One-Sample Tests and Confidence Intervals

In a letter to the New England Journal of Medicine, Dr. Sanders Frank reported that 20 of his male patients with creases in their earlobes had many of the risk factors (such as high cholesterol levels, high blood pressure, and heavy cigarette usage) associated with heart disease. For instance, the average cholesterol level for his patients with earlobe creases was 257 milligrams per 100 milliliters, compared to an average of 215 with a standard deviation of 10 for healthy middle-aged men. If these 20 patients were a random sample from a population with a mean of 215 and a standard deviation of 10, what is the probability that their average cholesterol level would be 257 or higher? Explain why these 20 patients may be a biased sample.

The random walk hypothesis says that the probability that stock prices will increase today is independent of past performance. If, on any given day, the percent change in the Dow Jones Industrial average is normally distributed with a mean of $.1 \%$ and a standard deviation of $1.5 \%$, what is the probability that the average of the daily percentage changes over 250 trading days in a year will be positive?

Explain the error in this Newsweek explanation of the margin of error in a public opinion poll:

The margin of error, calculated according to a textbook statistical formula, varies inversely with the sample size. In general, about 500 responses give a possible error of $5 \%$ either way; 2,500 responses decrease it to $1 \%$.

A 1995 survey for Fodor's, a publisher of travel books, asked 600 US travelers to name their least desirable vacation destination. The winner (with $9 \%$ of the votes) was Iraq/Iran; in second place (with 7) was New York City. A spokesman for New York's mayor scoffed, "They only interviewed 600 people. That's the size of an apartment building." Assuming this to be a random sample, give a $95 \%$ confidence interval for the fraction of all US travelers who would choose New York City as their least desirable vacation destination.

Suppose that the damage award favored by individual potential jurors in a particular personal injury case can be described by a normal distribution with a mean of $\$ 5$ million and a standard deviation of $\$ 2$ million. (This probability distribution is across randomly selected jurors.)

A certain grade of apple has a weight that is normally distributed with a mean of 10 ounces and a standard deviation of 2 ounces.

The population of Spain is approximately four times the population of Belgium. If random samples are taken of 2,000 Spaniards and 2,000 Belgians, will the margin for error in these polls be (a) four times larger for the Spanish poll; (b) four times larger for the Belgian poll; (c) twice as large for the Spanish poll; (d) twice as large for the Belgian poll; or (e) the same for both polls?

A July 1987 Wall Street Journal poll asked 35 economic forecasters to predict the interest rate on 3-month Treasury bills in June 1988. These 35 forecasts had a mean of 6.19 and a variance of .47. Assuming these to be a random sample, give a $95 \%$ confidence interval for the mean prediction of all economic forecasters and then explain why each of these interpretations is correct or not:

An article about computer forecasting software explained how to gauge the uncertainty in a prediction: "[Calculate] the standard deviation and mean (average) of the data. As a good first guess, your predicted value... is the mean value, plus or minus one standard deviation-in other words, the uncertainty is about one standard deviation." What did they overlook?

Does the width of a confidence interval depend on the size of the sample, the size of the population, or the ratio of the sample size to the population size? Explain your reasoning in words not equations.

Here is an excerpt from a statistical paper:

We asked 100 students to guess the number of beans in a glass jar. (There were 183.) Using their guesses, we constructed this $95 \%$ confidence interval:

Explain why you either agree or disagree with the following:

Explain the obvious error:

The null hypothesis is that the population mean is 0 ; the alternative hypothesis is that it is not zero. Here, $n=75 ; X$-bar $=.004$; standard deviation $=.0443, t=.781964$. Looking at a $t$ table with 74 degrees of freedom, the two-sided p-value is 1.9925.

Nationally, $39.6 \%$ of all Americans are either first-born or only children. A random-sample of 29 American students at an elite college found that 14 were either first-born or only children.

Identify the error(s):

We calculated the fraction pof the soccer players born in the first 6 months of the year by dividing the number born from January to June by the total number of players.

Then we set up a one-variable proportion test to figure out whether the fractions born in the first 6 months and last 6 months were equal $(p=.5)$ or not equal $(p \neq .5)$. We calculated the Z -value by using the equation:

Explain why this argument either does or does not make sense to you:

The distribution of individual test scores is bi-modal. This means that you cannot do a t-test of the sample mean of all test scores, no matter how large the sample, because that test is based on the assumption that individual scores are normally distributed.

An online college statistics class asked this question:

A true-false quiz with 10 questions was given to 100 students in a statistics class. Following is the distribution of the scores. Find the mean score and interpret the result. Round the answers to two decimal places as needed.

What is wrong with the given answer (below)? (Hint: The average score really is 7.48.)

The mean score is 7.48. This means that if we were to give this quiz to more and more students, the average score for these students would approach 7.48.

The Super Bowl Indicator says that the stock market goes up if a team in the National Football Conference or a team in the American Football Conference that used to be in the National Football League wins the Super Bowl; the market goes down otherwise. This indicator worked for 20 of the first 22 Super Bowls (1967-1988). In 1990 two finance professors published an article in the Journal of Finance, a premier peer-reviewed finance journal, arguing that, "although the theoretical relationship connecting the Super Bowl and subsequent stock market movements is not obvious," the 20 -out-of-22 record was highly statistically significant and would have been very profitable if followed by investors. To disprove the possibility that it might be a coincidence, they calculated the binomial probability of 20 out of 22 successes to be

The life of a Rolling Rock tire is normally distributed with a mean of 40,000 miles and a standard deviation of 5,000 miles. Assuming independence, if you buy four tires, what is the probability that

A 1981 study published in the New England Journal of Medicine looked at the use of cigars, pipes, cigarettes, alcohol, tea, and coffee by patients with pancreatic cancer and concluded that there was "a strong association between coffee consumption and pancreatic cancer." This study was immediately criticized on several grounds, including the fact multiple tests were conducted and the most statistically significant result was reported. Subsequent studies failed to confirm an association between coffee drinking and pancreatic cancer. Suppose that independent tests are conducted of six products that are unrelated to pancreatic cancer. What is the probability that at least one of these tests will have a p-value below .05?

Karl Pearson flipped a coin 24,000 times and obtained 12,012 heads. Use these data to estimate a $95 \%$ confidence interval for the probability of heads with this coin. If this coin's probability of a head is .5, what is the exact probability that the number of heads in 24,000 flips would be between 11,988 and 12,012?

A statistics textbook gives this example:

Suppose a downtown department store questions 49 downtown shoppers concerning their age.... The sample mean and standard deviation are found to be 40.1 and 8.6, respectively. The store could then estimate $\mu$, the mean age of all downtown shoppers, via a $95 \%$ confidence interval as follows:

Thus, the department store should gear its sales to the segment of consumers with average age between 37.7 and 42.5. Explain why you either agree or disagree with this interpretation: $95 \%$ of downtown shoppers are between the age of 37.7 and 42.5.

A study of 387 corporate stock splits that occurred during the 10 years (2007 through 2016) found that 245 stocks had price increases on the day the split was announced, while 142 had price decreases. Is this difference substantial and statistically persuasive?

A monthly survey of consumer sentiment was criticized because only 5,000 people were surveyed. In contrast, Twitter tweets can be data mined for millions of people.

An investor divided her money equally among 100 stocks. She believes that over the next 12 months, each stock's return will be independently determined from a probability distribution with a mean of $10 \%$ and a standard deviation of $30 \%$, What can we say about the probability distribution of the return on her portfolio?

It used to be that, when a National Football League (NFL) game was tied at the end of regulation, the teams played a 15-minute suddendeath overtime with a coin flip used to decide which team kicks off and which team receives. The first team to score wins. During the 2001-2006 seasons, there were 109 sudden-death overtimes, one of which was still tied at the end of the overtime period because neither team scored. In the other 108 games, the team that received the ball first won 64 games and the team that kicked off won 44 games. Use these data to calculate the exact two-sided p-value for a test of this null hypothesis: Of those games decided by sudden-death overtime, the kicking and receiving teams have an equal chance of winning. Does the difference seem substantial?

California birth records were used to obtain data on the ZIP code residence of foreign-born mothers when they gave birth to a daughter and the ZIP code residence of these daughters when they gave birth to a child; 996 of the adult daughters had the same ZIP code as their mothers and 4,298 had different ZIP codes.

A researcher asked 30 students to read a short article and then estimate how long it took them to read the article. The 30 differences (in seconds) between the actual and estimated times were used to see if people generally overestimated or underestimated. It was reported that a test of the null hypothesis that the population mean is zero gave a t -value of 2.05 and that a $95 \%$ confidence interval is $26.53 \pm 13.82$. How do you know that there was an error in these calculations?

The first American to win the Nobel prize in physics was Albert Michelson (1852-1931), who was given the award in 1907 for developing and using optical precision instruments. His October 12-November 14,

1882 measurements of the speed of light in air (in kilometers per second) were as follows:

Assuming that these measurements were a random sample from a normal distribution, does a $99 \%$ confidence interval include the value 299,710.5 that is now accepted as the speed of light?

The American Coinage Act of 1792 specified that the gold 10-dollar eagle should contain 247.5 grains of pure gold. An 1837 Act of Congress permitted a .25 grain deviation in the weight of a single eagle and a .048 grain deviation in the average weight of 1000 eagles. If the tested coins are a random sample from a normal distribution with a mean of 247.5 grains and an unknown standard deviation $\sigma$, would the Mint be more likely to fail the test of a single coin or the test of 1,000 coins?

In 1977, the US Supreme Court observed that persons with Spanish surnames constituted $79 \%$ of the population of Hildago county, Texas but only 339 of 870 recent jurors in this county. The Court calculated a Z-value of 29 and commented that "as a general rule for such large samples, if the difference between the expected value and the observed number is greater than 2 or 3 standard deviations, then the hypothesis that the jury drawing was random would be suspect to a social scientist."

A high school basketball coach said that a missed free throw by a right-handed player is likely to bounce to the right, while the reverse is true of a left-hander. To test this theory, a researcher asked two college varsity players to shoot 50 free throws. These players missed 21 free throws, of which 15 landed on the same side as their shooting hand, and 6 bounced to the opposite side. Is this difference substantial and statistically persuasive? (Do not use a normal approximation.)

Sixteen pregnant women were asked to read a children's story aloud three times a day during the last 6.5 weeks of pregnancy. These readings were tape recorded and, shortly after birth, each baby was allowed to choose (by varying the rhythm of their sucking on a special bottle) between a tape recording of their mother reading the familiar story and reading an unfamiliar one. In 13 of 16 cases, the babies chose the familiar story. Is this result substantial and statistically persuasive? (Do not use a normal approximation.)

In the 1989 and 1990 NCAA basketball tournaments, 1,891 three-point shots were attempted, of which 840 were taken by favored teams and 1,051 by underdog teams. Assuming that these data are a random sample from NCAA tournament games, calculate a $95 \%$ confidence interval for the probability that a randomly selected three-point shot was taken by an underdog team.

Fifty-six college students participated in an ESP test in which a professor flipped a coin ten times and each student attempted to identify each flip as a head or tail.

An economics professor told his students that instead of spending hundreds of dollars on a very accurate watch, they should wear 20 cheap watches and calculate the average time. Suppose that the time shown on each cheap watch is normally distributed with a mean equal to the actual time and a standard deviation of 10 seconds, and that the errors on the cheap watches are all independent of each other. What is the mean and standard deviation of the average time on these 20 watches?

A sociology professor argued that the stress associated with the number 4 might cause Chinese and Japanese persons to suffer an unusually large number of fatal heart attacks on the fourth day of the month. Mortality data for Chinese and Japanese Californians for the years 1989-1998 show that 472 of the 1391 fatal heart attacks that occurred on days 3, 4, or 5 of a month occurred on day 4. Is this substantial and statistically persuasive evidence that more than one-third of the fatal heart attacks that occur on days 3,4, or 5 of a month happen on day 4? (Use a normal approximation.)

In 1942, a seed company sold 20 bags of rye seeds with labels indicating a $90 \%$ germination rate. The War Food Administration took a sample of seeds from these bags and found that only 240 of the 400 seeds that they tested germinated. Is this difference of any practical importance? Use a normal approximation to calculate a two-sided p-value for testing the null hypothesis that each seed has a .90 probability of germinating.

Researchers calculated the difference between the actual and predicted daily high temperatures at the Los Angeles Civic Center for every day in 1996. Explain the error(s) in this interpretation of their results. "The t -value for the null hypothesis $\mu =0$ was 6.47, revealing that there is zero chance that the population mean is zero. Thus at the $1 \%$ level, we disproved the null hypotheses that weather forecasters make no errors in their predictions."

A test of the null hypothesis that an average college student gains 15 pounds during the first year at college surveyed 100 sophomores and obtained a sample average reported gain of 4.82 pounds with a standard deviation of 5.96 pounds. Explain why you either agree or disagree with each of these conclusions:

Americans drive about 3 trillion miles a year and average about 77 reported injuries and 1.09 fatalities per 100 million miles driven. A Google fleet of 55 cars had 11 crashes (2 injuries and no fatalities) while driving 1.3 million miles in autonomous mode between 2009 to 2015.

If a test of autonomous vehicles has no fatalities, how many miles would have to be driven fatality free to reject at the $5 \%$ level the null hypothesis that the probability of a fatality is 1 per 100 million miles?