STAT Modules 17-19

If n = 190 and X = 152, construct a 90% confidence interval for the population proportion, p. Give your answers to three decimals

0.7523 < p < 0.8477

We survey a random sample of American River College students and ask if they drink coffee on a regular basis. The 90% confidence interval for the proportion of all American River College students who drink coffee on a regular basis is (0.262, 0.438). What will be true about the 95% confidence interval for these data?

The 95% confidence interval is wider than the 90% confidence interval.

In September 2011, Gallup surveyed 1,004 American adults and asked them whether they blamed Barack Obama a great deal, a moderate amount, not much, or not at all for U.S. economic problems. The results showed that 53% of respondents blamed Barack Obama a great deal or a moderate amount. Calculate the 99% confidence interval for the proportion of all American adults who blame Barack Obama a great deal or a moderate amount for U.S. economic problems.

(0.489, 0.571) The estimated standard error is 0.0158 so the confidence interval is given by: 0.53 ± 2.576 (0.0158). Note that your answer may vary slightly due to rounding.

Research question: Do the majority of U.S. adults oppose the death penalty? Based on this research question, here are the hypotheses: H0: 50% of U.S. adults oppose the death penalty. Ha: A majority (more than 50% of adults) oppose the death penalty. Suppose a survey was conducted in which a random sample of 1,100 U.S. adults are asked about their opinions on the death penalty, and based on the data, the researchers state, "at the 5% significance level, this survey provides strong evidence that the majority of U.S. adults oppose the death penalty. Which of the following P-values support this conclusion?

0.03 The P-value must be less than the significance level of 0.05 in order for us to conclude that the alternative hypothesis is true.

At the 1% significance level, which P-values will lead to a conclusion of "fail to reject the null hypothesis"? One or more answer options may be correct.

0.10, 0.05, 0.03 A P-value greater than the significance level of 0.01 means that the sample results are not strong enough to reject the null hypothesis. The conclusion is "fail to reject Ho."

Suppose we take a survey and use the sample proportion to calculate a confidence interval. Which level of confidence gives the confidence interval with the largest margin of error? 90%, 95%, or 99%?

99% Since the for a 99% confidence interval is the largest of the three, the margin of error will be the greatest. You can also see this by picturing the middle 99% of the sampling distribution.

For many years "working full-time" was 40 hours per week. A business researcher gathers data on the hours that corporate employees work each week. She wants to determine if corporations now require a longer work week.

Testing a claim about a single population mean. We are testing the claim that the mean number of work hours per week for all corporate employees is greater than 40 hours.

At Valencia College are male students more likely to be smokers than female students?

Testing a claim that compares two population proportions. We are testing the claim that the proportion of male Valencia students who smoke is greater than the proportion of female Valencia students who smoke.

The following two hypotheses are tested: H0: The average number of miles driven per year is 12,000. Ha: The average number of miles driven per year is less than 12,000. In a survey, 1,600 randomly selected drivers were asked the number of miles they drive yearly. Based on the results, the P-value = 0.068. Use a 0.05 (5%) significance level. Based upon the p-value, which of the following conclusions can be made?

The data do not provide significant evidence that the average number of miles driven per year is less than 12,000. Since p > 0.05, we fail to reject the null hypothesis. We could also say that we fail to accept the alternative hypothesis.

Smoking levels: According to the Centers for Disease Control and Prevention, the proportion of U.S. adults age 25 or older who smoke is .22. A researcher suspects that the rate is lower among U.S. adults 25 or older who have a bachelor's degree or higher education level. What is the alternative hypothesis in this case?

The proportion of smokers among U.S. adults 25 or older who have a bachelor's degree or higher is less than .22. Indeed, the alternative hypothesis usually represents what we want to check, or what we suspect is really going on. In this case, what the researcher suspects is going on is that the proportion of smokers among U.S. adults age 25 and older who have a bachelor's degree or higher is lower than the rate of all adults in this age group.

The administration at Pierce College conducted a survey to determine the proportion of students who ride a bike to campus. Of the 125 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-hat) is not greater than 10.

IQ levels: A study investigated whether there are differences between the mean IQ level of people who were reared by their biological parents and those who were reared by someone else. What is the alternative hypothesis in this case?

There is a difference between the mean IQ level of people who were reared by their natural parents and those who were not. Indeed, the alternative hypothesis represents what we suspect or want to check. In this case, we want to check whether there is a difference between the mean IQ levels of the two groups.

The following two hypotheses are tested: H0: The average number of miles driven per year is 12,000. Ha: The average number of miles driven per year is less than 12,000. In a survey, 1,600 randomly selected drivers were asked the number of miles they drive yearly. Based on the results, the P-value = 0.068. Use a 0.05 (5%) significance level. When would you conclude that the average number of miles driven per year is less than 12,000?

When the p-value is small (less than 0.05) We can accept Ha if there is sufficient evidence, namely that the p-value < 0.05.

We wish to estimate what percent of adult residents in a certain county are parents. Out of 600 adult residents sampled, 42 had kids. Based on this, construct a 95% confidence interval for the proportion p of adult residents who are parents in this county. a. Express your answer in tri-inequality form. Give your answers as decimals, to three places. b. Express the same answer using the point estimate and margin of error. Give your answers as decimals, to three places.

a. 0.0509 < p < 0.0904 b. p = 0.07 ± 0.02

A local newspaper claims that 67% of the county's residents support a bond measure. We conduct a phone survey and find that 85 out of the 150 people contacted support the measure.

categorical The variable is support (or not) for a bond measure. It is categorical. So this type of scenario will be analyzed using proportions.

Which of the following P-values will give the strongest evidence against H0? P-value = 0.02

0.02 A small P-value (like 0.02) indicates that the sample result is not likely to occur in random sampling from a population in which H0 is true. So a small P-value provides strong evidence against H0.

Which of the following research questions asks us to test a claim about a population proportion?

A bond measure takes a 2⁄3 majority to pass. Do more than 67% of the voters support the measure?

In 2010 as part of the General Social Survey, 1295 randomly selected American adults responded to this question: "Some countries are doing more to protect the environment than other countries are. In general, do you think that America is doing more than enough, about the right amount, or too little?" Of the respondents, 473 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?

(0.339, 0.391) We can be 95% confident that the proportion of all American adults who feel that America is doing about the right amount to protect the environment is between 0.339 and 0.391.

For many years, working full-time has meant working 40 hours per week. Nowadays, it seems that corporate employers expect their employees to work more than this amount. A researcher decides to investigate this hypothesis. H0: The average time full-time corporate employees work per week is 40 hours. Ha: The average time full-time corporate employees work per week is more than 40 hours. To substantiate his claim, the researcher randomly selects 250 corporate employees and finds that they work an average of 47 hours per week with a standard deviation of 3.2 hours. In order to assess the evidence, we need to ask:

How likely it is that in a sample of 250 we will find that the mean number of hours per week corporate employees work is as high as 47 if the true mean is 40? Indeed, in hypothesis testing, in order to assess the evidence, we need to find how likely is it to get data like those observed assuming that the null hypothesis is true.

If we are testing an alternative hypothesis of Ha: p < p0, which of the following test statistics will give the smallest P-value? Z = -2

If Z = -2, the data's p-hat is 2 standard deviations below p0. So it is very unlikely that p-hats from random sampling will be located further away from p0 than the observed data. Hence the small P-value.

Statistics students surveyed 135 students at Tallahassee Community College. From their data we are 95% confident that between 44.7% and 61.9% of all TCC students are female. The students realize that this interval contains a large margin of error. What can they do to make a narrower interval?

Survey more students. A larger sample size will mean a smaller margin of error, and thus a narrower, confidence interval.

IQ levels: A study investigated whether there are differences between the mean IQ level of people who were reared by their biological parents and those who were reared by someone else. What is the null hypothesis in this case?

There is no difference between the mean IQ levels of people who were reared by their natural parents and those who were not. Indeed the null hypothesis claims that "nothing special is going on," which, in this context, means that there is no difference between the mean IQ levels of the two groups.

Community college students survey students at their college and ask, "Have you met with a counselor to develop an educational plan?" Of the 25 randomly selected students, 17 have met with a counselor to develop an educational plan. What is the 90% confidence interval for the proportion of all students at the college that have met with a counselor to develop an educational plan? (Answers may vary slightly due to rounding. Round SE to 2 decimal places before calculating the margin of error.)

We should not calculate the 90% confidence interval because normality conditions are not met. Normality conditions are not met. The sample is random but the count of failures (those not seeing a counselor) is 8, which is less than the requisite 10.

What percentage of U.S. adults oppose resident college tuition for the children of illegal immigrants?A CBS poll conducted at the beginning of October 2011 asked "Do you favor or oppose allowing the children of illegal immigrants to attend state college at the lower tuition rate of state residents?" Of the 1,012 U.S. adults in the sample, 68% were opposed with a margin of error of 3%. To answer this research question we estimate a population proportion.

estimate proportion

On September 25, 2011 Michael Vick, the quarterback for the Philadelphia Eagles broke his non-throwing hand in a football game against the New York Giants. ESPN then posted a poll on their website. The poll asked viewers to predict which team would win the NFC East Division. The Eagles play in the NFC East Division. By 4:15 pm, 914 fans had voted. 25% of them thought the Eagles would still win the division. What can we conclude from a 95% confidence interval about the opinions of ESPN viewers?

nothing because the sample was not randomly selected Inference procedures, such as confidence intervals, depend on a random sample. If the sample is not randomly selected, you should not use statistical inference. Why? Because statistical inference is based on probability and probability is based on random events.

A local newspaper claims that the county's residents commute an average of 18 miles each way to work. We conduct a phone survey in which we ask the number of miles the respondent drives each way to work.

quantitative The variable is number of miles commuted each way to work. It is quantitative. So this type of scenario will be analyzed using means, not proportions.

A poll of students at your college asks each student to give an estimate of the number of military causalities that have occurred since the United States invaded Iraq in March of 2003.

quantitative The variable is the estimate of the number of military casualties. It is quantitative. So this type of scenario will be analyzed using means, not proportions.

For their statistics project, a group of students want to determine the proportion of LMC students who eat fast food frequently. They select a random sample of 50 students and find that 22 report eating fast food frequently (more than 3 times a week.) What is the count of success? What is the count of failures? Can the students use this data to calculate a 95% confidence interval? yes, normality conditions are met

22 A success is "eats fast food frequently." Why? A success is defined by the proportion of interest. Here the students are interested in the proportion of LMC students who eat fast food frequently. 28 A success in this situation is "eating fast food frequently." There are 22 successes and 50 - 22 = 28 failures (those "NOT eating fast food frequently.) yes, normality conditions are met The sample is randomly selected and the counts of successes and failures in the sample are both greater than 10.

Valencia College: At Valencia College, are male students more likely to be smokers than female students? What is the alternative hypothesis in this case?

The proportion of male Valencia College students who are smokers is greater than the proportion of female Valencia College students who are smokers. If male students are more likely to be smokers than female students, then the proportion of males who smoke is greater than the proportion of females who smoke.

Which of the following research questions involves testing a claim about a single population proportion?

According to The College Board, 62% of students graduating from a community college with an associate degree in 2007-2008 had no student loan debt. Has this figure increased since then? We are testing the claim that the proportion of community college students with student loan debt when graduating with an associate degree is greater than 0.62.

What does the 99% confidence level in the previous problem tell us?

Of confidence intervals with this margin of error, 99% will contain the population proportion. Since 99% of sample proportions will fall within 2.576 standard errors of p, 99% of the time the confidence interval will contain p.

In 2011 the Journal of Family Practice published a review of 13 randomized controlled studies of zinc's effect on the common cold. The review concluded that zinc supplements produced a statistically significant reduction in the duration and severity of cold symptoms. What does the term "statistically significant" mean?

The reduction in duration and severity of cold symptoms is too large to be attributed to the role of chance in random assignment. A statistically significant difference is larger than expected in random sampling, or in this case, random assignment. It is too large to attribute to chance.

Valencia College: At Valencia College, are male students more likely to be smokers than female students? What is the null hypothesis in this case?

There is no difference between the proportion of male Valencia College students who are smokers and the proportion of female Valencia College students who are smokers. The null hypothesis in this case would be that the proportion of male students who smoke and the proportion of female students who smoke are the same.

A newsgroup is interested in constructing a 95% confidence interval for the proportion of all Americans who are in favor of a new Green initiative. Of the 547 randomly selected Americans surveyed, 377 were in favor of the initiative. Round answers to 4 decimal places where possible. a. With 95% confidence the proportion of all Americans who favor the new Green initiative is between __ and __ b. If many groups of 547 randomly selected Americans were surveyed, then a different confidence interval would be produced from each group. About __ percent of these confidence intervals will contain the true population proportion of Americans who favor the Green initiative and about __ percent will not contain the true population proportion.

a. 0.6504 and 0.728 b. 95 and 5

Throughout this activity, use a 0.05 (5%) significance level. Use the following information to answer questions 1 through 3. The following two hypotheses are tested: H0: The proportion of U.S. adults who support gay marriage is roughly 50%. Ha: The proportion of U.S. adults who support gay marriage is above 50% (i.e., the majority support). Suppose a survey was conducted in which a random sample of 1,100 U.S. adults were asked about their opinions on gay marriage, and based on the data, the P-value was found to be 0.002. a. The fact that the P-value = .002 means that b. Based on the P-value you can conclude that: c. When would you conclude that the data provide enough evidence that the proportion of U.S. adults who support gay marriage is 50%?

a. There is a probability of .002 (i.e., very unlikely) to observe data like those observed if the proportion of U.S. adults who support gay marriage were 50%. Indeed, the P-value is the probability of observing data like those observed assuming that the null hypothesis, H0, is true. b. The data provide significant evidence that the majority of U.S. adults support gay marriage. Indeed, since the P-value is small (less than .05) we have enough evidence to reject H0 and accept Ha, or in other words, to conclude that the majority of U.S. adults support gay marriage. c. Never. Indeed, we never conclude that the data provide enough information to accept H0.

A common accepted tradition is that college students study 2 hours outside of class for every hour in class. This means a full-time student taking 15 units (hours of class) studies for 30 hours per week. An educator suspects this figure is different now than in the past. H0: The average time full-time college students study outside of class per week is 30 hours. Ha: The average time full-time college students study outside of class per week is not 30 hours. To substantiate her claim, the educator randomly selects 1,500 college students and finds that they study an average of 27 hours per week with a standard deviation of 1.7 hours. In order to assess the evidence, we need to determine:

How likely it is in a random sample of 1,500 students to observe students studying an average of at most 27 or at least 33 hours per week outside of class, if the mean number is actually 30 hours per week. Indeed, in hypothesis testing, in order to assess the evidence, we need to find how likely it is to get data like those observed assuming that the null hypothesis is true.

Does the confidence interval indicate that a majority of all American adults blame Barack Obama a great deal or a moderate amount for U.S. economic problems?

No, because the interval extends below 0.5, meaning that it is reasonably possible that a minority of all American adults feel this way. In order to be convinced that a majority of American adults blame Barack Obama a great deal or a moderate amount, the entire confidence interval must be above 0.5. The 99% confidence interval we calculated in the previous problem extends down to 0.489, meaning that a minority is a reasonable possibility.

The following two hypotheses are tested: H0: The average number of miles driven per year is 12,000. Ha: The average number of miles driven per year is less than 12,000. In a survey, 1,600 randomly selected drivers were asked the number of miles they drive yearly. Based on the results, the P-value = 0.068. Use a 0.05 (5%) significance level. The fact that the P-value is .068 means that:

There is a probability of .068 (reasonably likely) of observing data like those observed if the average number of miles driven per year is 12,000. The P-value is the probability of observing data like those observed given that the null hypothesis, H0, is true.

In a random sample of 100 LMC students, 32 report reading the LMC Experience regularly. Use a 95% confidence interval to estimate the percentage of all LMC students who read the LMC Experience regularly. Round the standard error to 2 decimal places.

We are 95% confident that between 22% and 42% of all LMC students read the LMC Experience regularly.

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. When we select a lower confidence level, the margin of error will be smaller, which then decreases the width of the confidence interval. This would increase the precision of our estimate of the true population proportion. When we increase the sample size, the margin of error will decrease, which then decreases the width of the confidence interval. This would increase the precision of our estimate of the true population proportion.

According to the Centers for Disease Control (CDC), roughly 21.5% of all high school seniors in the United States have used marijuana. (The data were collected in 2002. The figure represents those who smoked during the month prior to the survey, so the actual figure might be higher.) A sociologist suspects that the rate among African American high school seniors is lower. In this case, then, H0: The rate of African American high-school seniors who have used marijuana is 21.5% (same as the overall rate of seniors). Ha: The rate of African American high-school seniors who have used marijuana is lower than 21.5%. To check his claim, the sociologist chooses a random sample of 375 African American high school seniors, and finds that 16.5% of them have used marijuana. In order to assess this evidence, we need to find:

How likely it is that in a sample of 375 we'll find that as low as 16.5% have used marijuana, when the true rate is actually 21.5%. Indeed, in hypothesis testing, in order to assess the evidence we need to find how likely it is to get data like those observed assuming that the null hypothesis is true.

Smoking levels: According to the Centers for Disease Control and Prevention, the proportion of U.S. adults age 25 or older who smoke is .22. A researcher suspects that the rate is lower among U.S. adults 25 or older who have a bachelor's degree or higher education level. What is the null hypothesis in this case?

The proportion of smokers among U.S. adults 25 or older who have a bachelor's degree or higher is .22. Indeed, the null hypothesis claims that "nothing special is going on" or "there is no change from the status quo." In this context, this means that there is nothing special about the smoking habits of adults age 25 and older who have a bachelor's degree or higher.

Marijuana legalization: In a 2015 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. This sample is only representative of California adult residents.

A psychologist is interested in constructing a 90% confidence interval for the proportion of people who accept the theory that a person's spirit is no more than the complicated network of neurons in the brain. 75 of the 836 randomly selected people who were surveyed agreed with this theory. Round answers to 4 decimal places where possible. a. With 90% confidence the proportion of all people who accept the theory that a person's spirit is no more than the complicated network of neurons in the brain is between __ and __ b. If many groups of 836 randomly selected people are surveyed, then a different confidence interval would be produced from each group. About __ percent of these confidence intervals will contain the true population proportion of all people who accept the theory that a person's spirit is no more than the complicated network of neurons in the brain and about __ percent will not contain the true population proportion.

a. 0.0740 and 0.1060 b. 90 and 10

In 2008 polls indicated that 54% of Americans thought that the death penalty was applied fairly in the U.S. This year in a poll of 1,000 Americans 58% feel the death penalty is applied fairly in the U.S. Has the percentage of the public with this opinion increased since 2008? We test the following hypotheses. H0: The proportion of Americans this year who feel the death penalty is applied fairly in the U.S. is 0.54. Ha: The proportion of Americans this year who feel the death penalty is applied fairly in the U.S. is greater than 0.54. The P-value is 0.036. The next three questions present three different interpretations of this P-value. Indicate if each interpretation is valid or invalid. a. The probability that more than 54% of Americans now feel the death penalty is applied fairly is 0.036. b. If 54% of Americans still feel the death penalty is applied fairly, then there is a 3.6% chance that poll results will show 58% or more with this opinion. c. There is a 3.6% chance that the null hypothesis is true if random poll results are greater than 54%.

a. invalid This statement says that the P-value is the probability that the alternative hypothesis is true. We cannot make a probability statement about whether a hypothesis is true or false. To make a probability statement we need a random event. The random event is random sampling. So the P-value makes a probability statement about random samples. b. valid This is a good interpretation of the P-value. The P-value is the probability that poll results will be more extreme than the latest poll if opinions have not changed since 2008. "More extreme" means in the direction of the alternative hypothesis. In this case, "more extreme" means "greater than." c. invalid This statement says that the P-value is the probability that the null is true. We cannot make a probability statement about whether a hypothesis is true or false. We could say, "there is a 3.6% chance that if the null hypothesis is true, random poll results will be greater than 58%."

College Students and Federal Grants According to the American Association of Community Colleges, 23% of community college students receive federal grants. The California Community College Chancellor's Office anticipates that the percentage is smaller for California community college students. They collect a sample of 1,000 community college students in California and find that 210 received federal grants. a. For which group are we making a hypothesis? b. Fill in the blanks to formulate a question that fits the scenario: Is p for California __ or is it __ the national number? c. Which are the correct hypotheses for this scenario? d. What does p represent in the hypotheses?

a. the population of community college students in California We are trying to answer a question about California community college students. b. 0.23 (We want to know if the proportion for California is the same as the national proportion) smaller than (The problem says that the Chancellor's Office anticipates that "the percentage is smaller.") c. H0: p = 0.23 Ha: p < 0.23 d. the proportion of California community college students who received federal grants The population of interest is California community college students. The variable is "receive federal grants" (yes/no).