# Basic Concepts of Probability

8.) Suppose that we have an urn with red and green balls with an X or Y on them (i.e. four different kinds of balls, namely, Red – X , Red – Y , Green – X , Green – Y ). Suppose that we know the following:
P ( Red ) = .2
P ( Red and X ) = .1
P ( X ) = .3
Find:
A.) P ( Green )
B.) P ( Red or X )
C.) P ( Red|X )
D.) P ( X|Red )
E.) Are events Red and X independent?
F.) Why or why not? If they are not independent, change one of the probabilities given to make them independent.

10.) Suppose the probability of being farsighted is .1. Suppose also that the probability of a farsighted person being dyslexic is .05 and the probability of a person who is not farsighted being dyslexic is .025 (1/2 as likely). What is the probability that a person with dyslexia is farsighted?

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11.) In baseball, suppose we are told that the probability of scoring a run on a double is .54. That is, given a play has generated a double, 54% of the time at least one run will score. However, we want to know how often when a run scores, it was generated by a double. This is not the same question. Do we see the difference? We are told that the probability of runs scoring on plays that are not doubles is .11 and the probability of hitting a double is 18%.

13.) One cab company in our city is named “Blue Cab Co.” And they have had some complaints about the driving behavior of their employees. But we know that all cab companies have some drivers who are a bit reckless. We know that the probability of getting a reckless driver if we are in a Blue Cab car is .25, but what we want to know is if we have a reckless driver, how likely is it a Blue Cab that we are in? Do we see the difference? We know the probability of getting a reckless driver if we are not in a Blue Cab car is .15, and we know the probability of getting a Blue Cab car is .4. So, what is the probability of being in a Blue Cab car if we have a reckless driver?

3.) For each of the following situations, specify the null and alternative hypotheses:
A.) The average respiration rate per minute is 8. Do smokers have an average rate different from 8?
B.) The average score on the Beck Depression Inventory is 12. Does the average depression score of mothers with young children deviate from the population mean?
C.) The average miles per gallon (mpg) of cars used in the United States is 20. Is the observed mpg of a sample of cars used in Japan different?

4.) When conducting an inferential test, when should we use the t distribution?

5.) `What if a researcher conducting a project using a single‐sample design has access to both the population and the sample standard deviation, which test should they use?

7.) What type of error corresponds to a “false positive?”

8.) What type of error corresponds to a “false negative?”

11.) For which type of error can we set the precise risk rate, and which one can we only increase or decrease the chance of making?

16.) Among trained typists, suppose it is known that the average typing speed using a standard keyboard is 60 words per minute (wpm), with a standard deviation of 5 wpm. The manufacturer of an ergonomically designed keyboard claims their device will improve typing speed. A random sample of 50 typists is tested on the ergonomically designed keyboard, and the sample mean wpm is 65. Test the hypothesis that using the new device affects typing speed. Set alpha at .05.
A.)Should we use the z distribution or t distribution? Why?
B.) State H 0 and H 1 .
C.) What are the critical values?
D.) What is the obtained statistic?
E.) Reject the null hypothesis?
F.) What type of decision error might have been made?
G.) Is there sufficient evidence to support the manufacturer’s claim?
H.) If so, what is the effect size?

17.) On one standardized measure of IQ, μ = 100 and σ = 15. Imagine we want to test the hypothesis that children of parents with college degrees have an average IQ that is greater than the national average. A sample of 100 students who have college‐educated parents is randomly selected, and the mean is 110 with a standard deviation of 12. Conduct a test of the null hypothesis and set alpha at .05.
A.) Should we use the z distribution or t distribution? Why?
B.) State H 0 and H 1 .
C.) What are the critical values?
D.) What is the obtained statistic?
E.)Reject the null hypothesis?
F.) What type of decision error might have been made?
G.) Interpret the finding.
H.) If the null is rejected, what is the effect size?

20.) An industrial/organizational psychologist believes that people who work at home experience greater job satisfaction. Imagine that a job satisfaction rating scale exists. The publishers of this scale claim the population is normally distributed with a mean of 50. The psychologist samples 20 people who work at home finding M = 63 and s = 17.
A.) Should we use the z distribution or t distribution? Why?
B.) State H 0 and H 1 .
C.) What are the critical values?
D.) What is the obtained statistic?
E.) Reject the null hypothesis?
F.) What type of decision error might have been made?
G.) Interpret the finding.
H.) If the null is rejected, what is the effect size?

23.) anthropologist hypothesizes that physical stress in childhood increases height (Landauer & Whiting, 1964 ). The researchers locate a tribe of people in which physical stress is a by‐product of frequent tribal rituals (e.g. piercing and molding body parts, exposure to extreme temperatures, etc.). The mean height of the people in the region who do not use physically stressful rituals with their young is used as the population mean. The following raw data are for adult biological males and women of the tribe in question. Conduct a t test for men and a t test for women. The population mean height for men is 65 and 59 in. for women.
Men  Women
67        59
69        63
72        65
70.      60
70.      59
72        62
64        61
70        66
A.)    What is t obt for men?
B.)    What is t obt for women?
C.)    What are the critical values for each test ( α = .05)?
D.)    Compare each t obt with its respective critical values and interpret the findings; present the findings in a professionally appropriate manner.

28.) health psychologist is interested in educating high school students about the negative effects of smoking. Fifty students who smoke are randomly selected to participate in the program. To measure the success of the program, the average number of cigarettes smoked per day among the participants is obtained 10 weeks after the end of the program. Assume that previous research had shown that, among all smoking students, the average number of cigarettes smoked in a day was 17. Set alpha at .05, and conduct a t test on the following data. Interpret the findings.
Average number of cigarettes consumed per day among participants
12 11 7 0 0 6 2 23 45
0 0 1 2 0 3 16 8 22 17 9
12 10 6 5 9 11 0 33 24 5
11 10 0 0 0 22 4 22 21 0
10 11 0 6 7 11 3 42 38 0

29.) An insurance company states that it takes them an average of 15 days to process an auto accident claim. A random sample of 40 claims is drawn from processed claims over the past six months. Based on the following data, is there any evidence that the mean number of days to pay claims is not 15? Set α = .05.
Number of days to process a claim
22 11 7 9 9 8 7 23 45 9
23 21 8 8 5 16 9 22 17 6
12 29 6 5 9 23 7 33 24 5
15 14 9 7 3 17 8 19 15 8. For more information on Basic Concepts of Probability see this: https://www.encyclopedia.com/science/encyclopedias-almanacs-transcripts-and-maps/probability-basic-concepts-mathematical-probability

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