Probability Theory and Statistical Inference: A Comprehensive Guide
1. Test
1.V The binomial model is characterized as dichotomous. (True).
2.F The normal curve is asymmetrical with respect to the mean. (False, depending on the definition of the curve).
3.F Probability theory works with deterministic experiments. (False, working with random type experiments and probabilistic).
4.V The probability theory leads to building a model of the random experiment. (True).
5.F The route of the standard model is defined on positive real numbers. (False, not only for real positive but also negative ones).
6.V Decisions based on probability theory are positive. (True).
7.F A random experiment is the process of gathering information for an event that shows a result when repeated several times. (False, is an information gathering process that can be repeated indefinitely, but cannot tell what the outcome will be, we can only describe the set of possible outcomes).
Probability theory 8.V delivers procedures for calculating the results of a randomized trial. (True).
9.V In a randomized experiment, we can describe a set of possible outcomes. (True).
10.F The set of possible outcomes of the random experiment is known as a random experiment. (False; it is called the sample space).
11.V Events are dependent in the binomial model. (False).
12.V If the experiment is to throw 3 coins, the number of possible outcomes shall be eight. (True, ccc, ccs, csc, css, sec, scs, ssc, sss).
13.V An event is a subset of a sample space. (True).
14.V The empty set is known as an impossible event. (True).
15.V If the probability of an event approaches one, the event is likely to happen. (True).
16.F The sample space is known as an event likely. (False, it is known as the set of possible outcomes).
17.V A probability can be defined as frequentist and subjective. (True).
18.V A Bayesian probability is the degree of certainty people have about an event. (True).
19.F The opposite of an event A (A’) is the event consisting of all elements in the sample space that are not in A. (False, AEL is composed of elements not in A).
20.V The notation P (A) means the probability that event A occurs (True).
21.F A sample space is finite and countable if it has a finite number of terms and these belong to the real numbers. (False, because not only are they finite, they are also finite and infinite countable many).
22.V The probability of the sample space equals 1. (True).
23.F The probability of an impossible event is equal to half. (False, it is 0).
24.V The probability of a subset is the relative size in total. (True).
25.V Let A be an event such that P (A’) = 1 – P (A). (True).
26.F Let A1, A2 be any events such that P (A1 – A2) = P (A1UA’2). (False).
27.V The probability of event A is calculated by P (A) = # A / # E. (True).
28.V The union of two events A and B is the event which consists of the experimental results that are in A or B or in both. (True).
29.V To calculate the conditional probability is to calculate the intersection between two events. (True).
30.V The Gaussian model is a bell-shaped curve. (True).
31.F Subjective probability of an event is the relative frequency of times that the event would happen to run an experiment again and again. (False, it is the degree of certainty people have about an event).
32.F A comprehensive and inclusive system if the union of events is a sample space, and their intersection is different from the vacuum. (False, a system is comprehensive and inclusive if the union of all the sample space are events and their intersections are disjoint).
33.V Two events are independent if the occurrence of one does not provide any information about the occurrence of the other. (True).
34.V Bayes’ theorem calculates a conditional probability of an event Ai (i = 1,2, …. N) of the partition of the sample space conditional on event B. (True).
35.F The specificity is determined by the probabilities of the true positives. (False, it is determined by the true negatives).
36.V The prevalence is the percentage of the population who has an illness. (True).
37.V P (Ill l +) = positive predictive index. (True).
38.V A random variable is a function which assigns each event a number. (True).
39.F Random variables can be described as discrete and discontinuous. (False, as discrete and continuous).
40.V A density function is a nonnegative function of area equal to one. (True).
41.F The incidence is the percentage of cases of disease present in the population. (False, the incidence is the percentage of new cases of disease in the population).
42.V In the probability density functions of an interval describes a certain area. (True).
43.F The expected value equals the median. (False, it is the average).
44.F The sensitivity is determined by the probabilities of the true negatives. (False, by the true positives).
45.F The parameters of a normal model are: the mean and proportion. (False, they are the mean and standard deviation).
46.V In a normal average model, the mean is the location of the curve. (True).
47.V Karl Gauss determined the standard model through observations. (True).
48.F P (Ill l -) = negative predictive index. (False, it is the positive predictive index).
49.V The normal model delivers the standard deviation curve shape. (True).
50.F The intersection of two events A and B is the event which consists of the elements that are in A or B. (False, they are those elements of both A and B).
Alternatives
1. SPSS means: Statistics, products, services, and solutions.
2. In SPSS Data View, you can: enter data.
In SPSS, the label is written: You write the variable data.
1. In SPSS, values are described as: Codes of qualitative variables.
2. In SPSS, a frequency table is made in the command: Analyze.
3. In SPSS, the variable type string is: Variable quality.
4. An example of a random experiment is, the psychologist treating patients with stress.
5. In a randomized experiment, it is said that throwing 3 coins has 8 elements (not 3).
6. Let A1 and A2 be mutually exclusive events, then P (A1 U A2) is: P (A1) + P (A2).
7. Let A1 and A2 be any events such that P (A1) = 0.2, P (A2) = 0.3, P (A1 A2) = 0.01, then P (AUB) is: 0.49.
8. Let P (A) = 0.04, P (B) = 0.07, then P (AUB) = 0.11.
2. Test
1.V In SPSS, the normality test indicates that Ho: variable is normal V / s H1: the variable is not normal. (True).
2.V The normal model is determined by the mean and standard deviation. (True).
3.V In the Gaussian model, it is known that between the mean and one standard deviation, there is a probability of about 68%. (True).
4.V The normal density function is symmetric, mesokurtic (presents an average concentration around the core values of the variable) and unimodal (occurs as a single peak). (True).
5.V To estimate the parameters of a linear regression model, we use the least squares estimation technique to minimize errors. (True).
The 6.V normal distribution with zero mean and standard deviation 1 is known as the standard normal distribution. (True).
7.V In the case of a normal X variable, the interpretation is: Assign to every value of N (u, o) a value of N (0,1) that leaves exactly the same probability below. (True).
8.V The average in the normal model is a translation factor. (True).
9.V The standard deviation in the normal pattern determines the shape of the curve. (True).
10.V If P (Z <1.85) = 0.968, then P (Z> 1.85) = 0.032. (True).
11.F The probability P (Z <0) = 0.5. (True).
12.V Although not a random variable, a statistic or estimator calculated on large random samples, if it possesses a normal distribution. (True).
13.F The standard deviation of the mean always equals the standard deviation of the variables. (False, it is the standard deviation of the variables divided by the square root of the sample size).
14.V The average of a random sample from a normal population will be normal anyway. (True).
15.F With the characterization measurements, we can compare different models of normal. (False, it is to compare two values of normal distributions).
16.F If N> 20, the mean will be normal. (False, it is N> 30).
17.F The Chi-square model is symmetrical. (False, it is asymmetrical).
18.V The Student’s T model is symmetrical about the mean. (True).
19.V The Gaussian model appears in the appearances of measurement errors. (True).
20.V If n> 30 and p is small (np> 5), n large, then the binomial model can approximate the Poisson curve. (True).
21.V The F Snedecor model has two parameters. (True).
22.V The ideal population for investigation is called the target population. (True).
23.V The Chi-square test is applied to verify the independence or association of variables in contingency tables. (True).
24.F The group that we can actually study is called the target population. (False, it is called the study population).
Probability sampling 25.V allows us to know the probability that an individual is selected for the sample. (True).
26.F Non-probabilistic sampling has bias. (False, because no matter what type of sampling, there is always a possibility of bias).
27.V Statistical inference techniques assume that the sample was selected using simple random sampling. (True).
28.V To avoid biases, we use random response techniques. (True).
29.V Cluster sampling is applied when it is difficult to have a list of all individuals who are part of the study population. (True).
30.V An estimator is a numerical quantity calculated on a sample and it is a good representation of a statistic. (True).
31.F The standard deviation of the sample mean is o / n. (False, it is u / sqrt(n)).
32.V The bias due to systematic differences between the target population and the study population is called selection bias. (True).
33.V The confidence interval estimate gives a set of estimates and the probability of error. (True).
34.F Statistical inference is the set of methods to obtain features from a probabilistic sample. (False, from a probability sample).
35.V A statistical hypothesis is a procedure to make a decision on random variables that are present in a probability model. (True).
36.V The covariance measures the strength of the linear relationship between two variables. (True).
37.V Pearson’s correlation coefficient r is between 1 and -1. (True).
38.V The coefficient r is dimensionless. (True).
39.V Regression analysis is used to predict the relationship of the dependent variable with the independent variable. (True).
40.F Assuming H1: the data can refute it. (False, the data can show evidence in favor).
41.F Assuming H0: There should be accepted with strong evidence in favor. (False, it should not be rejected without strong evidence in favor).
42.V The confidence level is 90%, then the probability of error is 0.10. (True).
43.V The coefficient r-squared interpretation is the percentage of variability of the dependent variable explained by the independent variable. (True).
44.V The scatter plot is a plot of data points, which measures the tendency of the data. (True).
45.F If p> Alpha, H1 is rejected. (False, H0 is not rejected).
46.V The contrast is not significant, when p> alpha. (True).
47.V The type II error, says, H0 is accepted when it is false. (True).
48.F The Mann-Whitney test is a test to compare the means of two related samples. (False, it is a nonparametric test to compare the medians of two independent samples).
49.V The p-value is known before the experiment. (False, it is calculated after the experiment).
50.F The Wilcoxon test is a nonparametric test to obtain the means of two related samples. (False, it is a nonparametric test to compare the medians of two related samples).
Decision Rule
H1: The value is less than 1%, 5% or 10%.
H0: The value is greater than or equal to 1%, 5% or 10%.
Chi-Square Test
H1: The (anxiety) is not associated with the (tension).
H0: The (anxiety) is associated with (tension).
There is insufficient evidence to say that anxiety is (not) associated with stress, considering an error of 1%, 5% or 10%.
Correlations
H1: The (current salary) is not associated with (starting salary).
H0: The (current salary) is associated with (starting salary).
There is insufficient evidence to say that there is a significant correlation (no significant correlation) between current salary and starting salary, considering an error of 1%, 5% or 10%.
Statisticians Group (Proof of homogeneity of variance Levene test)
H1: The variance in educational level (group A) equals the variance of educational level (group B)
HO: The educational level of variance (group A) is different from the variance of educational level (group B).
There is insufficient evidence to say that there is a difference (which are equal) Significant variances between the educational levels of groups A and B, considering an error of 1%, 5% or 10%.
T test for equality of means in independent samples
H1: The average educational level (group A) is equal to the average educational level (group B).
H0: The average educational level (group A) is different from the average educational level (group B).
There is insufficient evidence to say that there is a significant difference (no equality) between the average educational levels in groups A and B, considering an error of 1%, 5% and 10%.
Kolmogorov-Smirnov (a) and Shapiro-Wilk
H1: Current Salary is normal.
H0: Current Salary is not normal.
There is insufficient evidence to say that the variable current salary (no) is normal considering an error of 1%, 5% or 10%.