Statistics Test 3 Guide

Pictured below (in scrambled order) are three histograms. One of them represents a population distribution. The other two are sampling distributions of x-bar: one for sample size n = 5 and one for sample size n = 30. Based on the histograms, what is the most likely value of the population mean?

3.0

Suppose that battery life is a normal random variable with μ = 8 and σ = 1.2. Using the Standard Deviation Rule, what is the probability that a randomly chosen battery will last between 6.8 and 9.2 hours?

31+-(3*6)=13,49

In a study of the effects of acid rain, a random sample of 100 trees from a particular forest is examined. Forty percent of these show some signs of damage. Which of the following statements is correct?

40% is a statistic

Consider sampling heights from the population of all female college soccer players in the United States. Assume the mean height of female college soccer players in the United States is μ = 66 inches and the standard deviation is σ=3.8 inches. Suppose we randomly sample 100 values from this population and compute the mean, then repeat this sampling process 5000 times and record all the means we get. Which of the following is the best approximation for the mean of our 5000 sample means?

66

Pictured below (in scrambled order) are three histograms. One of them represents a population distribution. The other two are sampling distributions of x-bar: one for sample size n = 5 and one for sample size n = 40. Based on the histograms, what is the most likely value of the population mean?

8

The score of golfers for a particular course follows a normal distribution that has a mean of 73 and a standard deviation of 3. Suppose a golfer played the course today. Find the probability that her score is at least 74.

=1-NORM.DIST(74,73,3,1) =0.3694

In the months leading up to an election, news organizations conduct many surveys to help predict the results of the election. Often news organizations will increase the sample size in the last few weeks before the election. Which of the following is the primary reason they increase the sample size?

A larger sample size gives a narrower confidence interval.

When the population is not normally distributed, the sampling distribution of the mean approximates which of the following?

A normal distribution given a large enough sample

A social scientist wishes to conduct a survey. She plans to ask a yes/no question to a random sample from the U.S. adult population. One proposal is to select 100 people; another proposal is to select 900 people. If the study were conducted repeatedly (selecting different samples of people each time), which one of the following would be true regarding the resulting sample proportions of "yes" responses?

Different sample proportions, ^p, could result each time, but for either sample size, they would be centered (have their mean) at the true population proportion, p.

Marijuana legalization: In a Public Policy Institute of California (PPIC) poll, 53% of 1,706 California adult residents surveyed say that marijuana should be legal. Based on the results, the 95% confidence interval is (0.506, 0.554). Which of the following is an appropriate interpretation of this confidence interval?

We are 95% confident that between 50.6% and 55.4% of California residents say that marijuana should be legal.

Genetically modified foods: In a Pew Research Center report from January 2015, 37% of American adults say that genetically modified (GM) foods are generally safe to eat. The margin of error for the 95% confidence interval is 3.1%.

We are 95% confident that the population proportion is within 3.1% of the sample proportion of 37%.

Suppose that the distribution for total amounts spent by students vacationing for a week in Florida is normally distributed with a mean of 650 and a standard deviation of 120. Suppose you take a simple random sample (SRS) of 35 students from this distribution. What is the probability that a SRS of 35 students will spend an average of between 600 and 700dollars?

sd-s=120/sqrt35 600<x<700=normdist=0.98630

A machine is designed to fill 16-ounce bottles of shampoo. When the machine is working properly, the amount poured into the bottles follows a normal distribution with mean 16.05 ounces with a standard deviation of .2 ounces. If four bottles are randomly selected each hour and the number of ounces in each bottle is measured, then 95% of the means calculated should occur in what interval?

sd-xbar = .2/sqrt(4)=.1 lower= 16.05-2*.1 upper=16.05+2*.1

A factory produces plate glass with a mean thickness of 4mm and a standard deviation of 1.1mm. A simple random sample of 100 sheets of glass is to be measured, and the mean thickness of the 100 sheets is to be computed. What is the probability that the average thickness of the 100 sheets is less than 3.74 mm?

sd-xbar = 1.1/sqrt(100)=.11 x<3.74=normdist(3.74,4,.11,1)=.00905

The run times of a marathon runner are approximately normally distributed. The z-score for his run this week is - 2 . Which one of the following statements is a correct interpretation of his z-score? This week his time was 2 ___ ___ than his ___

standard deviations, lower, average time

The number of hours a light bulb burns before failing varies from bulb to bulb. The distribution of burnout times is strongly skewed to the right. The Central Limit Theorem says that

the average burnout time of a large number of bulbs has a distribution that is close to Normal.

The SAT is the most widely used college admission exam. (Most community colleges do not require students to take this exam.) The mean SAT math score varies by state and by year, so the value of µ depends on the state and the year. But let's assume that the shape and spread of the distribution of individual SAT math scores in each state is the same each year. More specifically, assume that individual SAT math scores consistently have a normal distribution with a standard deviation of 100. An educational researcher wants to estimate the mean SAT math score (μ) for his state this year. The researcher chooses a random sample of 673 exams in his state. The sample mean for the test is 484. Find the 95% confidence interval to estimate the mean SAT math score in this state for this year. (Note: The critical z-value to use, zc, is: 1.960.) (476.445 , 491.555)

xbar but use given zc

Suppose the time to complete a 200-meter backstroke swim for female competitive swimmers is normally distributed with a mean μ = 141 seconds and a standard deviation σ = 7 seconds. What is the completion time for the 200-meter backstroke for a female with a z-score of −1.64? (Round answer to 1 decimal place.)

141+(7*-1.64)= 129.5

"In 1973 the Roe versus Wade decision established a woman's constitutional right to an abortion, at least in the first three months of pregnancy. Would you like to see the Supreme Court completely overturn its Roe versus Wade decision, or not?" The results were: Yes—30%, No—62%, Unsure—8% Which of the following is true about this scenario?

30%, 62%, and 8% are all statistics.

In June 2015, Gallup conducted a poll of a random sample of 15394 adults to determine the well-being of people living in the United States. One question asked, "Did you exercise at least 30 minutes for 3 or more days in the past week?" In the survey, 55.3% of males and 44.7% of females responded yes to this question. Which of the following is true about this scenario?

55.3% and 44.7% are both statistics.

The weights of cockroaches living in a typical college dormitory are approximately normally distributed with a mean of 80 grams and a standard deviation of 4 grams. The percentage of cockroaches weighing between 77 grams and 83 grams is about

=NORM.DIST(83,80,4,1)-NORM.DIST(77,80,4,1) = 55%

The lifetime in miles for a certain brand of tire is normally distributed with a mean of 22,000 miles and a standard deviation of 3,100 miles The tire manufacturer wants to offer a money-back guarantee so that no more than 3%of tires will qualify for a refund. What is the minimum number of miles the manufacturer should guarantee that the tires will last?

=NORM.INV(.03,22000,3100) = 16,170 miles

Suppose the scores on an exam are normally distributed with a mean μ = 75 points, and standard deviation σ = 8 points. Suppose that the top 4% of the exams will be given an A+. In order to be given an A+, an exam must earn at least what score? Report your answer in whole numbers.

=NORM.INV(.96,75,8)= 89

Which of the following best describes the sampling distribution of a statistic?

A distribution of a single statistic from repeated random samples of the same size, from the same population.

Two bottling plants package a certain type of sports drink. Suppose the mean volume of all of this type of sports drinks is 20 fluid ounces. Bottling plant A bottles approximately 42084 sports drinks per day. Bottling plant B bottles approximately 172283 sports drinks per day. On a particular day, which bottling plant is less likely to record a mean volume of 21 fluid ounces for the day?

Bottling plant B (with 172283 sports drinks per day), because the daily mean will be closer to 20 fluid ounces with more sports drinks in the sample.

Suppose we take a poll (random sample) of 3974 students classified as Juniors and find that 3162 of them believe that they will find a job immediately after graduation. What is the 99% confidence interval for the proportion of GSU Juniors who believe that they will, immediately, be employed after graduation.

phat, zc method

Suppose two students from Georgia State University, working as interns for the American National Elections Studies agency (ANES), are both assigned to survey a random sample of registered voters and ask whether or not they will vote for a certain candidate. The first intern plans to select 500 voters and the second intern plans to select 1500 voters. If each intern conducted the study repeatedly (selecting different samples of people each time ... but using the same sample size), which one of the following would be true regarding the resulting sample proportion, ^p, of "yes" responses for each intern?

Different sample proportions, ^p, could result for each intern, but for either sample size, they would be centered (have their mean) at the true population proportion, p.

Students in a statistics class conduct a survey to estimate the mean number of units students at their college are enrolled in. The students took a random sample of 50 students from their college. The students calculated a 90% confidence interval to estimate the mean number of units students at their college are enrolled in. The confidence interval was too wide to provide a precise estimate. The students are strategizing about how to produce a narrower confidence interval. True or false? The students could produce a narrower confidence interval by increasing the confidence level to 95%.

False

Suppose that the mean birth weight of human babies is 3130g. Hospital A records an average of 52 births a day. Hospital B records an average of 9 births a day. On a particular day, which hospital is less likely to record an average birth weight of 3430g or more?

Hospital A (with 52 births a day), because the mean will typically be closer to 3130 g with more births.

The following displays two normal distributions. Which of the following are true? I. The mean of A is less than the mean of B. II. The standard deviation of A is less than B. III. The area under the curve of A is less than B.

I and II only

The average 30- to 39-year old man is 69.5 inches tall, with a standard deviation of 2.7 inches, while the average 30- to 39-year old woman is 64.2 inches tall, with a standard deviation of 3.2 inches. Who is relatively taller based on their comparison to their gender, LeBron James at 81 inches or Candace Parker at 76 inches?

LeBron is relatively taller because he has a larger z-score. z-score =(A-mu)/sd

Students in a statistics class take a random sample of 100 students at their college and record how many units each student is enrolled in. The students compute a 90% confidence interval for the mean number of units for students at their college and get (11.93, 12.47). Next, the students calculate a 95% confidence interval. As the confidence level increases, which of the following will happen to the interval width?

Increase

What do you expect for the variability (spread) of a sampling distribution with sample size n = 6?

Less variability than the population (a narrower distribution).

Roughly 8% of white males have some form of colorblindness, while the incidence among white females is only 1%. A random sample of 20 white males and 40 white females was chosen. Let X be the number of males (out of the 20) who are colorblind. Let Y be the number of females (out of the 40) who are colorblind. Let Z be the total number of colorblind individuals in the sample (males and females together). Which of the following is true regarding the random variables X and Y?

Neither X nor Y can be well approximated by a normal random variable. conditions: p ≥ 10 and n (1 - p) ≥ 10

A researcher is estimating the mean income of residents in a large city. The income variable is usually skewed to the right. She collects a random sample of 25 people. The resulting 95% confidence interval is ($26700, $35400). Which one of the following conclusions is valid?

No conclusion can be drawn.

Two different groups in a statistics class are conducting a survey to estimate the mean number of units students at their college are enrolled in. The population mean and standard deviation are unknown. The sample from the first group survey has 49 data values. The sample from the second group survey has 81 data values. For each sample, the groups construct a 90% confidence interval to estimate the population mean. Which confidence interval will have greater precision (smaller width) for estimating the population mean?

The confidence interval based on the sample of 81 data values will be more precise.

Which of the following statements about the sampling distribution of the sample mean, x-bar, is true? Check all that apply.

The distribution is normal regardless of the shape of the population distribution, as long as the sample size, n, is large enough. The distribution is normal regardless of the sample size, as long as the population distribution is normal. The distribution's mean is the same as the population mean. The distribution's standard deviation is smaller than the population standard deviation

Suppose we take repeated random samples of size 20 from a population with a mean of 60 and a standard deviation of 8. Which of the following statements is true about the sampling distribution of the sample mean (x̄)? Check all that apply.

The distribution will be normal as long as the population distribution is normal. The distribution's mean is the same as the population mean 60.

In April and May of 2011, the Pew Research Center surveyed cell phone users about voice calls and text messaging. They found that 55% of those who send 51 or more text messages per day prefer to be contacted by text message rather than by a voice call. The margin of error for this sample was 5.7%. What does this margin of error tell you about the results of the survey?

The population proportion is most likely within 5.7% of the sample proportion from a randomly selected sample.

Same-sex marriage: In a recent ABC News/Washington Post poll, 1328 adults nationwide answered the question, "Overall, do you support or oppose allowing gays and lesbians to marry legally?" Of the respondents, 455 support same-sex marriage. What is the 95% confidence interval for the proportion of all American adults who support same-sex marriage?

phat, zc method

Parking survey: For a class assignment, a group of statistics students set up a table near the student parking lot. They asked students who passed by to complete a quick survey about whether they support the building of a multi-level parking structure that would add 425 new spaces at the college. They used the information from the survey to calculate the 95% confidence interval: (0.53, 0.72). To which population does the confidence interval apply?

The results do not apply to any population because this was a convenience sample.

An interactive poll on the front page of the CNN website in October 2011 asked if readers would consider voting for Herman Cain, who at the time, was a Republican presidential candidate. A statistics student used the information from the poll to calculate the 95% confidence interval. He got (0.53,0.59). He also conducted a hypothesis test. He found very strong evidence that more than half of voters would consider voting for Herman Cain.

The results do not apply to any population because this was a voluntary response sample

The administration conducted a survey to determine the proportion of students who ride a bike to campus. Of the 129 students surveyed 6 ride a bike to campus. Which of the following is a reason the administration should not calculate a confidence interval to estimate the proportion of all students who ride a bike to campus? Check all that apply.

The sample needs to be random but we don't know if it is. The actual count of bike riders is too small. n*^p is not greater than 10.

Smoking habits: A group of statistics students conducted a survey about smoking habits on campus to determine the proportion of students who believe that smoking hookah (a water pipe) is less harmful than smoking cigarettes. Of the 50 students surveyed, 45 think that smoking hookah is less harmful than smoking cigarettes. Which of the following is a reason that this group of students should not calculate a confidence interval for the proportion of all students who believe that smoking hookah is less harmful than smoking cigarettes?

The sample needs to be random but we don't know if it is. The actual count of students who do not believe that smoking hookah is less harmful than smoking cigarettes is too small. n(1−^p) is not greater than 10

The administration at GSU wants to estimate the number of parking spaces they will need next year. They survey 80 students; 75 of the students in the sample drive to campus by themselves each day. Which of the following is a reason the administration should not calculate a confidence interval for the proportion of all students who drive to campus?

The sample needs to be random but we don't know if it is. The actual count of those who do not drive to campus is too small. n(1−^p) is not greater than 10.

Concert marketing: GSU's Rialto Center for the Performing Arts wanted to investigate why ticket sales for the upcoming season significantly decreased from last year's sales. The marketing staff collected data from a survey of community residents. Out of the 110 people surveyed, only 7 received the concert brochure in the mail. Which of the following is a reason that the marketing staff should not calculate a confidence interval for the proportion of all community residents who received the concert brochure by mail? Check all that apply.

The sample needs to be random, but we don't know if it is. The actual count of community residents who received the concert brochure by mail is too small. n^p is not greater than 10

It plans to ask a yes/no question to determine if those surveyed plan to vote for a certain candidate. One proposal is to randomly select 400 people and another proposal is to randomly select 1600 people. Which of the following is true regarding the sample proportion ^p of "yes" responses?

The sample proportion from sample of 1,600 is more likely to be close to the true population proportion, p.

A social scientist wishes to conduct a survey. She plans to ask a yes/no question to a random sample from the U.S. adult population. One proposal is to select 100 people; another proposal is to select 900 people. Which of the following is true regarding the sample proportion, ^p, of "yes" responses?

The sample proportion from the sample of 900 is more likely to be close to the true population proportion, p.

A doctor is measuring the mean systolic blood pressure of female students at a large college. Systolic blood pressure is known to have a skewed distribution. The doctor collects systolic blood pressure measurements from random sample of 28 female students. The resulting 90% confidence interval is (100.4, 159.6). Units of systolic blood pressure are mmHg. Which one of the following conclusions is valid?

The sampling distribution of means will probably not follow a normal distribution, so we cannot draw a conclusion.

Consider random samples selected from the population of all female college soccer players in the United States. Assume the mean height of female college soccer players in the United States is 66 inches and the standard deviation is 3.5 inches. Which do you expect to have less variability (spread): a sampling distribution with sample size n = 118 or a sampling distribution with sample size n = 17?

The sampling distribution with sample size n = 118 will have less variability.

A social scientist wishes to conduct a survey. She plans to ask a yes/no question to a random sample from the U.S. adult population. One proposal is to select 100 people; another proposal is to select 900 people. Which one of the following is true regarding the standard deviation of the sampling distribution of the sample proportion, ^p, of "yes" responses?

The standard deviation of the sampling distribution will be 3 times larger with sample size 100.

A researcher took a random sample of 100 students from a large university. She computed a 95% confidence interval to estimate the average weight of the students at this university. The confidence interval was too wide to provide a precise estimate. True or false? The researcher could produce a narrower confidence interval by increasing the sample size to 150.

True

Students in a statistics class conduct a survey to estimate the mean number of units students at their college are enrolled in. The students took a random sample of 50 students from their college. The students calculated a 90% confidence interval to estimate the mean number of units students at their college are enrolled in. The confidence interval was too wide to provide a precise estimate. The students are strategizing about how to produce a narrower confidence interval. True or false? The students could produce a narrower confidence interval by increasing the sample size to 100.

True

Race relations: A New York Times/CBS poll surveyed 1,027 adults nationwide about race relations in the United States. Of the sample, 61% responded that race relations in this country are generally bad. The 95% confidence interval is (0.58, 0.64). Which of the following is an appropriate interpretation of the 95% confidence interval?

We are 95% confident that the proportion of all Americans who say that race relations in this country are generally bad is between 58% and 64%.

In April and May of 2011, the Pew Research Center surveyed cell phone users about voice calls and text messaging. They surveyed a random sample of 1914 cell phone users. 75% of the sample use text messaging. The 95% confidence interval is (73.1%, 76.9%). Which of the following is an appropriate interpretation of the 95% confidence interval?

We can be 95% confident that the proportion of all cell phone users who use text messaging is between 73.1% and 76.9%.

Confidence interval precision: We know that narrower confidence intervals give us a more precise estimate of the true population proportion. Which of the following could we do to produce higher precision in our estimates of the population proportion?

We can select a lower confidence level and increase the sample size.

In which of the following scenarios would the distribution of the sample mean x-bar be normally distributed? Check all that apply.

We take repeated random samples of size 15 from a population that is normally distributed. We take repeated random samples of size 50 from a population of unknown shape.

The Environmental Protection agency requires that the exhaust of each model of motor vehicle be tested for the level of several pollutants. The level of oxides of nitrogen (NOX) in the exhaust of one light truck model was found to vary among individually trucks according to a Normal distribution with mean 1.45 grams per mile driven and standard deviation 0.40 grams per mile. (a) What is the 90th percentile for NOX exhaust, rounded to four decimal places? (b) Find the interquartile range for the distribution of NOX levels in the exhaust of trucks rounded to four decimal places.

a)x if z=90% = 1.96262 =NORM.INV(0.29,1.45,.4) b)IQR=Q3-Q1 = .5396

The distribution of scores on a recent test closely followed a Normal Distribution with a mean of 22 points and a standard deviation of 2 points. For this question, DO NOT apply the standard deviation rule. (a) What proportion of the students scored at least 24 points on this test, rounded to five decimal places? (b) What is the 29 percentile of the distribution of test scores, rounded to three decimal places?

a)z>=24=0.15866 1-NORM.DIST(24,22,2,1) b)x if z=29% = 20.893 =NORM.INV(.29,22,2)

Suppose we take a random sample of 41 state college students. Then we measure the length of their right foot in centimeters. We compute a 95% confidence interval for the mean foot length for students at this college. We get (21.71, 25.09). Suppose that we now compute a 90% confidence interval. As confidence level decreases, the interval width

decreases

A researcher took a random sample of 100 students from a large university. She computed a 95% confidence interval to estimate the average weight of the students at this university. The confidence interval was too wide to provide a precise estimate. True or false? The researcher could produce a narrower confidence interval by increasing the confidence level to 99%.

false

The package of a particular brand of rubber band says that the bands can hold a weight of 7 lbs. Suppose that we suspect this might be an overstatement of the breaking weight. So we decide to take a random sample of 36 of these rubber bands and record the weight required to break each of them. The mean breaking weight of our sample of 36 rubber bands is 6.6 lbs. Assume that the standard deviation of the breaking weight for the entire population of these rubber bands is 2 lbs. True or false? Finding a random sample with a mean this low in a population with mean 7 and standard deviation 2 is very unlikely.

false mean-s= 7 sd-s= 2/sqrt(36) xbar>6.6= 1-normdist(6.6,7,sd-s,1)= 0.885

Suppose that 20% of the residents in a certain state support an increase in the property tax. An opinion poll will randomly sample 400 state residents and will then compute the proportion in the sample that support a property tax increase. How likely is the resulting sample proportion, ^p, to be within 0.04 of the true proportion, p (i.e., between 0.16 and 0.24)?

mean= 0.2 sd= sqrt(0.2*((1-0.2)/400)=0.02 0.16<x<0.24 NORMDIST=95%

A Gallup survey of 2322 adults (at least 18 years old) in the U.S. found that 408 of them have donated blood in the past two years. Construct a 90% confidence interval for the population proportion of adults in the U.S. who have donated blood in the past two years. Round your answer to three decimal places.

phat, zc method

In 2015 as part of the General Social Survey, 1141 randomly selected American adults responded to this question: "Some countries are doing more to protect the environment than other countries. In general, do you think that America is doing more than enough, about the right amount, or too little?" Of the respondents, 466 replied that America is doing about the right amount. What is the 95% confidence interval for the proportion of all American adults who feel that America is doing about the right amount to protect the environment.

phat, zc method

A survey asks a random sample of 1500 adults in Ohio if they support an increase in the state sales tax from 5% to 6%, with the additional revenue going to education. Let ^p denote the proportion in the sample who say they support the increase. Suppose that 12% of all adults in Ohio support the increase. The standard deviation of the sampling distribution is

sd=SQRT(.12*((1-.12)/1500)=.0084

In a study of the effects of acid rain, a random sample of 100 trees from a particular forest is examined. Forty percent of these show some signs of damage. Which of the following statements is correct?

the sampling distribution of the proportion of damaged trees is approximately normal

Is smoking during pregnancy associated with premature births? To investigate this question, researchers selected a random sample of 142 pregnant women who were smokers. The average pregnancy length for this sample of smokers was 265 days. From a large body of research, it is known that length of human pregnancy has a standard deviation of 16 days. The researchers assume that smoking does not affect the variability in pregnancy length. Find the 90% confidence interval to estimate the length of pregnancy for women who smoke. (Note: The critical z-value to use, zc, is: 1.645) (262.791 , 267.209)

xbar but use given zc

A group of engineers developed a new design for a steel cable. They need to estimate the amount of weight the cable can hold. The weight limit will be reported on cable packaging. The engineers take a random sample of 40 cables and apply weights to each of them until they break. The 40 cables have a mean breaking weight of 775.3 lb. The standard deviation of the breaking weight for the sample is 14.9 lb. Find the 90% confidence interval to estimate the mean breaking weight for this type cable. (771.33 , 779.27)

xbar, tc method

A group of engineers developed a new design for a steel cable. They need to estimate the amount of weight the cable can hold. The weight limit will be reported on cable packaging. The engineers take a random sample of 47 cables and apply weights to each of them until they break. The 47 cables have a mean breaking weight of 770.1 lb. The standard deviation of the breaking weight for the sample is 16.2 lb. Find the 95% confidence interval to estimate the mean breaking weight for this type cable. (765.34 ,774.86)

xbar, tc method

Students in a statistics class are conducting a survey to estimate the mean number of units students at their college are enrolled in. The students collect a random sample of 47 students. The mean of the sample is 12.3 units. The sample has a standard deviation of 1.9 units. What is the 95% confidence interval for the average number of units that students in their college are enrolled in? Assume that the distribution of individual student enrollment units at this college is approximately normal. (11.74, 12.86)

xbar, tc method