Tests for proportions

It is well known that infants born to mothers who smoke tend to be small and prone to a range of ailments. It is conjectured that also they look abnormal. Nurses were shown selections of photographs of babies, one half of whom had smokers as mothers; the nurses were asked to judge from a baby's appearance whether or not the mother smoked. In 1500 trials the correct answer was given 910 times. Is the conjecture plausible? If so, why?

At Bell Labs in the early 1950s, David Hagelbarger and Claude Shannon built machines to predict whether a human coin-flipper would call heads or tails. Hagelbarger's machine (a "sequence extrapolating robot", or "SEER") was correct on 5218 trials of 9795, and Shannon's machine (a "mind-reading (?) machine") was correct on 5010 trials of 8517. In each case, what is the probability of doing at least as well by chance?

To test the effect of a new drug on blood pressure, the blood pressure of 100 patients was measured before and after taking the drug. It turned out that in 32 cases the pressure increased after taking the drug, and in 68 cases it decreased. Can it be considered established that this drug affects blood pressure? What is the probability that purely random fluctuations in pressure would produce a deviation from 50 at least as large?

An urn contains white and black balls. The only thing known about the composition of the urn is that the proportion of white balls is either 0.5 or 0.4. From the urn, 100 balls are drawn with replacement, and it is found that white balls make up the larger part of the sample. On the basis of this observation it is concluded that the proportion of white balls in the urn is 0.5. What is the probability that an erroneous conclusion has been made?

Suppose $n$ Bernoulli trials with success probability $p$ have been carried out. Suppose the number of successes $S_{n}$ turns out to equal $m$, $0 < m < n$, and we need to check whether this is consistent with some hypothesis about the unknown value of $p$. We can use the following criterion. Fix a number $\alpha , 0 < \alpha < 1$, and, assuming the hypothesis $p = p_{0}$ is true, compute the probability $\mathbb {P}\left(S_{n} \geqslant m\right)$. If this probability is less than $\alpha$, we reject the hypothesis. Otherwise, we consider the hypothesis to be consistent with the statistical data.

In 1000 independent coin tosses, heads came up 540 times. Test the hypothesis that the coin is fair at $\alpha = 0.05$.

(Continuation) Let $\alpha$ be given. Assuming that some hypothesis $p = p_{0}$ is true, find the smallest integer $m_{\alpha }$ such that $\mathbb {P}\left(S_{n} \geqslant m_{\alpha }\right) \leqslant \alpha$. A criterion for testing the hypothesis $p = p_{0}$ can be as follows: if $S_{n} \geqslant m_{\alpha }$, we reject the hypothesis $p = p_{0}$; if $S_{n} < m_{\alpha }$, we consider the hypothesis to be consistent with the data. The probability of an erroneous rejection of the hypothesis does not exceed $\alpha$.

Test the hypothesis about the probability of a boy being born, from problem 6.138.

Let $X_1$ and $X_2$ be independent observations from the binomial distributions with sizes $n_1$ and $n_2$ and probabilities $p_1$ and $p_2$, respectively, where $n_i$'s are known and $p_i$'s are unknown.

Let $X_1$ and $X_2$ be independent observations from the binomial distributions with sizes $n_1$ and $n_2$ and probabilities $p_1$ and $p_2$, respectively, where $n_i$'s are known and $p_i$'s are unknown.

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:

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

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?

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.

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.)

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.

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.

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?

An A/B test is like an internet laboratory experiment. Two versions of a web page are created-typically the current page and a proposed modification, perhaps with a different banner, headline, or testimonial. A random number generator determines which page a user sees. A prespecified metric, like purchases or mailing-list signups, is used to determine the winning page.

The Obama 2008 presidential campaign used an A/B test to compare the number of web-page visitors who signed up for more information with these two buttons: "Sign Up" and "Learn More."

The results:

A French hospital put 412 patients who had suffered one heart attack on a traditional Mediterranean diet (including olive oil, fruit, and bread); the control group consisted of 358 heart-attack patients who were given a recommended low-fat diet. Four of the patients on the Mediterranean diet and 17 of the patients on low-fat diet suffered a second heart attack during the 2 years of the study. Is the observed difference substantial and statistically persuasive?

Evidence from many of sports indicate that performance is enhanced by our personal assessment of our ability. For example, when both contestants in an arm-wrestling study incorrectly believed the weaker person was stronger than his opponent, the weaker person won 10 of 12 matches. When the contestants had correct information about who was stronger, the stronger person won all 12 matches. Is this difference substantial and statistically persuasive?

A student played 100 games of Roshambo (rock, scissors, paper) against a computer software program, winning 32 games, losing 38, and drawing 30. The computer tried to figure out the player's strategy. This table separates the results into the first 50 games and the last 50 games:

A study of 4 years of accident reports by the Dallas Fire Department found that fire trucks painted red made 153,348 runs and had 20 accidents, while fire trucks painted yellow made 135,035 runs and had 4 accidents. Assuming these to be independent random samples, is this difference substantial and statistically persuasive?

In 1983 22,000 doctors participated in an experiment testing the effects of aspirin on the incidence of heart attacks. Half of the doctors were given a single aspirin tablet every other day; the other half took a placebo every other day. Neither the volunteers or the scientists conducting the experiment knew which volunteers were taking aspirin. After 5 years of what was intended to be a 7 -year test, the experiments were stopped by an outside committee monitoring the results. During this period, the doctors taking placebos had 18 fatal heart attacks and the doctors taking aspirin had 5. Is this difference substantial and statistically persuasive? Why do you suppose the experiment was stopped 2 years early?

In horseshoes, each player pitches two shoes in each inning. If the player pitches two ringers, this is called a double. A "hot hands" study of the 2000 World Horseshoe Pitching championships looked at whether players were more likely or less likely to throw a double after throwing a double in the previous inning. For example, female pitchers threw 1641 doubles and 1691 nondoubles after a nondouble inning, and threw 2142 doubles and 1666 nondoubles after a doubles inning. Is the difference substantial and statistically persuasive?

Students in one California school district are classified as either a gifted and talented education (GATE) student or a regular student. A survey of seventh- and eighth-graders found that 10 of 86 GATE students had mothers who did not attend college and that 35 of 124 regular students had mothers who did not attend college. Is this difference substantial and statistically persuasive?

Explain why the proposed test is wrong and identify the correct test:

My question is "Does playing on home ice in the National Hockey League (NHL) give the home team a better chance to win?" I will choose six NHL teams and, for each team, use a Z single-sample categorical test of the null hypothesis that the chance of winning at home is equal to one half.

California birth records were used to obtain data on the ZIP code residence of mothers when they gave birth to a daughter and the ZIP code residence of these daughters when they gave birth to a child. Of the 4,298 daughters of foreign-born mothers who lived in different ZIP codes, 2,579 lived in ZIP codes with higher median income than their mother's ZIP code and 1,719 lived in ZIP codes with lower income. For the 5,935 daughters of California-born mothers who lived in different ZIP codes, 3,181 lived in ZIP codes with higher median income than their mother's ZIP code and 2,754 lived in ZIP codes with lower median income. Is this difference between the daughters of foreign-born and California-born mothers substantial and statistically persuasive?

The 794 students in Pomona College's freshmen classes of 1987-1988 and 1988-1989 were separated into two groups: 552 who had attended public high schools and 242 who had attended private schools. Of those from public schools, $48(8.7 \% )$ were Pomona Scholars (an A-grade point average); of those from private schools, $13(5.4 \% )$ were Pomona Scholars. Does the observed difference seem substantial? The author calculated the Z-value to be 1.64 and concluded that although "two classes proved the null hypothesis to be true, ...the sample size is minuscule." Did she prove the null hypothesis to be true? Are her results invalidated by the small sample size?

In the 1980s Kahneman and Tversky reported that when 200 people were asked the following question, 98 answered yes and 108 answered no:

Imagine that you have decided to see a play and paid the admission price of $\$ 10$ per ticket. As you enter the theater, you discover that you have lost the ticket. The seat is not marked and the ticket cannot be recovered. Would you pay $10 for another ticket? When 183 people were asked the following question, 161 answered yes and 22 answered no:

Imagine that you have decided to see a play where admission is $10 per ticket. As you enter the theater, you discover that you have lost a $10 bill. Would you still pay $10 for a ticket for the play? Is the difference in responses substantial and statistically persuasive?

Two polling organizations report these voter preferences:

Can the difference in these poll results be explained by sampling error? If you have no reason to believe one poll is more reliable than the other, what would you use for a prediction of each candidate's vote percentage?

Basketball players are often told that a missed shot usually bounces to the opposite side of the basket from which the shot is taken; for example, that a missed shot from the right side of the basket usually bounces to the left of the basket. A college player suspected that this tendency may be reversed at the end of a game, when the players are tired and their missed shots bounce off the front of the rim. A study of the first and last 10 minutes of videotapes of four different Division III male basketball teams yielded the data shown below. Use these data to see if the difference between the first and last 10 minutes is substantial and statistically persuasive.

A random sample of 25 male college students and 25 female college students were each asked to name their favorite singer; 21 of the males named a male singer and 16 of the females named a male singer. Explain the error in this test of the null hypothesis that male and female students are equally likely to name a male singer: "The pooled proportion is $(21+16) /(25+25)=0.74$. We tested the null hypothesis $\mathrm{p}=0.74$ with the following Z statistic:"

A 2002 study compared radical prostatectomy to watchful waiting for the treatment of localized prostate cancer: 53 of 347 patients in the prostatectomy group died, as did 62 of 348 patients in the control group. The p -value for a test of the null hypothesis that there was no difference in the death rates was .367. Explain why you either agree or disagree with the conclusion of the US Preventive Services Task Force that there was "no difference between the two groups in overall mortality."

Explain why you either agree or disagree with each of these statements:

Paul the Octopus went 8 for 8 in predicting the winner of soccer games at the 2010 World Cup. Assuming independence, use these data to test the null hypothesis that Paul has a .5 probability of picking the winner of a soccer game. Why might p-hacking and HARKing be problems here?