Statistics Confidence Intervals and Significance Test

A political polling organization conducted a survey by selecting 100 random samples, each consisting of 500 registered voters. The registered voters in each sample were asked whether they planned to vote for the Republican or Democratic candidate in the next presidential election. For each of the 100 samples, the polling organization created a 99 percent confidence interval for the proportion of all registered voters who planned to vote for the Republican candidate. Which of the following statements is the best interpretation of the 99 percent confidence level?

• We would expect about 99 of the 100 confidence intervals to contain the sample proportion of the registered voters who plan to vote for the Republican candidate.

• We would expect about 99 of the 100 confidence intervals to contain the proportion of all registered voters who plan to vote for the Republican candidate.

• We would expect the margin of error to be 0.01 because the polling organization constructed 99 percent confidence intervals.

• We would expect the margin of error to be less than 0.01 because the polling organization constructed 100 different 99 percent confidence intervals.

• We would expect only 1 of the 100 confidence intervals to reveal that the Democratic candidate is favored.

A pharmaceutical company claims that side effects will be experienced by fewer than 20% of the patients who use medication X. A clinical trial with a random sample of 400 patients was conducted in which half of the patients were randomly assigned to take medication X and the other half received a placebo. Of those that received medication X they find 68 who experienced side effects. Assuming all conditions for inference were met, which of the following is a 90 percent confidence interval for the proportion of patients that will experience side effects while taking medication X?

• (0.1391, 0.2009)

• (0.2744, 0.4057)

• (0.2849, 0.3951)

• (0.6257, 0.7343)

• (0.6416, 0.7184)

Household income in the United States is strongly skewed to the right. The current presidential administration claims that the mean household income is greater than it has ever been in the past. An independent contractor will obtain a random sample of 100 households in the United States and will calculate the mean household income. Which of the following statements is true?

• The sampling distribution of the sample mean household income is approximately normal because the sample size of 100 is greater than 30.

• The distribution of household income for the sample is approximately normal because the sample size of 100 is greater than 30.

• The sampling distribution of the sample mean household income is strongly skewed to the right because the population standard deviation is unknown.

• The distribution of household income is strongly skewed to the left because the population sample size of 100 is greater than 30.

• The sampling distribution of the sample mean household income is strongly skewed to the right because the population distribution is strongly skewed to the right.

In a month-long randomized comparative experiment, participants were assigned to one of two treatments. 35 of the 50 participants who received the first treatment, a fitness watch and training on how to meet their daily goals as measured by the watch, lost weight. However, only 16 of the 40 participants who received the second treatment, instruction to recite positive affirmations each morning, lost weight. Which of the following is an appropriate margin of error for a 99 percent confidence interval to estimate the difference in the population proportion of all users of each treatment who would lose weight? Click to enlarge the answer choices at left.

• A

• B

• C

• D

• E

A Statistics class from a high school with 4,000 students took a survey of the first 35 students who walked through the front door of the school, and asked how far they traveled to school that day. The class plans to run a one-sample t-test to determine if the average travel distance has increased since last year. The students notice that the sample data are right skewed. Which conditions have been satisfied for the t-test?

I. The sample is from a random sample or randomized experiment.

II. The sampling distribution of sample means is approximately normal. III. The sample size is small relative to the population.

• III

• I and II

• I and III

• II and III

• I, II, and III

An advertiser wants to find convincing evidence that television viewers remember more than 4 commercials, on average, after watching a 30 minute TV program. They take a random sample of 100 television viewers and ask them how many commercials they could remember after watching a 30 minute TV program. The appropriate t-test was conducted, which resulted in a P-value of 0.15. Assuming all conditions were met, which of the following is an appropriate conclusion?

• Because the P-value is less than 0.05, at the 5% significance level, there is not convincing evidence that television viewers remember more than 4 commercials, on average, after watching a 30 minute TV program.

• Because the P-value is greater than 0.05, at the 5% significance level, there is convincing evidence that television viewers remember fewer than 4 commercials, on average, after watching a 30 minute TV program.

• Because the P-value is greater than 0.01, at the 1% significance level, there is not convincing evidence that television viewers remember more than 4 commercials, on average, after watching a 30 minute TV program.

• Because the P-value is greater than 0.01, at the 1% significance level, there is convincing evidence that television viewers remember exactly 4 commercials, on average, after watching a 30 minute TV program.

• Because the P-value is less than 0.01, at the 1% significance level, there is convincing evidence that television viewers remember fewer than 4 commercials, on average, after watching a 30 minute TV program.

After completing a statistical analysis of a survey of 40 students, the principal of North High School made the following conclusion: reject the null hypothesis; there is convincing evidence that more than 50% of students support a schedule change to have lunch occur earlier in the day. Which error could have been committed?

• Type I error: Conclude that more than 50% of students want earlier lunch, when 50% or less want earlier lunch.

• Type I error: Conclude that more than 50% of students want earlier lunch, when more than 50% want earlier lunch.

• Type II error: Fail to reject that 50% of students want earlier lunch, when more than 50% want earlier lunch.

• Type II error: Fail to reject that 50% of students want earlier lunch, when 50% or less want earlier lunch.

• Type II error: Fail to reject that more than 50% of students want earlier lunch, when 50% or less want earlier lunch.

•  There is a 24% probability that the null hypothesis is true. 

• If the test were repeated many times, we would correctly reject the null hypothesis 24% of the time.

• If the test were repeated many times, we would incorrectly fail to reject the null hypothesis 24% of the time.

• If the null hypothesis is true, there is a 24% probability of getting a sample difference in means as far or farther below 0 as the difference found in the samples. 

• If the null hypothesis is false, there is a 24% probability of getting a sample difference in means as far or farther below 0 as the difference found in the samples.

Is the proportion of adults who watch the nightly news dropping? In a survey taken in 2013, 24 out of 40 adults surveyed responded that they had watched the local TV news at least once in the last month. In a similar survey in 2010, 40 out of 50 adults said they had watched the local TV news at least once in the last month. Is this convincing evidence that the proportion of adults watching the local TV news dropped between 2010 and 2013? For a significance test with the hypotheses Ho: p₁ - p₂ = 0, Ha: p₁ - p₂ < 0, where p₁ is the proportion of all adults who watched the local news at least once a month in 2013, and p₂ is the proportion for 2010, which of the following is closest to the standard error for the test statistic? Enlarge the answer choices at left by clicking on them.

• A

• B

• C

• D

• E

A 20-ounce soda bottle is supposed to contain 20 ounces of soda. The distribution of the actual amount of soda in a 20-ounce soda bottle is approximately normal. A curious student randomly selects fifteen 20-ounce soda bottles from various retailers and carefully measures their content to see if the soda company is cheating the customers. The mean and standard deviation of the fifteen bottles is 19.4 ounces and 0.25 ounces, respectively. Which of the following is the test statistic for the appropriate test to determine if the average amount of soda that is contained in their 20-ounce soda bottles is significantly less than 20-ounces? Click on the answer choices at left to enlarge them.

• A

• B

• C

• D

• E