Psychic Power To test whether my friend\'s fish Googly had psychic powers, I wro

Question

Psychic Power

To test whether my friend's fish Googly had psychic powers, I wrote R code to display two windows. I entered either "Left" or "Right" depending on which way Googly was facing. Then the random number generator in R selected either the left or the right window, with probability 0.5 for each, in which to display a star. Let p be the probability Googly guesses correctly on a given trial (assume this is constant.) In 80 trials, Googly correctly guessed the window with the star 41 times.

1. Using mathematical notation, write down null and alternative hypotheses for a one-sided test.

2. If the test statistic is the number of correct guesses (41) in 80 trials, write down R code to nd the P-value of a one-sided test.

3. Even without R, we can see that Googly's success rate was close to its expected value under the null, so the one-tailed P-value will be close to 0.5. State your conclusion about the fish's psychic powers.

4. Continued from part (b). If you only know that the R code dbinom(40, 80, 0.5) gives the number 0.0889, how would you find the exact P-value of the test?

Explanation / Answer

1) Ho : p = 0.5

Ha: p > 0.5

2)

p-value = P(X >= 41) =   1-pbinom(40,80,0.5)

= 0.4555361

3) p-value is close to 0.5

since p-value > 0.05 (level of significance}

we fail to reject the null

and conclude that there is not enough evidence for fish's psychic powers.

4) P(X> = 41)

since p = 1/2 ,the distribution in symmetric

hence

P(X < 40) = P(X> 40)

and P(X <40)+P(X=40) +P(X>40) = 1

hence p-value = (1 - 0.0889)/2 = 0.45555

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