stats unit 10

machine is used to fill bags with a popular brand of trail mix. The machine is calibrated so the distribution of the weights of the bags of trail mix is normal, with mean 240 grams and standard deviation 3 grams. Of the following, which is the least weight of a bag in the top 5 percent of the distribution?

234 grams

For which of the following conditions is it not appropriate to assume that the sampling distribution of the sample mean is approximately normal?

A random sample of 10 taken from a population distribution that is skewed to the right

A recent survey indicated that the mean time spent on a music streaming service is 210 minutes per week for the population of a certain country. A simulation was conducted to create a sampling distribution of the sample mean for a population with a mean of 210. The following histogram shows the results of the simulation. Which of the following would be the best reason why the simulation of the sampling distribution is not approximately normal?

The sample size was not sufficiently large.

city health officials; fleas; cats; dogs

120-100 divided by the square root of 22 squared divided by 22 plus 30 squared over 15

At a large corporation, 6,000 employees from department A and 4,000 employees from department B are attending a training session. A random sample of 500 employees attending the session will be selected. Consider two sampling methods: with replacement and without replacement. How will the methods affect the standard deviations of the sampling distribution of the sample proportion of employees from department B?

Sampling without replacement will result in a standard deviation less than but close to 0.4(0.6)500.5000.4(0.6)​​.

A pollster is interested in comparing the proportions of women and men in a particular town who are in favor of a ban on fireworks within town borders. The pollster plans to test the hypothesis that the proportion of women in favor of the ban is different from the proportion of men in favor of the ban. There are 4,673 women and 4,502 men who live in the town. From a simple random sample of 40 women in the town, the pollster finds that 38 favor the ban. From an independent simple random sample of 50 men in the town, the pollster finds that 27 favor the ban. Which of the following statements is true about this situation?

A two-proportion z-test would not be valid for these data.

To investigate whether there is a significant difference between two regions of a state in the percent of voters who intend to vote for the incumbent governor in the next election, a polling agency interviewed 300 randomly selected voters from the north of the state and 400 randomly selected voters from the south of the state. Of those interviewed, 200 from the north and 325 from the south indicated they intended to vote for the incumbent governor in the next election. Which of the following is the most appropriate method for analyzing the results?

A two-sample z-test for a difference in population proportions

seeds; water; proportion of seeds

I and III only

Clara recorded 50 numerical observations on a certain variable and then calculated the mean x‾x and the standard deviation s for the observations. To help decide whether a normal model is appropriate, she created the following chart. In Clara's chart, the letters a, b, c, d, and e represent the number of observations falling in each interval. Which of the following list of counts for a, b, c, d, and e respectively, is the best indicator that the variable can be modeled with a normal approximation?

1, 7, 34, 7, 1

A sample of size n will be selected from a population with population proportion p. Which of the following must be true for the sampling distribution of the sample proportion to be approximately normal?

Both np and n(1 - p) are at least 10

The mean number of pets owned by the population of students at a large high school is 3.2 pets per student with a standard deviation of 1.7 pets. A random sample of 16 students will be selected and the mean number of pets for the sample will be calculated. What is the mean of the sampling distribution of the sample mean for samples of size 16 ?

3.2

Which of the following is not a condition for constructing a confidence interval to estimate the difference between two population proportions?

The data must come from populations with approximately normal distributions.

A national charity contacted 100 randomly selected people by phone, and 7 percent of those contacted made a donation to the charity. The population proportion of those who make a donation when contacted by phone is known to be p = 0.05. For samples of size 100, which of the following best interprets the mean of the sampling distribution of the sample proportion of people who make a donation when contacted by phone?

The mean of all sample proportions of those who make a donation from all random samples of 100 people contacted by phone is 0.05.

A two-sample t-test for a difference in means was conducted to investigate whether defensive players on a football team can bench-press more weight, on average, than offensive players. The conditions for inference were met, and the test produced a test statistic of t = 1.083 and a p-value of 0.15. Based on the p-value and a significance level of α=0.05α=0.05 which of the following is the correct conclusion?

Fail to reject the null hypothesis because 0.15 > 0.05. There is not convincing evidence that defensive players can bench-press more weight, on average, than offensive players.

symphony one

For all random samples of 200 adults from the city and 200 students from the city, the sample proportion of adults who prefer pop music will be greater than the sample proportion of students who prefer pop music in about 2.2% of samples.

poldactyl cats; toes; region A; region B

H0​:pa​−pb​=0 Ha:pa−pb>0Ha​:pa​−pb​>0

A recent survey concluded that the proportion of American teenagers who have a cell phone is 0.27. The true population proportion of American teenagers who have a cell phone is 0.29. For samples of size 1,000 that are selected at random from this population, what are the mean and standard deviation, respectively, for the sampling distribution of the sample proportion of American teenagers who have a cell phone?

0.29,1000(0.29)(0.71)​​

An experiment will be conducted to test the effectiveness of a weight-loss supplement. Volunteers will be randomly assigned to take either the supplement or a placebo for 90 days, with 12 volunteers in each group. The subjects will not know which treatment they receive. At the end of the experiment, researchers plan to calculate the mean weight loss for each of the two groups and to construct a two-sample t-confidence interval for the difference of the two treatment means. Which of the following assumptions is necessary for the confidence interval to be valid?

The distributions of weight loss of the two treatments are approximately normally distributed.

The marketing director for an ice cream company investigated whether there was a difference in preference for two new ice cream flavors—cotton candy and mango. Each participant from a large group of people was randomly assigned to taste one of the two flavors. After tasting, each person rated the flavor on a numerical scale from 1 to 5, where 1 represented strongly dislike and 5 represented strongly like. A two-sample t-interval for a difference between means (cotton candy minus mango) was constructed. Based on the interval, there was convincing statistical evidence of a difference in population mean flavor ratings, with mango having the greater sample mean rating. Which of the following could be the constructed interval?

( -2.1, -1.3 )

Carly commutes to work, and her commute time is dependent on the weather. When the weather is good, the distribution of her commute times is approximately normal with mean 20 minutes and standard deviation 2 minutes. When the weather is not good, the distribution of her commute times is approximately normal with mean 30 minutes and standard deviation 4 minutes. Suppose the probability that the weather will be good tomorrow is 0.9. Which of the following is closest to the probability that Carly's commute time tomorrow will be greater than 25 minutes?

0.0950

The distribution of time needed to complete a certain programming task is approximately normal, with mean 47 minutes and standard deviation 6 minutes. Which of the following is closest to the probability that a randomly chosen task will take less than 34 minutes or more than 60 minutes to complete?

0.0303

Researchers on car safety studied driver reaction time and cell phone use while driving. Participants in the study talked on either a hands-free phone or a handheld phone while driving in a car simulator. A two-sample t-test for a difference in means was conducted to investigate whether the mean driver reaction time between the two groups of participants was different. All conditions for inference were met, and the test produced a test statistic of t = -2.763 and a p-value of 0.03. Which of the following is a correct interpretation of the p-value?

Assuming that the mean reaction times for hands-free and handheld phones are equal, the probability of obtaining a test statistic greater than 2.763 or less than -2.763 is 0.03.

A two-sample t-test for a difference in means will be conducted to investigate mean gasoline prices in two states. From each state, 45 gasoline stations will be selected at random. On the same day, the price of regular gasoline will be recorded for each selected station and the sample mean price for each state will be calculated. Have all conditions for inference been met?

Yes, all conditions have been met.

According to government data, 22 percent of children in the United States under the age of 6 years live in households with incomes that are classified at a particular income level. A simple random sample of 300 children in the United States under the age of 6 years was selected for a study of learning in early childhood. If the government data are correct, which of the following best approximates the probability that at least 27 percent of the children in the sample live in households that are classified at the particular income level? (Note: z represents a standard normal random variable.)

P(z>300(0.22)(0.78)​​0.27−0.22​)

From a random sample of 185 children from school G, 108 indicated they wanted to study science in college. From a different random sample of 165 children from school H, 92 indicated they wanted to study science in college. Assuming all conditions for inference are met, which of the following is closest to the standard error for a confidence interval for the difference in population proportions between the two schools of children who want to study science in college?

two parentheses, over a divider, plus another of the same, square rooted

Based on records kept at a gas station, the distribution of gallons of gas purchased by customers is skewed to the right with mean 10 gallons and standard deviation 4 gallons. A random sample of 64 customer receipts was selected, and the sample mean number of gallons was recorded. Suppose the process of selecting a random sample of 64 receipts and recording the sample mean number of gallons was repeated for a total of 100 samples. Which of the following is the best description of a dotplot created from the 100 sample means?

The dotplot is approximately normal with mean 10 gallons and standard deviation 0.5 gallon.

A state educational agency was concerned that the salaries of public school teachers in one region of the state,region A, were higher than the salaries in another region of the state, region B. The agency took two independent random samples of salaries of public school teachers, one from region A and one from region B. The data are summarized in the table below. Assuming all conditions for inference are met, do the data provide convincing statistical evidence that the salaries of public school teachers in region A are, on average, greater than the salaries of public school teachers in region B?

Yes, there is evidence at the significance level of α = 0.05 but not at α = 0.01.

Two non-profit organizations, L and M, accept donations from people. In a certain month, 140 people donated to organization L, with an average donation amount of x‾L=$113,xL​=$113, and 42 people donated to organization M, with an average donation amount of x‾M=$390.xM​=$390. What is the correct unit of measure for the mean of the sampling distribution of x‾L−x‾M?xL​−xM​?

Dollars

A fitness center piloted two new programs to help people reduce stress levels and maintain a healthy lifestyle. After one month, 112 of the 125 people who volunteered for a program in mindfulness reported a reduction in stress levels, and 110 of the 135 people who volunteered for a yoga program reported a reduction in stress levels. The fitness center wants to investigate whether there is a significant difference between the proportions of all people in the two programs who would report reductions in stress levels. Have the conditions for inference been met?

No, because the samples were not selected or assigned using a random method.

The mean and standard deviation of the sample data collected on continuous variable x are -0.25 and 0.03, respectively. The following table shows the relative frequencies of the data in the given intervals. IntervalRelative Frequency−0.34≤x≤−0.31−0.34≤x≤−0.310.02−0.31≤x≤−0.28−0.31≤x≤−0.280.15−0.28≤x≤−0.25−0.28≤x≤−0.250.33−0.25≤x≤−0.22−0.25≤x≤−0.220.36−0.22≤x≤−0.19−0.22≤x≤−0.190.11−0.19≤x≤−0.16−0.19≤x≤−0.160.03 Based on the table, do the data support the use of a normal model to approximate population characteristics?

Yes, because the distribution of relative frequencies is very close to the empirical rule for normal models.

A manufacturer claims its Brand A battery lasts longer than its competitor's Brand B battery. Nine batteries of each brand are tested independently, and the hours of battery life are shown in the table below. Provided that the assumptions for inference are met, which of the following tests should be conducted to determine if Brand A batteries do, in fact, last longer than Brand B batteries?

A one-sided, two-sample t-test

Researchers will conduct a study of the television-viewing habits of children. They will select a simple random sample of children and record the number of hours of television the children watch per week. The researchers will report the sample mean as a point estimate for the population mean. Which of the following statements is correct for the sample mean as a point estimator? A sample of size 25 will produce less variability of the estimator than a sample of size 50.

A sample of size 25 will produce more variability of the estimator than a sample of size 50.

There were 5,317 previously owned homes sold in a western city in the year 2000. The distribution of the sales prices of these homes was strongly right-skewed, with a mean of $206,274 and a standard deviation of $37,881. If all possible simple random samples of size 100 are drawn from this population and the mean is computed for each of these samples, which of the following describes the sampling distribution of the sample mean?

Approximately normal with mean $206,274 and standard deviation $3,788

On any given day, the proportion of workers at a factory who are more than 5 minutes late to work is 0.11. A random sample of 20 workers will be selected. Which of the following is the best interpretation of the mean of the sampling distribution of the sample proportion of workers in the sample who are more than 5 minutes late to work for samples of size 20 ?

For all samples of size 20, the mean of all possible sample proportions is equal to 0.11.

Two siblings, Alice and Sean, are both convinced that they are faster than the other at solving a puzzle cube. They recorded the length of time it took them to solve the cube 18 times each during a one-month period. Then each calculated the mean amount of time and standard deviation, in minutes, for their times. Let μAμA​ equal the mean time it took Alice to solve the puzzle cube and μSμS​equal the mean time it took Sean. Which of the following are the appropriate null and alternative hypotheses to test for a difference in time for the siblings to solve the cube?

Ho​:μA​−μS​=0 Ha:μA−μS≠0Ha​:μA​−μS​​=0 yes it's the one with the funny not equal sign

Zoologists studying two populations of tigers conducted a two-sample t-test for the difference in means to investigate whether the tigers in population X weigh more, on average, than the tigers in population Y. Two independent random samples were taken, and the difference between the sample means was calculated. All conditions for inference were met, and the test produced a p-value of 0.02. Which of the following is a correct interpretation of the p-value?

Assuming that the mean weights for populations X and Y are equal, the probability of observing a difference as great or greater than the sample difference is 0.02.

Researchers investigated whether a new process for producing yarn could reduce the mean amount of volatile organic compounds (VOCs) emitted by carpet. From random samples of carpets, the researchers found the mean reduction of VOCs emitted by carpets made with yarn produced by the new process compared with that of carpets made with yarn produced by the traditional process was 13 parts per million (ppm). All conditions for inference were met, and the p-value for the appropriate hypothesis test was 0.095. Which of the following statements is the best interpretation of the p-value?

If the null hypothesis is true, the probability of observing a mean reduction of at least 13 ppm is 0.095.

At a large university, the division of computing services surveyed a random sample of 45 biology majors and 55 business majors from populations of over 1,000 biology and 1,000 business majors. The sampled students were asked how many hours they spend per week using any university computer lab. Let x‾1x1​ represent the average hours per week spent in any university computer lab by the 45 biology majors, and let x‾2x2​ represent the average hours per week spent in any university computer lab by the 55 business majors. Which of the following is the best explanation for why the sampling distribution of x‾1−x‾2x1​−x2​ can be modeled with a normal distribution?

The sample sizes are both sufficiently large.

A group of 80 people who had been diagnosed as prediabetic because of high blood glucose levels volunteered to participate in a study designed to investigate the use of cinnamon to reduce blood glucose to a normal level. Of the 80 people, 40 were randomly assigned to take a cinnamon tablet each day and the other 40 were assigned to take a placebo each day. The people did not know which tablet they were taking. Their blood glucose levels were measured at the end of one month. The results showed that 14 people in the cinnamon group and 10 people in the placebo group had normal blood glucose levels. For people similar to those in the study, do the data provide convincing statistical evidence that the proportion who would be classified as normal after one month of taking cinnamon is greater than the proportion who would be classified as normal after one month of not taking cinnamon?

There is not convincing statistical evidence at any reasonable significance level.

A study will be conducted to investigate whether there is a difference in mean tail lengths between two populations of snow leopards. Random samples of leopards will be selected from both populations, and the mean sample tail length will be calculated for each sample. Which of the following is the appropriate test for the study?

A two-sample t-test for a difference between population means

A political scientist claims that negative advertising on television affects younger voters more than it affects older voters. To test this claim, the scientist obtained data from two random samples of voters categorized into two age-groups, older and younger. The null hypothesis was that there was no difference in the proportions of voters in the two age-groups who would be affected by negative ads. The alternative hypothesis was that the proportion of younger voters affected would be greater than the proportion of older voters affected. Assuming all conditions for inference were met, the scientist conducted the test at a significance level of α=0.05α=0.05. The resulting p-value was 0.206. Which of the following is the correct decision for the test?

The p-value is greater than \(\alpha)\, and the null hypothesis is not rejected. There is not convincing evidence to support the claim that younger voters are more affected by negative ads than are older voters.

City officials estimate that 46 percent of all city residents are in favor of building a new city park. A random sample of 150 city residents will be selected. Suppose that 51 percent of the sample are in favor of building a new city park. Which of the following is true about the sampling distribution of the sample proportion for samples of size 150 ?

The distribution is approximately normal, and the mean is 0.46.

Researchers investigated whether there is a difference between two headache medications, R and S. Researchers measured the mean times required to obtain relief from a headache for patients taking one of the medications. From a random sample of 75 people with chronic headaches, 38 were randomly assigned to medication R and the remaining 37 were assigned to medication S. The time, in minutes, until each person experienced relief from a headache was recorded. The sample mean times were calculated for each medication. Have the conditions been met for inference with a confidence interval for the difference in population means?

Yes, all conditions have been met.

To determine whether employees at Site X have higher salaries, on average, than employees at Site Y of the same company do, independent random samples of salaries were obtained for the two groups. The data are summarized below. Based on the data, which of the following statements is true?

At the 5% significance level, employees at Site X have a significantly higher mean salary than employees at Site Y do.

A statistics student wants to compare the mean times needed to access flight information for two major airlines. Twenty randomly selected students accessed one airline's Web site, and the time required to locate the flight information using the Web site had a mean of 2.5 minutes and a standard deviation of 0.8 minute. Twenty different randomly selected students accessed the other airline's Web site, and the time required to locate the flight information using the Web site had a mean of 2.1 minutes and a standard deviation of 1.1 minutes. Assuming that the conditions for inference are met, which of the following statements about the p-value obtained from the data and the conclusion of the significance test is true?

The p-value is greater than 0.10; therefore, there is no significant difference in mean search times on the two Web sites.

Researchers are studying two populations of wild horses living in the western regions of a country. In a random sample of 32 horses taken from the first population, the mean age of the sample was 21 years. In a random sample of 41 horses from the second population, the mean age of the sample was 19 years. Is the sampling distribution of the difference in sample mean ages approximately normal?

Yes, because the sample sizes are both greater than 30.