1 and 2 Sample Proportions and Hypothesis Testing
Of the 94 adults selected randomly from one town, 68 have health insurance. Construct a 90% confidence interval for the percentage pf all adults who have health insurance.
(64.8%, 79.9%)
Of the 346 items tested, 12 are found to be defective. Construct a 98% confidence interval for the percentage of all such items that are defective.
(1.18%, 5.76%)
Of the 131 adults selected randomly from one town, 33 of them smoke. Construct a 99% confidence interval for the percentage of all adults in the town who smoke.
(15.4%, 35.0%)
Of the 369 randomly selected medical students, 23 said that they planned to work in rural community. Construct a 95% confidence interval for the percentage of all medical students who plan to work in a rural community
(3.77%, 8.70%)
A survey of 865 voters in one state reveals that 408 favor approval of an issue before the legislature. Construct a 95% confidence interval for the percentage of all voters in the state who favor approval.
(43.8%, 50.5%)
In a survey of 3200 TV viewers, 20% said they watch network news programs. Find the standard error for the sample proportion.
.0071
In a sample of 150 children selected randomly from one town, it is found that 24 of them suffer from asthma. Find the p-value for a test of the claim that the proportion of all children in the town who suffer from asthma is equal to 11%. (the alternative is right-tailed)
.025
500 random samples of size n=900 are taken from a large population in which 10% are left handed. The proportion of the sample that is left-handed is found for each sample and a histogram of these 500 proportions is drawn. Which interval covers the range into which 68% of the values in the histogram will fail?
.1 +- .010
We have calculated a confidence interval based on a sample size of n=100. Now we want to get a better estimate with a margin of error that is only one-fourth as large. How large does our new sample need to be?
1600
The National Research Council of the Philippines reported that 210 of 361 members in biology are women, but only 34 of 86 members in math are women. Establish a 95% confidence interval estimate of the difference in proportions of women in biology and women in math in the Philippines.
.187 +- .115
Given Ha: p=/ po. What is the p-value if the test statistics is calculated to be z= 1.08?
.28
In a survey of 9700 TV viewers , 40% said they watch network new programs. Find the margin of error for this survey if we want 95% confidence in our estimate of the percent of TV viewers who watch network news programs.
.975%
A 95% confidence interval for the difference between 2 population proportions is found to be (.07, .19). Which of the following statements are true? 1. We are 95% confident that the true difference between the population proportions is between .07 and .19 2. The probability is .95 that the true difference between the population proportions is between .07 and .19 3. It is unlikely that the 2 populations have the same proportions
1 and 3 only
Under a null hypothesis, a sample value yields a p-value of .015. Which of the following statements is true? 1. This finding is statistically significant at the .05 level of significance 2. This finding is statistically significant at the .01 level of significance 3. The probability of getting a sample value or more extreme as the one obtained by chance alone if the null hypothesis is true is .015
1 and 3 only
Which of the following statements is true? 1. The p-value of a test is the probability of obtaining a result or more extreme as the one obtained assuming the null hypothesis is true 2. If the p-value for a test is.015, the probability that the null hypothesis is true is .015 3. When the null hypothesis is rejected, it is because it is not true
1 only
Power Formula
1-P(Type 2 Error)
A survey of shoppers is planned to see what percentage use credit cards. Prior surveys suggest 53% of shoppers use credit cards. How many randomly selected shoppers must we survey in order to estimate the proportion of shoppers who use credit cards within 3% with 95% confidence?
1064
An advertising firm wants to confirm estimates that 30% of households will buy something on the Internet next year. What sample size will produce a margin of error of 7% with 90% confidence level?
117
In a simple random sample of 300 elderly men, 65% were married, while in an independent simple random sample of 400 elderly women, 48% were married. Determine a 99% confidence interval estimate for the difference between the percentages of elderly men and women who are married.
17% +- 9.6%
Which is true about a 99% confidence interval based on a given sample? 1. The interval contains 99% of the population 2. Results from approximately 99% of all samples will capture the true parameter in their respective intervals. 3. The interval is wider than a 95% confidence interval would be
2 and 3 only
A certain population is strongly skewed to the left. We sample repeatedly from that population, hoping to estimate its proportion. If we increase our sample size, which will happen? 1. The population distribution of proportions will become closer to normal 2. The sampling distribution of the sample proportions will become closer to normal 3. The variability of the sample proportions will increase
2 only
We have calculated a 95% confidence interval and would prefer for our next confidence interval to have a smaller margin of error without losing any confidence. In order to do this, we can 1. Change the z value to a smaller number 2. Take a larger sample 3. Take a smaller sample
2 only
Which is true about a 98% confidence interval for a population proportion based on a given sample? 1. We are 98% confident that other sample proportions will be in our interval 2. There is a 98% chance that our interval contains the population proportion 3. The interval is wider than a 95% confidence interval would be
3 only
410 people were asked if they were satisfied with their jobs. 37% said they were. It is wished to test the following null hypothesis: Ho: p= .3. Find the test statistic
3.093
You are going to create a 95% confidence interval for a population and want the margin of error to be no more than .05. Historical data indicate that the population proportion has remained constant at about .7. What is the minimum sample size you need to construct this interval?
323
A USA Today "Lifeline" column reported that in a survey of 500 people, 39% said they watch their bread while it is being toasted. Establish a 90% confidence interval estimate for the percentage of people who watch their bread being toasted.
39% +- 3.6%
A politician wants to know what percentage of the voters who support her position on the issue of a state income tax. What size voter sample should be obtained to determine with 90% confidence the support level to within 4%?
423
In a survey of 280 adults over 50, 80% said that they were taking vitamin supplements. Find the margin of error for this survey if we want a 99% confidence in our estimate of the percent of adults over 50 who take vitamin supplements.
6.16%
A pollster wishes to estimate the true proportion of U.S. voters who oppose capital punishment. How many voters should be surveyed in order to be 95% confident that the true proportion is estimated to within 4%? (When p-hat is not stated, assume p-hat=.5)
601
A newspaper reports that the governor's approval rating stands at 60%. The article adds that the poll is based on a random sample of 1476 adults and has a margin of error of 2.5%. What confidence level did the pollsters use?
95%
In 1993, Los Angeles Times poll of 1703 adults revealed that only 17% thought the media was doing a "very good" job. With what degree of confidence can the newspaper say that 17% +- 2% of adults believe the media is doing a "very good" job?
97.2%
Which statement is NOT true about confidence intervals?
A confidence interval between 20% and 40% means that the population proportion lies between 20% and 40%
Power Definition
A test's ability to detect a false hypothesis; correctly rejecting a false null hypothesis
Data in 1980 showed that about 40% of the adult population have never smoked cigarettes. In 2004, a national health survey interviewed a random sample of 2000 adults and found that 50% had never been smokers. Create a 95% confidence interval for the proportion of adults (in 2004) who had never been smokers.
Based on the data, we are 95% confident that proportion of adults in 2004 who had never smoked cigarettes is between 47.8% and 52.2%
In general, how does doubling the sample size change the confidence interval size?
Divides the interval size by 1.414
A researcher claims that 62% of voters favor gun control. Determine the null and alternative hypotheses
Ho: p=.62 vs. Ha: p =/ .62
A randomly selected sample of 400 students at a university with 15-week semesters were asked whether or not they think they semester should be shortened to 14 weeks (with longer classes). 46% of the 400 students surveyed answered "yes." Which one of the following statements about the number 46% is correct?
It is a sample statistic
We are about to test a hypothesis using data from a well-designed study. Which is true? 1. A large p-value would be strong evidence against the null hypothesis 2. We can set a higher standard of proof by choosing alpha=1-% instead of 5% 3. If we reduce the risk of committing a Type 1 error, then the risk of Type 2 error will also decrease
None
Type 2 Error
Occurs when we fail to reject Ho, when in fact Ha is true
Type 1 Error
Occurs when we reject Ho, when in fact Ho is true
We will test the hypothesis that p=60% versus p>60%. We do not know it, but actually p is 70%. With which sample size and significance level will our test have the greatest power?
a= .05 n=500
Effect Size Definition
The difference between the null hypothesis value and the true value of a model parameter
The mayor of a large city will run for governor if he believes that more than 30% of the voters in the state already support him. He will have a survey firm ask a random sample of n voters whether or not they support him. He will use a large sample test for proportions to test the null hypothesis that the proportion of all voters who support him is 30% or less against the alternative that the percentage is higher than 30 percent. Suppose that 35% of all voters in the state actually support him.In which of the following situations would the power for this test be highest?
The mayor uses a significance level of .05 and n=1000 voters
The corn rootworm is a pest that can cause significant damage to corn, resulting in a reduction in yield and thus in farm income. A farmer will examine a random sample of plant from a field in order to decide whether or not the number of corn rootworms in the whole field is at a dangerous level. If the farmer concludes that it is, the field will be treated. The farmer is testing the null hypothesis that the number of corn rootworms is not at a dangerous level against the alternative hypothesis that the number is at a dangerous level. Suppose that the number of corn rootworms in the whole field is actually at a dangerous level. Which of the following is equal to the power of the test?
The probability that the farmer will decide to treat the field
A researcher investigating whether runners are less likely to get colds than non-runner found a p-value of .03. This means that:
There's a 3% chance that the sample statistic or more extreme will occur assuming there is not difference between the number of colds whether a runner or not
In a test of the null hypothesis, Ho: p=.35 with a=.01, against the alternative hypothesis Ha: p<.35, a large random sample produced a z-score of -2.05. Based on this, which of the following conclusions can be drawn?
We do not have sufficient evidence to reject the null hypothesis
Suppose that a device advertised to increase a car's gas mileage really does not work. We test it on a small fleet of cars (with Ho: not effective) and our data results in a P-value of .0004. What probably happens as a result of our experiment with an alpha of .05?
We reject Ho; making a Type 1 error
A building inspector believes that the percentage of new construction with serious code violations may be even greater than the previously claimed 7%. She conducts a hypothesis test on 200 new homes and finds 23 with serious code violations. Is this strong evidence against the .07 claim?
Yes, because the p-value is .006
In a random sample of 50 men, 40% said they preferred to walk up stairs rather than take the elevator. In a random sample of 40 women, 50% said they preferred the stairs. The difference between the two sample proportions (m-w) is to be calculated. Which of the following choices correctly denotes the difference between the 2 sample proportions that is desired?
p-hat of men - p-hat of women = -.10
In an effort to curb certain diseases, especially autoimmune (AIDS), San Francisco has a program whereby drug users can exchange used needles for fresh ones. As reported in the Journal of the American Medical Association, 35% of 5644 intravenous drug users in San Francisco admitted to sharing needles. Is this sufficient evidence to say that the rate of sharing needles has dropped from the pre-needle exchange rate of 66%?
p-value< .001, so this is very strong evidence that the rate has dropped
Amos Tversky and Thomas Gilovich, in their study on the "Hot Hand" in basketball found that in a random sample of games, Larry Bird hit a second free throw in 48 of 53 attempts after the first free throw was missed, and hit a second free throw in 251 of 285 attempts after the first free throw was made. Is there sufficient evidence to say that the probability that Bird will make a second free throw is different depending on whether or not he made the first free throw?
p-value> .10, so there is little or no evidence that the probabilities are different
