interpretation of the p-value?
If the population mean is 120 calories, the p-value of 0.00093 is the probability of observing a sample mean of 155.6 calories or more.
If all else is constant, which of the following would result in a decrease of the probability of a Type II error?
The sample size is increased.
A researcher is conducting a study of charitable donations by surveying a simple random sample of households in a certain city. The researcher wants to determine whether there is convincing statistical evidence that more than 50 percent of households in the city gave a charitable donation in the past year. Let p represent the proportion of all households in the city that gave a charitable donation in the past year. Which of the following are appropriate hypotheses for the researcher?
H0: p = 0.5 and Ha: p > 0.5
The mean length μμ of male geckos is 9.5 inches. A researcher studying a population of geckos in a certain region will conduct a hypothesis test to investigate whether there is evidence that the mean length is greater than 9.5 inches. A random sample of geckos was selected, and the sample mean x−x− was calculated as 10 inches. Which of the following is the correct set of hypotheses?
H0:μ=9.5 Ha:μ>9.5Ha:μ>9.5
Suppose that on a hypothesis test for a single population mean, Ha: μ < 10. Assume that Ha is true. For a fixed sample size and significance level α, the power of the test will be greatest if the actual mean is which of the following?
8
A company currently uses Brand A lightbulbs, which have a mean life of 1,000 hours. A salesperson marketing Brand B, a new brand of bulb, contacts the company. The company will switch to the new brand of bulb only if there is convincing evidence that the mean life of Brand B is greater than 1,000 hours. Which of the following hypotheses should the company test?
H0 : The mean life of Brand B bulbs is 1,000 hours. Ha : The mean life of Brand B bulbs is more than 1,000 hours.
The plant manager of a company that makes pillows claims that only 8 percent of the pillows made have a stitching defect. The quality control director thought that the percent might be different from 8 percent and selected a random sample of pillows to test. The director tested the hypotheses H0:p=0.08H0:p=0.08 versus Ha:p≠0.08Ha:p=0.08 at the significance level of α=0.08α=0.08 .The p-value of the test was 0.03. Assuming all conditions for inference were met, which of the following is the correct conclusion?
The pp-value is less than αα, and the null hypothesis is rejected. There is convincing evidence to suggest the true proportion of stitching defects is not 0.08.
A university will add fruit juice vending machines to its classroom buildings if the student body president is convinced that more than 20 percent of the students will use them. A random sample of n students will be selected and asked whether or not they would use the vending machines. A large-sample test for proportions at the significance level of α = 0.05 will be performed. The null hypothesis that the proportion of all students who would use the vending machines is 20 percent will be tested against the alternative that more than 20 percent of all students would use them. For which of the following situations would the power of the test be highest?
The sample size is n = 1,000, and 50 percent of all students use the vending machines.
For a one-sample test for a population proportion p and sample size n, why is it necessary that np0np0 and n(1−p0)n(1−p0) are both at least 10 ?
The sample size must be large enough to support an assumption that the sampling distribution of the sample proportion is approximately normal.
An environmental scientist wants to test the null hypothesis that an antipollution device for cars is not effective. Under which of the following conditions would a Type I error be committed?
The scientist concludes that the antipollution device is effective when it actually is not.
When a virus is placed on a tobacco leaf, small lesions appear on the leaf. To compare the mean number of lesions produced by two different strains of virus, one strain is applied to half of each of 8 tobacco leaves, and the other strain is applied to the other half of each leaf. The strain that goes on the right half of the leaf is decided by a coin flip. The lesions that appear on each half are then counted. The data are given below. What is the number of degrees of freedom associated with the appropriate t-test for testing to see if there is a difference between the mean number of lesions per leaf produced by the two strains?
7
In high school X, approximately 9 percent of the students saw a certain movie on opening night. From a random sample of 200 students from high school Y, 22 saw the movie on opening night. Consider a hypothesis test to investigate whether the proportion of all students in high school Y who saw the movie on opening night is greater than that of high school X. Which of the following is the standard deviation used to calculate the test statistic for the one-sample z-test?
(0.09)(0.91)200200(0.09)(0.91)
The process of producing pain-reliever tablets yields tablets with varying amounts of the active ingredient. It is claimed that the average amount of active ingredient per tablet is at least 200 milligrams. The Consumer Watchdog Bureau tests a random sample of 70 tablets. The mean content of the active ingredient for this sample is 194.3 milligrams, while the standard deviation is 21 milligrams. What is the approximate p-value for the appropriate test?
0.012
Makers of a new pain-relieving medication claim that it relieves chronic pain faster than the current top-selling pain reliever on the market. A double-blind experiment was conducted in which 10 people who experience chronic pain were randomly selected to take either the new or the current medication. Each of the 10 people recorded the time, in minutes, from taking the medication until pain relief. After an appropriate time period, each of the 10 people took the other medication and recorded the time from taking the medication until pain relief. The medication each person took first was randomly determined, and because both medications look the same, the people in the study did not know which medication was taken first. The table below shows summary statistics for the results. Which of the following values is closest to the p-value of the appropriate t-test?
0.1802
A recent study indicated that 17 percent of adults in the country actively seek out science news sites to keep current on topics in science. A university researcher believes that percent is too low. From a random sample of adults in the country, the researcher found that 22 percent of the sample actively seek out science news sites. Which of the following is the most appropriate method for the researcher's study?
A one-sample zz-test for a population proportion
A six-week fitness program was designed to decrease the time it takes retired individuals to walk one mile. At the beginning of the program, 20 randomly selected retired individuals were invited to participate, and their times to walk a mile were recorded. After the six-week program, their times to walk a mile were again recorded. Most participants saw little to no improvement in their times to walk one mile; however, a few participants saw drastic improvements in their times to walk one mile. The program director would like to perform a hypothesis test to determine if the program reduces the mean time for retired individuals to walk a mile. Which of the following statements is true?
Because the sample size of 20 is less than 30 and the improvements in walk times in the sample data appear to be skewed, the distribution of sample means should not be assumed to be approximately normal.
Perchlorate is a chemical used in rocket fuel. People who live near a former rocket-testing site are concerned that perchlorate is present in unsafe amounts in their drinking water. Drinking water is considered safe when the average level of perchlorate is 24.5 parts per billion (ppb) or less. A random sample of 28 water sources in this area produces a mean perchlorate measure of 25.3 ppb. Which of the following is an appropriate alternative hypothesis that addresses their concern?
Ha : μ > 24.5
After a tropical storm in a certain state, news reports indicated that 19 percent of households in the state lost power during the storm. A state engineer believes that estimate is too low. The engineer will collect data to perform a hypothesis test on the proportion of all households without power. Which of the following are the appropriate hypotheses for such a test?
Ho:p=0.19 Ha:p>0.19Ha:p>0.19
An automobile manufacturer claims that the average gas mileage of a new model is 35 miles per gallon (mpg). A consumer group is skeptical of this claim and thinks the manufacturer may be overstating the average gas mileage. If µ represents the true average gas mileage for this new model, which of the following gives the null and alternative hypotheses that the consumer group should test?
Ho:μ=35 mpg Ha:μ<35Ha:μ<35 mpg
In order to make statistical inferences when testing a population proportion p, which of the following conditions verify that inference procedures are appropriate? The data are collected using a random sample or random assignment. The sample size is less than 10 percent of the population size. np0≥10np0≥10 and n(1−p0)≥10n(1−p0)≥10 for sample size nn and hypothesized proportion p0p0
I, II, and III
Past studies indicate that about 60 percent of the trees in a forested region are classified as softwood. A botanist studying the region suspects that the proportion might be greater than 0.60. The botanist obtained a random sample of trees from the region and conducted a test of H0:p=0.6 versus Ha:p>0.6. The pp-value of the test was 0.015. Which of the following is a correct interpretation of the pp-value?
If it is true that 60 percent of the trees in a forested region are classified as softwood, 0.015 is the probability of obtaining a sample proportion as large as or larger than the one obtained by the botanist.
A research organization reported that 41 percent of adults who were asked to describe their day responded that they were having a good day rather than a typical day or a bad day. To investigate whether the percent would be different for high school students, 600 high school students were randomly selected. When asked to describe their day, 245 students reported that they were having a good day rather than a typical day or a bad day. Do the data provide convincing statistical evidence that the proportion of all high school students who would respond that they were having a good day is different from 0.41?
No, because the p-value is greater than any reasonable significance level.
For a school fund-raiser, 600 raffle tickets were sold by students at the school, of which 88 were sold by one student, Audrey. Of the 600 tickets sold, 30 were randomly selected to receive prizes, and 7 of the 30 tickets selected were tickets sold by Audrey. To investigate how likely it was by chance alone that at least 7 of the 30 selected tickets could have been sold by Audrey, students in a statistics class ran a simulation. One trial of the simulation is described by the following steps. Step 1: From 600 chips, assign 88 red and the rest blue. Step 2: Select 30 chips at random without replacement. Step 3: Record the number of red chips in the selection of 30. The results of 1,000 trials of the simulation are shown in the histogram. Based on the results of the simulation, is there convincing statistical evidence at the significance level of 0.05 that the event of Audrey selling at least 7 of the 30 selected tickets is unlikely to have occurred by chance alone?
No, because the simulation suggests that Audrey selling at least 7 of 30 selected tickets would occur about 13.8% of the time.
A significance test is conducted for which the alternative hypothesis states that more than 85 percent of adult sea turtles on a certain beach are female. The pp-value for the test is 0.4158. If the null hypothesis is true, which of the following statements is a correct interpretation of the pp-value?
Of all possible samples of the same size, 41.58 percent will result in 85 percent or more of adult sea turtles on the beach being female.
In a physics experiment, two different methods were used to measure the angle of deflection when a subatomic particle collides with a certain material. Ten specimens of the material were used to compare the two methods. For each specimen, the angle of deflection was measured using both methods. For each specimen, the method used first was determined by the flip of a fair coin. The difference between the measured angles was calculated for each specimen. A test of the hypothesis that the population mean difference is zero had a p-value of 0.082. The hypothesis test described had three components: the number of specimens, the sample standard deviation of the differences, and the magnitude of the sample mean difference. Compared to the test described, which of the following would have resulted in a smaller p-value?
The number of specimens remained the same, but the magnitude of the sample mean difference was larger and the sample standard deviation of the difference was smaller.
A car company claims that its new car, the GoFast2000, has a gas mileage of 35 miles per gallon (mpg). A consumer group suspects that the true mean gas mileage of the new cars is less than 35 mpg. The group tests 50 randomly selected GoFast2000 cars and finds a sample mean of 34.8 mpg. With all assumptions for inference met, a hypothesis test resulted in a p-value of 0.324. For a significance level of α=0.05,α=0.05, which of the following is a correct conclusion?
The p-value is greater than 0.05, and the null hypothesis is not rejected. There is not convincing statistical evidence that the mean is less than 35 mpg.
A two-sided t-test for a population mean is conducted of the null hypothesis H0 : μ = 100. If a 90 percent t-interval constructed from the same sample data contains the value of 100, which of the following can be concluded about the test at a significance level of a = 0.10 ?
The p-value is greater than 0.10, and H0 should not be rejected.
A company that ships glass for a glass manufacturer claimed that its shipping boxes are constructed so that no more than 8 percent of the boxes arrive with broken glass. The glass manufacturer believed the actual percent is greater than 8 percent. The manufacturer selected a random sample of boxes and recorded the proportion of boxes that arrived with broken glass. The manufacturer tested the hypotheses Ho:p=0.08Ho:p=0.08 versus Ha:p>0.08Ha:p>0.08 at the significance level of α=0.01α=0.01. The test yielded a p-value of 0.001. Assuming all conditions for inference were met, which of the following is the correct conclusion?
The p-value is less than αα and the null hypothesis is rejected. There is convincing evidence that the proportion of all boxes that contain broken glass is greater than 0.08.
A one-sided hypothesis test is to be performed with a significance level of 0.05. Suppose that the null hypothesis is false. If a significance level of 0.01 were to be used instead of a significance level of 0.05, which of the following would be true?
The probability of a Type II error would increase and the power of the test would decrease.
Machines at a bottling plant are set to fill bottles to 12 ounces. The quality control officer at the plant periodically tests the machines to be sure that the bottles are filled to an appropriate amount. The null hypothesis of the test is that the mean is at least 12 ounces. The alternative hypothesis is that the mean is less than 12 ounces. Which of the following describes a Type I error that could result from the test?
The test provides convincing evidence that the mean is less than 12 ounces, but the actual mean is at least 12 ounces.
A large clinical trial was designed to determine whether a certain vitamin improves the general health of adults. The investigators first identified 85 variables that measure various aspects of the general health of adults. Because each adult in the clinical trial was to serve as his or her own control, the 85 variables were measured for each adult, both before taking the vitamin and after taking the vitamin for three months. The investigators then performed 85 matched-pair t-tests, one for each variable. They found statistically significant results at the 0.05 level in 2 of the variables, both in the direction of improved general health. Which of the following should the investigators conclude?
There is insufficient evidence that the vitamin improves the health of adults because at the 0.05 significance level, one could easily get statistically significant results in 2 out of 85 tests just due to chance variability.
The president of a large company recommends that employees perform, on average, 24 hours of community service each year. The president believes that the mean number of hours of community service performed last year was different from the recommended 24 hours. To estimate the mean number of hours of community service performed last year, the president obtained data from a random sample of employees and used the data to construct the 95 percent confidence interval (20.37, 23.49). If all conditions for inference were met, does the interval provide convincing statistical evidence, at a level of significance of α=0.05,α=0.05, to support the president's belief that the mean number of hours of community service performed last year is different from what is recommended?
Yes, the interval supports the president's belief because 24 is not contained in the interval.
A century ago, the average height of adult women in the United States was 63 inches. Researchers believe that the average might be greater today. A random sample of 40 adult women was selected from the population. The sample had mean 64.2 inches and standard deviation 2.9 inches. Assuming all conditions for inference are met, the researchers will perform an appropriate hypothesis test to investigate their belief. Which of the following is the correct test statistic for the hypothesis test?
t=402.964.2−63
For a certain brand of canned corn, the company claims that the mean weight of the contents of the cans is 15.25 ounces. A random sample of 36 cans were selected. The sample was found to have mean 15.18 ounces and standard deviation 0.12 ounce. A hypothesis test will be conducted to investigate whether there is evidence to support the belief that the mean is less than 15.25 ounces. Which of the following is the correct test statistic for the hypothesis test?
t=60.1215.18−15.25