Once again, there is a relevant quote from Jeffreys , p. Then it is worth while to examine the alternative [hypothesis] further and see what limits can be set to the new parameter, and thence to the consequences of introducing it.
Since weather and climate extremes have significant societal impacts, it is no surprise that many of the most severe impacts of climate change are expected to occur through changes in extreme events. If climate is understood as the distribution of all possible meteorological states, then the effect of climate change on extreme events is manifest in the changes in that distribution. This is the subject of a large literature. Over the last 20 years, the different topic of extreme event attribution has emerged, which seeks to answer the question of whether, or how, a particular extreme event can be attributed to climate change.
In contrast to the two previous examples, which concerned clear climate-science questions, here, it is far from obvious how to even pose the question within a climate-science framework, since every extreme event is unique NAS The most popular approach, first implemented by Stott et al.
Clearly, there is a trade-off involved here, which will depend on a variety of pragmatic factors. For example, in Stott et al. Such an anomaly was very rare for that highly aggregated statistic, but clearly nobody dies from temperatures that are only 1. The first point to make is that from the more general perspective of probability theory discussed in Sect. As Jeffreys says , p. A frequency is at best a useful mathematical model of unexplained variability.
The analogy that is often made of increased risk from climate change is that of loaded dice. But if a die turns up 6, whether loaded or unloaded, it is still a 6.
On the other hand, if an extreme temperature threshold is exceeded only very rarely in pre-industrial climate vs quite often in present-day climate, the nature of these exceedances will be different. One is in the extreme tail of the distribution, and the other is not, so they correspond to very different meteorological situations and will be associated with very different temporal persistence, correlation with other fields, and so on.
Since pretty much every extreme weather or climate event is a compound event in one way or another, this seems like quite a fundamental point.
It is perfectly sensible to talk about the probability of a singular event, so we should not feel obliged to abandon that concept. The fact is that climate change changes everything ; the scientific question is not whether, but how and by how much. When the null hypothesis is logically false, as here, use of NHST is especially dangerous. From NAS ,. This simple equation, which is based on the fundamental laws of probability theory, shows that the risk ratio factorizes into the product of two terms.
The second expresses how the probability of the conditioning factor might itself change. The scientific challenge here is that for pretty much any relevant dynamical conditioning factor for extreme events, there is very little confidence in how it will change under climate change Shepherd This lack of strong prior knowledge arises from a combination of small signal-to-noise ratio in observations, inconsistent projections from climate models, and the lack of any consensus theory.
If one insists on a frequentist interpretation of this second factor, as in PEA, then this can easily lead to inconclusive results, and that is indeed what tends to happen for extreme events that are not closely tied to global-mean warming NAS But there is an alternative.
We can instead interpret the second factor on the right-hand side of 5 as a degree of belief — which is far from inappropriate, given that the uncertainty here is mainly epistemic — and consider various hypotheses, or storylines Shepherd b.
The simplest hypothesis is that the second factor is unity, which can be considered a reasonable null hypothesis. One should of course be open to the possibility that the second factor differs from unity, but in the absence of strong prior knowledge in that respect, that uncertainty would be represented by a prior distribution centred around unity.
The advantage of this partitioning is that the first term on the right-hand side of Eq. This approach is actually used implicitly in much PEA. For example, in the analogue method e. Cattiaux et al. Both methods have been considered as perfectly acceptable within the PEA framework Stott et al. This assumption is very often not even discussed, and if it is, the argument is typically made that there is no strong evidence in favour of a value other than unity see e.
Yet for some reason, when exactly the same approach was proposed for the detailed dynamical situation of a highly unusual meteorological configuration Trenberth et al. For example, Stott et al. By always finding a role for human-induced effects, attribution assessments that only consider thermodynamics could overstate the role of anthropogenic climate change, when its role may be small in comparison with that of natural variability, and do not say anything about how the risk of such events has changed.
There is a lack of logical consistency here. Second, this approach is not biased towards overstating the role of anthropogenic climate change, as it could equally well understate it. As Lloyd and Oreskes have argued, whether one is more concerned about possible overstatement or understatement of an effect is not a scientific matter, but one of values and decision context. For example, in van Garderen et al. Instead, the attribution in PEA studies is invariably explained in terms of well-understood thermodynamic processes.
That seems like a pretty good justification for the storyline approach. In this way, the two approaches can be very complementary see Table 2 of van Garderen et al.
And if there are strong grounds for considering changes in dynamical conditions as in Schaller et al. Yet again, there is a relevant quote from Jeffreys , p. In induction there is no harm in being occasionally wrong; it is inevitable that we shall be. But there is harm in stating results in such a form that they do not represent the evidence available at the time when they are stated, or make it impossible for future workers to make the best use of that evidence.
In an application where there is little in the way of prior knowledge, and a lot of data, the Bayes factor rapidly overpowers the influence of the prior knowledge, and the result is largely insensitive to the prior. However, many aspects of climate-change science, especially although not exclusively in the adaptation context, are in the opposite situation of having a large amount of prior knowledge, and being comparatively data-poor in terms of data matching what we are actually trying to predict.
In particular, the observed record provides only a very limited sample of what is possible, and is moreover affected by sources of non-stationarity, many of which may be unknown. Larger data sets can be generated from simulations using climate models, but those models have many failings, and it is far from clear which aspects of model simulations contain useful information, and which do not.
Physical reasoning is therefore needed at every step. Statistical practice in climate-change science simply has to change. A statistician might at this point argue that the answer is to use Bayesian statistics. Indeed, Bayesian methods are used in particular specialized areas of climate science, such as inverse methods for atmospheric sounding Rodgers including pollution-source identification Palmer et al.
Mostly, this involves introducing prior probability distributions on the estimated parameters, but Sherwood et al. There have been brave attempts to employ Bayesian methods more widely, e. The difficulty is that Bayesian calibration for climate-change projections requires knowing the relationship between model bias in present-day climate which is measurable and the spread in a particular aspect of model projections.
Given the huge number of potential relationships, data mining can easily lead to spurious but apparently statistically significant relationships Caldwell et al. Indeed, several published emergent constraints have subsequently been debunked by Pithan and Mauritsen ; Simpson and Polvani ; Caldwell et al. Hall et al. This may help explain why it has been so challenging to find emergent constraints for circulation aspects of climate change relevant for adaptation , since there is no consensus on the relevant mechanisms and the circulation responses appear to involve multiple interacting factors, and potential nonlinearity.
For climate information to be useable, its uncertainties must be comprehensible and salient, especially in the face of apparently conflicting sources of information, and the connection between statistical analysis and physical reasoning must be explicit rather than implicit. They are useful heuristics, which researchers have some experience interpreting. And we need to make sure that we are not chasing phantoms.
Neuroscience has shown that human decision-making cannot proceed from facts alone but involves an emotional element, which provides a narrative within which the facts obtain meaning Damasio Narratives are causal accounts, which in the scientific context can be regarded as hypotheses. To return to the quote from Jeffreys at the beginning of this piece, we need to recognize that data does not speak on its own; there is no answer without a question, and the answer depends not only on the question but also on how it is posed.
Ambaum MHP Significance tests in climate science. J Clim — Google Scholar. Nature — Nature online version. Geophys Res Lett — Geophys Res Lett L Nature Clim Chang — Use the mean and standard deviation of a random variable to describe likely or unlikely events. Explain how a density function is used to find probabilities involving continuous random variables. Find probabilities associated with the normal distribution.
Recognize the features of a probability distribution and use probability distributions for discrete random variables to estimate probabilities and identify unusual events. Module Sampling Distributions Apply the sampling distribution of the sample mean as summarized by the Central Limit Theorem when appropriate. In particular, be able to identify unusual samples from a given population.
Apply the sampling distribution of the sample proportion when appropriate. Explain the concepts of sampling variability and sampling distribution.
Identify and distinguish between a parameter and a statistic. Unit 5: Inference Module Estimation Determine point estimates in simple cases, and make the connection between the sampling distribution of a statistic, and its properties as a point estimator. Explain what a confidence interval represents and determine how changes in sample size and confidence level affect the precision of the confidence interval.
Find confidence intervals for the population mean and the population proportion when certain conditions are met , and perform sample size calculations. Module Hypothesis Testing Apply the concepts of: sample size, statistical significance vs. Carry out hypothesis testing for the population proportion and mean when appropriate , and draw conclusions in context.
Determine the likelihood of making type I and type II errors, and explain how to reduce them, in context. Explain the logic behind and the process of hypotheses testing. In particular, explain what the p-value is and how it is used to draw conclusions. In a given context, specify the and alternative hypotheses for the population proportion and mean.
Module Inference for Relationships Identify and distinguish among cases where use of calculations specific to independent samples, matched pairs, and ANOVA are appropriate.
In a given context, carry out the inferential method for comparing groups and draw the appropriate conclusions. Specify the and alternative hypotheses for comparing groups. Accounting Information Systems 12th Edition. Accounting Principles. Advanced Accounting.
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Do not reject the null hypothesis. A positive t -statistic would mean that participants, on average, gained weight over the six months. The table has five rows and two columns. H 0 : The variables are independent. H 0 : The populations have the same distribution. To complete a painting job requires four hours setup time plus one hour per 1, square feet.
How would you express this information in a linear equation? Express this information in an equation. One job takes 2. What is the difference in labor costs for these two jobs? Describe the pattern in this scatter plot, and decide whether the X and Y variables would be good candidates for linear regression. Write the regression equation predicting weight from height in this data set, and calculate the predicted weight for someone 68 inches tall.
The correlation between body weight and fuel efficiency measured as miles per gallon for a sample of 2, model cars is —0. Calculate the coefficient of determination for this data and explain what it means. Rounded to two decimal places what correlation between two variables is necessary to have a coefficient of determination of at least 0.
Write the null and alternative hypotheses for a study to determine if two variables are significantly correlated. In a sample of 30 cases, two variables have a correlation of 0.
In a sample of 25 cases, two variables have a correlation of 0. A study relating the grams of potassium Y to the grams of fiber X per serving in enriched flour products bread, rolls, etc. For a product with five grams of fiber per serving, what are the expected grams of potassium per serving? Comparing two products, one with three grams of fiber per serving and one with six grams of fiber per serving, what is the expected difference in grams of potassium per serving?
In the context of regression analysis, what is the definition of an outlier, and what is a rule of thumb to evaluate if a given value in a data set is an outlier? In the context of regression analysis, what is the definition of an influential point, and how does an influential point differ from an outlier? You are conducting a one-way ANOVA comparing the effectiveness of four drugs in lowering blood pressure in hypertensive patients.
What are the null and alternative hypotheses for this study? You are comparing the results of three methods of teaching geometry to high school students.
Each sample includes students, and the final exam scores have a range of 0— Assuming the samples are independent and randomly selected, have the requirements for conducting a one-way ANOVA been met? Explain why or why not for each assumption.
You conduct a study comparing the effectiveness of four types of fertilizer to increase crop yield on wheat farms. When examining the sample results, you find that two of the samples have an approximately normal distribution, and two have an approximately uniform distribution. You are conducting a study of three types of feed supplements for cattle to test their effectiveness in producing weight gain among calves whose feed includes one of the supplements.
You have four groups of 30 calves one is a control group receiving the usual feed, but no supplement. You will conduct a one-way ANOVA after one year to see if there are difference in the mean weight for the four groups. What is SS within in this experiment, and what does it mean? What is SS between in this experiment, and what does it mean? What are MS between , and MS within , for this experiment? If there had been 35 calves in each group, instead of 30, with the sums of squares remaining the same, would the F Statistic be larger or smaller?
Histograms F 1 and F 2 below display the distribution of cases from samples from two populations, one distributed F 3,15 and one distributed F 5, Which sample came from which population? What assumptions must be met to perform the F test of two variances? You believe there is greater variance in grades given by the math department at your university than in the English department. You collect all the grades for undergraduate classes in the two departments for a semester, and compute the variance of each, and conduct an F test of two variances.
The independent variable is the hours worked on a car. The dependent variable is the total labor charges to fix a car. Because the intercept is included in both equations, while you are only interested in the difference in costs, you do not need to include the intercept in the solution.
The difference in number of hours required is: 6. The X and Y variables have a strong linear relationship. These variables would be good candidates for analysis with linear regression. The X and Y variables have a strong negative linear relationship. There is no clear linear relationship between the X and Y variables, so they are not good candidates for linear regression.
The X and Y variables have a strong positive relationship, but it is curvilinear rather than linear. These variables are not good candidates for linear regression. The coefficient of determination is the square of the correlation, or r 2. This means that 31 percent of the variation in fuel efficiency can be explained by the bodyweight of the automobile.
The amount that cannot be explained is 1 — 0. Your value is less than this, so you fail to reject the null hypothesis and conclude that the study produced no evidence that the variables are significantly correlated.
Using the calculator function tcdf, the p -value is 2tcdf 1. Do not reject the null hypothesis and conclude that the study produced no evidence that the variables are significantly correlated. Your value is greater than this, so you reject the null hypothesis and conclude that the study produced evidence that the variables are significantly correlated.
Using the calculator function tcdf, the p-value is 2tcdf 2. Reject the null hypothesis and conclude that the study produced evidence that the variables are significantly correlated. Because the intercept appears in both predicted values, you can ignore it in calculating a predicted difference score. An outlier is an observed value that is far from the least squares regression line.
A rule of thumb is that a point more than two standard deviations of the residuals from its predicted value on the least squares regression line is an outlier.
An influential point is an observed value in a data set that is far from other points in the data set, in a horizontal direction. Unlike an outlier, an influential point is determined by its relationship with other values in the data set, not by its relationship to the regression line. The value of 6.
The value of 2. The independent samples t -test can only compare means from two groups, while one-way ANOVA can compare means of more than two groups. Each sample appears to have been drawn from a normally distributed populations, the factor is a categorical variable method , the outcome is a numerical variable test score , and you were told the samples were independent and randomly selected, so those requirements are met.
However, each sample has a different standard deviation, and this suggests that the populations from which they were drawn also have different standard deviations, which is a violation of an assumption for one-way ANOVA. Further statistical testing will be necessary to test the assumption of equal variance before proceeding with the analysis.
One of the assumptions for a one-way ANOVA is that the samples are drawn from normally distributed populations. Since two of your samples have an approximately uniform distribution, this casts doubt on whether this assumption has been met.
Further statistical testing will be necessary to determine if you can proceed with the analysis. SS within is the sum of squares within groups, representing the variation in outcome that cannot be attributed to the different feed supplements, but due to individual or chance factors among the calves in each group. SS between is the sum of squares between groups, representing the variation in outcome that can be attributed to the different feed supplements.
The mean squares in an ANOVA are found by dividing each sum of squares by its respective degrees of freedom df. It would be larger, because you would be dividing by a smaller number. The value of MS between would not change with a change of sample size, but the value of MS within would be smaller, because you would be dividing by a larger number df within would be , not Dividing a constant by a smaller number produces a larger result.
All but choice c, —3. F Statistics are always greater than or equal to 0. As the degrees of freedom increase in an F distribution, the distribution becomes more nearly normal. Histogram F 2 is closer to a normal distribution than histogram F 1, so the sample displayed in histogram F 1 was drawn from the F 3,15 population, and the sample displayed in histogram F 2 was drawn from the F 5, population.
The samples must be drawn from populations that are normally distributed, and must be drawn from independent populations. Use the following information to answer the next two exercises: An experiment consists of tossing two, sided dice the numbers 1—12 are printed on the sides of each die.
Which of the following are TRUE when we perform a hypothesis test on matched or paired samples? Use the following information to answer the next two exercises: One hundred eighteen students were asked what type of color their bedrooms were painted: light colors, dark colors, or vibrant colors. The results were tabulated according to gender. Find the probability that a randomly chosen student is male or has a bedroom painted with light colors.
Use the following information to answer the next two exercises: We are interested in the number of times a teenager must be reminded to do his or her chores each week.
A survey of 40 mothers was conducted. Find the expected number of times a teenager is reminded to do his or her chores. Use the following information to answer the next two exercises: On any given day, approximately We randomly survey 22 cars. We are interested in the number of cars that are parked crookedly. For every 22 cars, how many would you expect to be parked crookedly, on average? What is the probability that at least ten of the 22 cars are parked crookedly. Using a sample of 15 Stanford-Binet IQ scores, we wish to conduct a hypothesis test.
It is known that the standard deviation of all Stanford-Binet IQ scores is 15 points. The correct distribution to use for the hypothesis test is:.
Use the following information to answer the next three exercises: De Anza College keeps statistics on the pass rate of students who enroll in math classes.
In a sample of 1, students enrolled in Math 1A 1st quarter calculus , 1, passed the course. In a sample of students enrolled in Math 1B 2nd quarter calculus , passed. In general, are the pass rates of Math 1A and Math 1B statistically the same?
If you were to conduct an appropriate hypothesis test, the alternate hypothesis would be:. Kia, Alejandra, and Iris are runners on the track teams at three different schools. Their running times, in minutes, and the statistics for the track teams at their respective schools, for a one mile run, are given in the table below:. Use the following information to answer the next two exercises: The following adult ski sweater prices are from the Gorsuch Ltd.
Assume the underlying sweater price population is approximately normal. The null hypothesis is that the mean price of adult ski sweaters from Gorsuch Ltd. Sara, a statistics student, wanted to determine the mean number of books that college professors have in their office. She randomly selected two buildings on campus and asked each professor in the selected buildings how many books are in his or her office. Sara surveyed 25 professors.
The type of sampling selected is. A clothing store would use which measure of the center of data when placing orders for the typical "middle" customer?
Use the following information to answer the next three exercises: A community college offers classes 6 days a week: Monday through Saturday.
Maria conducted a study of the students in her classes to determine how many days per week the students who are in her classes come to campus for classes. In each of her 5 classes she randomly selected 10 students and asked them how many days they come to campus for classes.
Each of her classes are the same size. The results of her survey are summarized in [link]. Combined with convenience sampling, what other sampling technique did Maria use? Use the following information to answer the next two exercises: The following data are the results of a random survey of Reservists called to active duty to increase security at California airports.
The lifetime of a computer circuit board is normally distributed with a mean of 2, hours and a standard deviation of 60 hours. What is the probability that a randomly chosen board will last at most 2, hours? A survey of reservists called to active duty as a result of the September 11, , attacks was conducted to determine the proportion that were married. Eighty-six reported being married. Winning times in 26 mile marathons run by world class runners average minutes with a standard deviation of 14 minutes.
A sample of the last ten marathon winning times is collected. The distribution for x is:. Suppose that Phi Beta Kappa honors the top one percent of college and university seniors.
Assume that grade point means GPA at a certain college are normally distributed with a 2. The number of people living on American farms has declined steadily during the 20 th century. Here are data on the farm population in millions of persons from to What was the expected farm population in millions of persons for ? In regression analysis, if the correlation coefficient is close to one what can be said about the best fit line? Use the following information to answer the next three exercises: A study of the career plans of young women and men sent questionnaires to all members of the senior class in the College of Business Administration at the University of Illinois.
One question asked which major within the business program the student had chosen. Here are the data from the students who responded. The p -value is 0. The conclusion to the test is:. A random sample of San Jose residents indicated 15 professional, 15 clerical, 40 skilled, 10 service, and 20 semiskilled laborers.
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