Practice Problems 11
Problem 1: Extrasensory Perception (ESP)
In an ESP test, one person writes down one of the letters A, B, C, D, or E and tries to telepathically communicate the choice to a partner. The partner then tries to guess what letter was selected.
(a). Repeat this process several times and then switch roles with your partner. Do this up to 10 times in total for each person. How often did you guess correctly?”
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Answer: Answers will vary!
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Answer: \(p = 0.2\) (since there are five choices and they are randomly guessing)
(c). Which sample proportion correct would provide the greatest evidence that people have ESP: (If we assume the sample size is the same in every case.)
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Answer: \(\hat{p} = 3/4\) since this means more correct.
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Answer: \[H_0: p = 0.2\] \[H_a: p > 0.2\] where \(p\) is the proportion correct for all people’s guesses. Since we are looking for evidence that the proportion is significantly above 0.2 (random guesses), the alternate hypothesis is larger than.
Problem 2: Sleep vs Caffeine
In an experiment, students were given words to memorize, then were randomly assigned to either take a 90 minute nap or take a caffeine pill. A couple hours later, they were tested on their recall ability. We wish to test to see if the sample provides evidence that there is a difference in mean number of words people can recall depending on whether they take a nap or have some caffeine.
(a). What is the explanatory variable? Is it categorical or quantitative?
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Answer: Explanatory = nap or caffeine (categorical)
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Answer: Response = number of words recalled (quantitative)
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Answer: Quantitative = mean responses, where \(\mu_1\) and \(\mu_2\) are the mean words recalled in the two different conditions
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Answer:
\[H_0: \mu_1 = \mu_2\] \[H_a: \mu_1 \neq \mu_2\]
The alternate hypothesis is not equals to since we are looking for evidence that the means are different (We do not know which one is larger!)Problem 3: Hand Dominance and Gender
Researchers are curious to find out if there is a significant difference in the proportion of left-handed individuals between males and females. They conduct a survey among a sample population to determine hand dominance for each gender.
Given the collected data, answer the following questions.
| Male | Female | Total | |
|---|---|---|---|
| Right-Hand | 120 | 150 | 270 |
| Left-Hand | 30 | 20 | 50 |
| Total | 150 | 170 | 320 |
(a). What proportion of males are left-handed?
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Answer:
Proportion of males who are left-handed = Number of left-handed males / Total males = 30/150 = 0.20 or 20%.(b). What proportion of females are right-handed?
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Answer:
Proportion of females who are right-handed = Number of right-handed females / Total females = 150/170 = 0.88 or 88%.(c). Among all left-handed person in the survey, what proportion of them are male?
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Answer:
Proportion that a randomly selected left-handed person is male = Number of left-handed males / Total left-handed individuals = 30/50 = 0.60 or 60%.(d). Formulate the null and alternative hypotheses for testing whether the proportion of left-handed individuals for males and females are different.
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Answer:
Null Hypothesis \(\left(H_0\right)\) : The proportion of left-handed individuals is the same for both males and females.
Alternative Hypothesis \(\left(H_a\right)\) : The proportion of left-handed individuals is different for males and females.